HEAD: Torsten Sillke's index of Martin Gardner's book with futher references. HEAD: last update 2000-10-14. HEAD: mailto:Torsten.Sillke@uni-bielefeld.de HEAD: URL: http//www.mathematik.uni-bielefeld.de/~sillke/ HEAD: Comments from me are tagged 'sio'. MGSA: Martin Gardner's Scientific American Mathematical Recreations Books MGSA: A which is where index to be used with the 'grep' command. MGSA: MG$1SA$2[.$3][.$4] with $1=booknumber, $2=Chapter, $3=Section, $4=Ref-Tag MGSA: Another index http://www.ms.uky.edu/~lee/ma502/gardner5/gardner5.html MG1SA: The Scientific American Book of MG1SA: Mathematical Puzzles and Diversions MG1SA: Simon & Schuster (1959) MG1SA.1 Hexaflexagons MG1SA.1.a Symmetries of the Trihexaflexagon, M. Mag. 49 (1976) 189-192 MG1SA.1.b V-Flexing the Hexahexaflexagon, AMM 86 (1979) 457-466 MG1SA.1.c Classification of flexagons, Eureka 53 (March 1993) MG1SA.1.c hexa- tetraflexagons, flexaflops, higher dimensions MG1SA.2 Magic with a Matrix MG1SA.3 Nine Problems MG1SA.3.1 The Returning Explorer MG1SA.3.2 Poker MG1SA.3.3 The Mutilated Chessboard (parity) MG1SA.3.4 The Fork in the Road (logic, truth-teller and liar, one question) MG1SA.3.5 Scrambled Box Tops MG1SA.3.6 Bronx v. Brooklyn (probability) MG1SA.3.7 Cutting the Cube MG1SA.3.7.a MM 25 (1952) 219, minimal number of cuts for a a*b*c box. MG1SA.3.7.b AMM (Aug. 1957) E 1279, Solution: (Mar. 1958), cuttung the n-cube MG1SA.3.8 The Early Commuter (time point) MG1SA.3.9 The Counterfeit Coins (weighing) -> MG6SA.11 MG1SA.4 Ticktacktoe, or Nougths and Crosses MG1SA.5 Probability Paradoxes MG1SA.5.a Birthday paradoxa, coupon collectors, caching algorithms and MG1SA.5.a self-organizing search, Disc. Appl. Math. 39 (1992) 207-229 MG1SA.5.b Majorization and the Birthday Inequality, M Mag. 64 (1991) 183-188 MG1SA.5.c On birthday, collector's, occupancy and other classical urn problems MG1SA.5.c Internat. Statistical Review 54 (1986) 15-27 (L. Holst) MG1SA.5.d The General Birthday Problem (L. Holst) MG1SA.5.d Random Structures and Algorithms 6:2/3 (1995) 201-208 MG1SA.5.e M. Klamkin and D. Newman, Extensions of the birthday MG1SA.5.e surprise, J. Comb. Th. 3 (1967), 279-282. MG1SA.5.f Birthday problems - a search for elementary solutions MG1SA.5.f Math. Gaz. 82 (1998) issue 493 111-114 MG1SA.5.g The birthday distribution Math. Gaz. 68 (1984) 204 MG1SA.5.g Prob(the rth is duplicate) = r/n*exp(-r*r/(2n)); (asymptotic) MG1SA.5.g E(R) = Sqrt(Pi*n/2); Var(R) = n*(2-Pi/2) MG1SA.6 The Icosian Game and the Tower of Hanoi MG1SA.7 Curious Topological Models MG1SA.7. M"obius-Band, double layer M"obius-Band, 2 M"obius-Band->Klein Bottle MG1SA.7.a A Non-singular Polyhedral M"obius-Band Whose Boundary is a Triangle MG1SA.7.a AMM 55:5 (1948) 309-311, B Tuckerman, made from 9 triangles MG1SA.8 The Game of Hex MG1SA.8. Piet Hein invented Hex 1942 (Parker Brothers Inc. 1952) 11*11 board MG1SA.8. John F. Nash independently invented Hex 1948 (14*14 board) MG1SA.8. there exists a first player winning strategy. Rex -> MG12SA.12 MG1SA.8. Rex (reverse Hex) odd board second even board first player win MG1SA.8. C. Shannon suggest an equilateral triangle. the aim is to connect MG1SA.8. the three sides with a tree (the corner belong to both sides). MG1SA.8. Hex and Shannon's Hex is a first player win. n*(n+1) Hex (pairing) MG1SA.8. variations of hex: Schensted, Madcrack Y and Poly Y -> MG75.12SA.117 MG1SA.8.a Beck, et al, Excursions into Mathematics, 1970, 327-339 MG1SA.8.a Hex 14*14 board, Beck's Hex, Hex has a first player move that loses MG1SA.8.b a winning opening in Reverse Hex, JoRM 7 (1974) 189-192 MG1SA.8.c some variants of Hex (Vex, Tex), JoRM 8 (1975/76) 120-122 MG1SA.8.d Hex must have a winner: an inductive proof, M. Mag. 49 (1976)85-86 MG1SA.8.e C. Berge, Some Remarks about a Hex Problem (14*14 board) MG1SA.8.e in: (ed. D. A. Klarner) The mathematical Gardner, p25-27 MG1SA.8.f Hex and the Brouwer Fixed-Point Theorem, AMM 86 (1979) 818-827 MG1SA.8.f there exists a connection in Hex <-> Brouwer Fixed-Point Theorem MG1SA.8.f only one pair can be connected <-> discrete Jordan Curve Theorem MG1SA.8.g Havannah (C. Freeling), Games&Puzzles No 79 (Winter 1980) 4-7,32 MG1SA.8.g Hexagon (length 8), connect two corners, three sides or build a ring MG1SA.8.h Hex is PSPACE-Complete, Acta Infomatica 15 (1981) 167-191 (German) MG1SA.8.i Three Person Winner-Take-All Games with McCarthy's Revenge Rule, MG1SA.8.i The College Math. Jour. 16:5 (Nov 1985) 386-394 (P. D. Straffin) MG1SA.8.i Three person Hex played on a hexagon (connect opposite edges), as MG1SA.8.i soon as it is no longer possible for a player to connect his two MG1SA.8.i edges, that player is eliminated and may not place any more marks MG1SA.8.j Hex Games and Twist Maps on the Annulus, AMM 98 (1991) 803-811 MG1SA.9 Sam Loyd: America's Greatest Puzzlist MG1SA.10 Mathematical Card Tricks MG1SA.11 Memorizing Numbers MG1SA.12 Nine More Problems MG1SA.12.1 The Touching Cigarettes MG1SA.12.2 Two Ferryboats (arithmetic) MG1SA.12.3 Guess the Diagonal (geometry) MG1SA.12.4 The Efficient Electrican MG1SA.12.5 Cross the Network MG1SA.12.5. Eulerian path problem solvable on a toric MG1SA.12.6 The Twelve Matches MG1SA.12.6. polygon with area 4. MG1SA.12.7 Hole in the Sphere MG1SA.12.7.a G Polya, Mathematik und plausibles Schliessen, 1962/63 MG1SA.12.7.a the length of a conic or parabolic hole determines the volume MG1SA.12.8 The Amorous Beetles MG1SA.12.8. rotating square, length of a logarithmic spiral -> MMR.8.1 MG1SA.12.9 How Many Children? MG1SA.12.9. a<b<c<d, a+b+c+d<18, a*b*c*d is the housenumber, MG1SA.12.9. not enough information sum-product problem MG1SA.12.9.a AMM (Mar. 1948) Prob E 776 (Lester R Ford) MG1SA.13 Polyominoes MG1SA.13. trominoes: 21 (...) cover the 8*8 square. (3-color proof) MG1SA.13. trominoes: (:.) cover the 2^n*2^n square. (induction) MG1SA.13. pentominoes: 6*10, 5*12, 4*15, 3*20, 8*8-center MG1SA.13. pentominoes: two 5*6 rectangles (unique) (uxvfnp, itwzyl) MG1SA.13. hexominoes (24 odd, 11 even): no rectangle possible (parity proof) MG1SA.13.a Mathematics Subject Classification: 05B50 Polyominoes MG1SA.13.b Impossible proofs (groups), Quantum 2:2 (1991) 20-23,61 MG1SA.13.c W. P. Thurston, Conway's Tiling Groups, AMM 97 (1990) 757-773 MG1SA.13.d Polyominoes, Puzzles and Problems in Tiling, (G. Martin) MAA 1991 MG1SA.13.d review: AMM 100 (1993) 414-416 MG1SA.13.e Tiling with polyominoes and combinatorial group theory, MG1SA.13.e JoCT A 53:2 (1990) 183-208 (Conway; Lagarias) triangle tiling MG1SA.13.f Impossible Six Six by Six, JoRM 23 (1991) 304-305 MG1SA.13.g A Polyomino Tiling Problem of Thurston and Its Configurational MG1SA.13.g Entropy (Tribones), JoCT 63:2 (1993) 338-358 (Lagarias, ..) MG1SA.13.h pentominoes: two 5*5 squares, PM 35:6 (1993) 277, 4 partitions MG1SA.13.h (lxufp, ityvn) (lufpw, ityvn) (lufyz, pitvw) (livnz, ufpty) MG1SA.13.i pentomino (problem types) alpha 29:2-3 (1995) 8-9, 58, 68-69 MG1SA.13.i pentomino (problem types) alpha 29:4 (1995) 23 MG1SA.14 Fallacies MG1SA.15 Nim and Tac Tix MG1SA.16 Left or Right? MG2SA: The Second Scientific American Book of MG2SA: Mathematical Puzzles and Diversions MG2SA: Simon & Schuster (1961) MG2SA.1 The five Platonic Solids (polyhedron) MG2SA.1.a Gener. and rendering four dimensional polytopes, Mathematica J. 3 MG2SA.1.b infinite convex polytopes with equiangular faces, Zbl 762:52002 MG2SA.1.c The Multifaceted Polyhedron, Quantum 3:5 (1993) 32-34 MG2SA.1.c shaddock-polyhedron: starlike, not convex, can't be puffed out MG2SA.1.c unique conves polyhedron (Euler char., vertex angles < 360) MG2SA.1.d Five space-filling polyhedra, Math. Gaz. 80 (Nov 1996) 466-475 MG2SA.1.sio Resistance of the Cube : (a+b+c+1/(1/a+1/b+1/c))/4 MG2SA.2 Tetraflexagons MG2SA.3 Henry Ernest Dudeney: England's Greatest Puzzlist MG2SA.3. four-piece dissection of equilateral triangle to square MG2SA.3. a pentagon reassebles into a square (6 pieces) MG2SA.3. spider and fly problem (shortest path, geodesic, box, cylinder) MG2SA.3. 3*3 square 2+2 knights turn their places (C_8, 16 moves) MG2SA.3. diophantine equ. x^3+y^3=9 (other than 2, 1). MG2SA.4 Digital Roots MG2SA.4. Sum of the digits, casting nines, divisibility test by 9, neunerprobe MG2SA.5 Nine Problems MG2SA.5.1 The Twiddled Bolts MG2SA.5.2 The Flight Around the World (-> MG2SA.14.1) MG2SA.5.3 The Circle on the Chessboard MG2SA.5.4 The Cork Plug (Volume, principle of Chavalieri) MG2SA.5.5 The Repeptitious Numbers (7 11 13 = 1001) MG2SA.5.6 The Colliding Missiles (head Calculation) MG2SA.5.7 The Sliding Pennies (Triangle - Ring) MG2SA.5.8 Handshakes and Networks (Parity) MG2SA.5.9 The Triangular Duel (probability) MG2SA.6 The Soma Cube MG2SA.7 Recreatinal Topology MG2SA.7. Euler-tours with the minimal number of bends (a triangular graph) MG2SA.7. a string and ring puzzle: move the ring to the other side, two MG2SA.7. solutions possible, if the ends are not knotted. An impossible variant MG2SA.7. -> MG7SA.9.3 The Key and the Keyhole (topological equivalent problem) MG2SA.7. winning Bridg-it = Gale, connection game, strategy -> MG3SA.18 MG2SA.8 Phi: The Golden Ratio MG2SA.9 The Monkey and the Coconuts MG2SA.10 Mazes MG2AS.10.a R. Abbott, Mad Mazes, Bob Adams Inc. Publishers, Holbrook, 1990 MG2AS.10.b Mazes... How to Get Out! 1, A. Treep, CFF 37 (June 1995) 18-21 MG2AS.10.b E(step)=#edges for tree graphs for Algo. Tarry, Tremaux, Minirepli MG2AS.10.c Mazes... How to Get Out! 2, A. Treep, CFF 38 (Oct. 1995) 22-26 MG2AS.10.d Rouse Ball, Math. Recreations and Essays, 12th Ed., 254-260, Mazes MG2AS.10.p the labyrinth of Altjessnitz (Bitterfeld) PM 35:6 (1993) 252-255 MG2SA.11 Recreational Logic MG2SA.12 Magic Squares MG2SA.13 James Hugh Riley Shows, Inc. MG2SA.13. cover a circle by five equal circles, cover any area of diameter 1 MG2SA.13.a H. Melissen, Lowest circle covering of an equilateral triangle, MG2SA.13.a Math Mag 70:2 (Apr 1997) 118-124, 5 circle covering correction MG2SA.13.b The Worm Problem of Leo Moser I, Quantum 3:3 (1993) 41 MG2SA.13.b which region (min area) covers a worm of length 1? MG2SA.13. three dice game: Chuck-a-luck or Bird Cage MG2SA.13. sandwich theorem, number of pieces by cutting a torus MG2SA.14 Nine More Problems MG2SA.14.1 Crossing the Desert (-> MG2SA.5.2) MG2SA.14.1.a Jeeper the Deeper, (-> MG6SA.12.1) MG2SA.14.2 The Two Children (probability) MG2SA.14.2. one child is a boy - the probability that both are boys is 1/3 MG2SA.14.2. the older is a girl - the probability that both are girls is 1/2 MG2SA.14.3 Lord Dunsany's Chess Problem MG2SA.14.4 Professor on the Escalator MG2SA.14.5 The Lonesome 8 MG2SA.14.6 Dividing the Cake MG2SA.14.6.a Minimal number of cuts for fair divisions MG2SA.14.6.a Ars Combinatoria 31 (1991) 191-197, #MR 92c:05016, Zbl 761:05009 MG2SA.14.6.a each person believes he obtains at least his fair share, MG2SA.14.6.a an divide and conquer algorithm with O(n log n) cuts is given MG2SA.14.6.b How to Cut a Cake Fairly, AMM 87 (1980) 640-644; 68 (1961) 1-17 MG2SA.14.6.c Die gerechte Teilung, Math.Kabinet.3.3.6 MG2SA.14.6.d toetjes, AMM 97 (1990) 785-794 MG2SA.14.6.d a number is secretly chosen from the interval [0,1], and n player MG2SA.14.6.d try in turn to guess this number. when the secret number is MG2SA.14.6.d revealed, the player with the closest guess wins. MG2SA.14.6.e Ramsey Partitions of Integers and Fair Divisions (Fair Division MG2SA.14.6.e Algo. for n Person Unequal Shares) Combinatorica 12 (1992)193-201 MG2SA.14.6.f An Envy-Free Cake Division Protocol, AMM 102:1 (Jan 1995) 9-18 MG2SA.14.6.g Dividing a Cake, Math Intell 15:1 (1993) 50-52 MG2SA.14.7 The Folded Sheet -> MG10SA.7 MG2SA.14.8 The Absent-minded Bank Clerk (Diophantine equation) MG2SA.14.9 Water and Wine MG2SA.14.9. give some water into the wine and the same amont back to the water MG2SA.14.9. each mixture has the same amount of the other, proof by volume. MG2SA.14.9. can you get 50%? Only if the liquid is not infinitly divisible MG2SA.15 Eleusis: The Induction Game MG2SA.16 Origami MG2SA.17 Squaring the Square MG2SA.17. tiling a square (rectangle) with different squares, -> MG7SA.11 MG2SA.17.a compound perfect squares, AMM 89 (1982) 15-32 MG2SA.18 Mechanical Puzzles MG2SA.19 Probability and Ambiguity MG2SA.19. Paradoxes arising in geometric probability, random cord in a circle MG2SA.19.a Concurrent Lines and Acute Angles, M. Mag. 64 (1991) 204-205 MG3SA: New Mathematical Diversions from Scientific American MG3SA: Simon & Schuster (1966) MG3SA.1 The Binary System MG3SA.2 Group Theory and Braids MG3SA.2.a A Random Ladder Game: Permutaions, Eigenvalues, and Convergence MG3SA.2.a of Markov Chains, College Math J. 23:5 (1992) 373-385 MG3SA.2.b PSL(2,7) = PSL (3,2), MR 89f:05094 elegant proof MG3SA.3 Eight Problems MG3SA.3.1 Acute Dissection MG3SA.3.1. Triangle cut into seven acute ones (or eight acute isosceles) MG3SA.3.1. acute dissection of a square (8), pentagram (5), Greek cross (20) MG3SA.3.1.a NU-Configurations in tiling the square, Math Comp 59 (1992)195-202 MG3SA.3.1.a tiling a square with integer triangles MG3SA.3.2 How Long is a "Lunar"? MG3SA.3.2. radius of the sphere, such that surface = volume MG3SA.3.3 The Game of Googol (probability) MG3SA.3.3. maximising the chance of picking the largest objekt MG3SA.3.3. maximizing the value of the selected object (proposed be Cayley) MG3SA.3.3.a On a Problem of Cayley, Scripta Mathematica (1956) 289-292 MG3SA.3.3.b An Optimal Mainteanance Policy of a Discrete-Time Markovian MG3SA.3.3.b Deterioration System, Comp. & Math. with Appl 24 (1992) 103-108 MG3SA.3.3.c A Secretary Problem with Restricted Offering Chances and Random MG3SA.3.3.c Number of Applications, Comp. & Math. with Appl 24 (1992) 157-162 MG3SA.3.3.d On a simple optimal stopping problem, Disc. Math. 5 (1973) 297-312 MG3SA.3.3.e Stopping time techniques for analysts and probabilits (L. Egghe) MG3SA.3.3.e LMS LNS 100 MG3SA.3.3.f Algebraic Approach to Stopping Variable Problems, JoCT 9 (1970) MG3SA.3.3.f 148-161, distributive lattices <-> stopping variable problems MG3SA.3.3.g secretary problem, Wurzel 27:12 (1993) 259-264 MG3SA.3.3.h Ferguson, Who solved the secretary problem? MG3SA.3.3.h Statistical Science 4 (1989) 282-296 MG3SA.3.3.i Freeman, the secretary problem and its extensions: a review MG3SA.3.3.i International Statistical Review 51 (1983) 189-206 MG3SA.3.4 Marching Cadets and a Trotting Dog MG3SA.3.5 Barr's Belt MG3SA.3.6 White, Black and Brown (logic) MG3SA.3.7 The Plane in the Wind MG3SA.3.8 What Price Pets? (linear Diophantine equation) MG3SA.4 The Games and Puzzles of Lewis Carroll MG3SA.5 Paper Cutting MG3SA.5. theorem of Pythagoras, dissection proof, MG3SA.6 Board Games MG3SA.7 Packing Spheres MG3SA.7.a MG9SA.3, figurative numbers, square, triangular, tetrahedral MG3SA.8 The Transcendental Number Pi MG3SA.9 Victor Eigen: Mathemagician MG3SA.10 The Four-Color Map Theorem MG3SA.11 Mr. Apollinax Visits New York MG3SA.12 Nine Problems MG3SA.12.1 The Game of Hip MG3SA.12.1. two color the 6*6 square, s. t. there is no monochromatic square MG3SA.12.1. the number of different squares in the n*n square is n^2(n^2-1)/12 MG3SA.12.1.a enumerating 3-, 4-, 6-gons with vertices at lattice points, MG3SA.12.1.a Crux Math 19:9 (1993) 249-254 MG3SA.12.2 A Switching Puzzle MG3SA.12.2. change two cars with a locomotive (circle and tunnel) MG3SA.12.3 Beer Signs on the Highway (calculus, speed, time, distance) MG3SA.12.4 The Sliced Cube and the Sliced Doughnut (geometry) MG3SA.12.4. cut the cube (regular hexagon), doughnut (two intersecting cirles) MG3SA.12.5 Bisecting Yin and Yang (geometry) MG3SA.12.5.a Bisection of Yin and of Yang, Math. Mag. 34 (1960) 107-108 MG3SA.12.6 The Blue-Eyed Sisters (probability) MG3SA.12.7 How old is the Rose-Red City? (linear equations) MG3SA.12.8 Tricky Track (logic, reconstruct a table) MG3SA.12.9 Termite and 27 Cubes (hamiltonian circle, parity) MG3SA.13 Polyominoes and Fault-Free Ractangles MG3SA.13.a On folyominoes and feudominoes, Fib. Quart. 26 (1988) 205-218 MG3SA.13.b Rookomino (Kathy Jones) JoRM 23 (1991) 310-313 MG3SA.13.c Rookomino (K. Jones) JoRM 22 (1990) 309-316 (Problem 1756) MG3SA.13.d Polysticks, JoRM 22 (1990) 165-175 MG3SA.13.e Fault-free Tilings of Rectangles (Graham) The Math. Gardner 120-126 MG3SA.14 Euler's Spoilers: The Discovery of an Order-10 Graeco-Latin Square MG3SA.14.a Universal Algebra and Euler's Officer Problem, AMM 86 (1979)466-473 MG3SA.15 The Ellipse MG3SA.15.a robust rendering of general ellipses and elliptic arcs, MG3SA.15.a ACM Trans. on Graphics, 12:3 (1993) 251-276 MG3SA.16 The 24 Color Squares and the 30 Color Cubes (MacMahon) MG3SA.16. 12261 solutions of the 4*6 rectangle, 3*8 is impossible MG3SA.17 H. S. M. Coxeter MG3SA.17. Coxeter's book Introduction to Geometry 1961 MG3SA.17. appl. of the M"obius band, contructions for 257, 65537 gon MG3SA.17. Morley's triangle, equal bisectors - Steiner-Lehmus Thm MG3SA.17.a Angle Bisectors and the Steiner-Lehmus Thm, Math. Log 36:3 (1992)1&6 MG3SA.17.b equal external bisectors, not isoscele, M. Math. 47 (1974) 52-53 MG3SA.17.c A quick proof of a generalized Steiner-Lehmus Thm, MG3SA.17.c Math Gaz. 81:492 (Nov. 1997) 450-451 MG3SA.17.h Morley's triangle (D.J.Newman's proof), M In 18:1 (1996) 31-32. MG3SA.17. kissing circles, Soddy's formular - Descartes' Circle Theorem MG3SA.17.d Circles, Vectors, and Linear Algebra, Math. Mag. 66 (1993) 75-86 MG3SA.17. semiregular tilings of the plane, the 17 cristallographic groups MG3SA.17. tilings of Escher: Heaven-Hell, Verbum MG3SA.17.e The metamorphosis of the butterfly problem (Bankoff) MG3SA.17.e Math. Mag. 60 (1987) 195-210 (47 refs) MG3SA.17.f A new proof of the double butterfly theorem, M. Mag. 63 (1990) 256-7 MG3SA.17.g Schaaf, Bibliography of Rec. Math. II.3.3 The butterfly problem MG3SA.18 Bridg-it and Other Games MG3SA.18. winning Bridg-it, pairing stategy (Shannon switching game) MG3SA.18. Connections (ASS, 1992) = Bridg-it board: connect or circle MG3SA.18.b Directed switching games on graphs and matroids, JoCT B60 (1986)237 MG3SA.18.c Shannon switching games without terminals, draft (I), see II, III MG3SA.18.c Graphs and Combinatorics 5 (1989) 275-82 (II), 8 (1992) 291-7 (III) MG3SA.19 Nine More Problems MG3SA.19.1 Collating the Coins (coin moving xyxyx -> xxxyy) MG3SA.19.2 Time the Toast (optimal shedule) MG3SA.19.3 Two Pentomino Posers MG3SA.19.3. 6*10 Rectangle with all pentominoes touch the border (unique) MG3SA.19.4 A Fixed Point Theorem MG3SA.19.5 A Pair of Digit Puzzles (cryptarithms) MG3SA.19.6 How did Kant Set His Clock (calculus, time, speed) MG3SA.19.7 Playing Twenty Questions when Probability Values are Known MG3SA.19.7. Huffman coding, data compression MG3SA.19.8 Don't Mate in One (chess) MG3SA.19.9 Find the Hexahedrons MG3SA.19.9. there are seven varieties of convex hexahedrons (six faces) MG3SA.20 The Calculus of Finite Differences MG3SA.20.d Symmetry Types of Periodic Sequences, Illionois J. of Math. MG3SA.20.d 5:4 (Dec 1961) 657-665, appl. to music and switching theory MG3SA.20.a generating two color necklaces, Disc. Math. 61 (1986) 181-188 MG3SA.20.b Generating Necklaces, J. of Algorithms 13:3 (1992) 414 MG4SA: The Numerology of Dr. Matrix, Chap. 1-7 MG4SA: Simon & Schuster (1967) MG4SA: The Incredible Dr. Matrix, Chap. 1-18 MG4SA: Scribner (1976) MG4SA: The Magic Numbers of Dr. Matrix, Chap. 1-22 MG4SA: Prometheus Books (1985) MG4SA.1 New York MG4SA.1. numerology, Wagner and 13, Plutonium 94 <-> 49 (Manhatten project) MG4SA.1. Dewey decimal classification of "number Theory": 512.81 (2^9.3^4) MG4SA.1. ELEVEN & TWO \ ONE is an anagram of TWELVE MG4SA.1. Number of the Beast 666 with Roman numerals, hundret system MG4SA.1. american president and the double letter (Rockefeller-Nixon) MG4SA.1. sequence OTTFFSSENT, cryptarithm FORTY+TEN+TEN=SIXTY, Bach 14-41 MG4SA.2 Los Angeles MG4SA.2. rookwise-connected antimagic 3*3 square (unique, 10-n transform) MG4SA.2.a A Remarkable Group of Antimagic Squares, Math. Mag. 44 (1971)13, 236 MG4SA.2. Triskaidekaphobia: an irrational fear of the number 13. numerology MG4SA.2. 10^2 + 11^2 + 12^2 = 13^2 + 14^2 = 365 and generalizations MG4SA.2. A. S. Eddington's work on the fine-structure constant 137 MG4SA.2. Thue-Morse sequence a(1)=01, a(k+1)=a(k) & Complement a(k) MG4SA.2.b Unending Chess, Symbolic Dynamics & Semigroups, Duke Math 11 (1944)1 MG4SA.2.c Is there a sequence of four symbols in which no two adjacent MG4SA.2.c segments are permutations of one another, AMM 78 (1971) 886-888 MG4SA.2.d On Nonrepetitive Sequences, JoCT A 16 (1974) 159-164 MG4SA.2.g Guy, Unsolved Problems in Number Theory, E21 MG4SA.2.h Dejean's conjecture on 5..11 letter alphabets solved,TCS 95(1992)187 MG4SA.2.h repetition threshold 2->2, 3->7/4, 4->7/5, k->k/(k-1) (k>=5) MG4SA.2.i Tiling the Morse Sequence, TCS 94:2 (1992) 215-221 MG4SA.2.j Every binary Pattern of Length 6 is Avoidable on the 2-Letter MG4SA.2.j Alphabet, Acta Informatica 29:1 (1992) 95-107 (P. Roth) MG4SA.2.k Overlap free words and finite automata, TCS 115:2 (1993) 243-260 MG4SA.2.l Enumeration of irreducible binary words, c n^1.155 <g_n< C n^1.587 MG4SA.2.l Disc. Appl. Math. 20 (1988) 221-232, overlap free word (XXx free) MG4SA.2.e On Placing the Last Digit First, AMM 48 (1941) 251 MG4SA.2.f Division of Integers by Transposition, JoRM 1 (1968) 180 MG4SA.3 Sing Sing MG4SA.4 Lincoln and Kennedy MG4SA.5 Chicago MG4SA.5.a Schaaf, Bibliography of Rec. Math. II.1.11 Four Fours MG4SA.5.b Schaaf, Bibliography of Rec. Math. III.1.5 Four Fours, Four Digits MG4SA.5.c representing numbers using only one 4, M. Mag. (1964) 308-310 MG4SA.5.d Guy, Unsolved Problems in Number Theory, F26 MG4SA.5.e \Pi in four 4's, Eureka 25 (1962) 18-19 MG4SA.6 Miami Beach MG4SA.6. Kaprekar problem, ruler problems (different differences) MG4SA.6. unique Leech ruler: length 36, 8 marks at 1, 3, 6, 13, 20, 27, 31, 35 MG4SA.6.a comment of rulers: T.H. O'Beirne, Puzzles and Paradoxes (1965) 110-1 MG4SA.6.b Schaaf, Bibliography of Rec. Math. III.1.5 ..., Hundret Problem MG4SA.7 Philadelphia MG4SA.8 Pi MG4SA.8. the millionth digit of pi MG4SA.9 Wordsmith College MG4SA.10 Squaresville MG4SA.10. Properties of the square numbers. Odd next to last digit then last 6. MG4SA.10. G-numbers: (12^2=144, 21^2=441) (13^2=169, 31^2=961) MG4SA.10.c V. Feser, Rec Math Mag. (Aug. 1962) 39, G-numbers MG4SA.10. Automorphic numbers: n^2 ends with n in base d. MG4SA.10. base 10: ...8212890625, ...1787109376 (simple genetating rule) MG4SA.10.a JoRM 1 173-79, 2 172-74, 5 27, 6 75-76, 7 177 MG4SA.10.b automorphic numbers, Austral. Math. Soc. Gaz. 20 (1993) 37-44 MG4SA.11 Left Versus Right MG4SA.12 Fifth Avenue MG4SA.13 The Moon MG4SA.13. squares ends rotated (19 = rot(61)): 19^2=361, 1969^2=3876961 MG4SA.13. landing on the mars alphametic: SPIRO * 7 = AGNEW. MG4SA.13.a M Mag. 42 (1969) 102-103; solutions of A = kB for number systems MG4SA.13.a 3..15 and k=2..14 and A and B hold all (or zerofree) digits once. MG4SA.13. n points on the sphere with maximal minimal distances MG4SA.13.b The problem of Packing a number of equal nonoverlapping circles on MG4SA.13.b a sphere, Transactions of the New York Acad. of Science, II MG4SA.13.b 24 (1962) 320-331, H S M Coxeter MG4SA.13.c L Fejes Toth, Regular Figures, 1964, Chap. 7 MG4SA.13. we have a friday the 13th one upto three time a year (-> CAL) MG4SA.13. each year has 2..4 perverse (needs six calendar lines) months MG4SA.14 Honolulu MG4SA.14. highly composite numbers (hc): sigma0(7!)=60, sigma0(7560)=64 is hc MG4SA.14. n!+1=m*m has the solutions: (n,m) equals (4,5), (5,11), (7,71). MG4SA.14.sio n!!+1=m*m has the solutions: (n,m) equals (3,2),(4,3),(5,4),(6,7). MG4SA.15 Houston MG4SA.15. 2001: a space odyssey (A. C. Clarke), HAL - IBM (Ceasar chipher 1) MG4SA.16 Clairvoyance Test MG4SA.17 Pyramid Lake MG4SA.17.a A New Series: F_n+1 = F_n + F_n-2, Fib. Quart. 16 (1978) 335-343 MG4SA.17.a great pyramide of Gizeh (dimensions), p^3 = p^2 + 1, MG4SA.17.a the triangle (1,p,p^2) has a 120 degree vertex MG4SA.18 The King James Bible MG4SA.19 Calcutta MG4SA.20 Stanford MG4SA.20. calendar cubes problem: three cubes, three letters for the 12 months MG4SA.20. ascending primes MG4SA.20. alphabetic number tables: eight..zero (english), C..XXXVIII (roman) MG4SA.20. english numbers with letters in ascending order: forty (unique) MG4SA.20. same in descending order: one MG4SA.20. lowest number with contains a,e,i,o,u,y ("and" does not count) MG4SA.21 Chautauqua MG4SA.22 Istanbul MG4SA.22. cube vetex labelings: unique labeling 0..7 such that the sum of MG4SA.22. labels at each edge is a prime (is composite); MG4SA.22. label with different square numbers, such that all sums are primes MG4SA.22. cube dissection into three congruent skew pyramids (yangmas) MG4SA.22. unfold the cube (six pyramids) to form a rhombic dodecahedron MG5SA: The Unexpected Hanging and Other Mathematical Diversions MG5SA: Simon & Schuster (1968) MG5SA.1 The Paradox of the Unexpected Hanging MG5SA.1.a The surprise examination or unexpected hanging paradox, MG5SA.1.a AMM 105:1 (1998) 41-51, T Y Chow MG5SA.2 Knots and Borromean Rings MG5SA.2.a Borromean Squares, AMM 99:4 (1992) 377; no circles possible MG5SA.3 The Transcendential Number e MG5SA.4 Geometric Dissections MG5SA.4.a More Geometric Dissections, JoRM 7 (1974) 206-212, n-gons: a <-> b MG5SA.4.b Proof without words: Fair Allocation of a Pizza, MM 67:4 (1994) 267 MG5SA.4.c Dissection 5*5-gons give a 5-gon (15 pieces), alpha 29:6 (1995) 23,38 MG5SA.5 Scarne on Gambling MG5SA.6 The Church of the Fourth Dimension MG5SA.7 Eight Problems MG5SA.7.1 A Digit-Placing Problem MG5SA.7.1. hamiltonian path in the complementary graph MG5SA.7.2 The Lady or the Tiger (probability) MG5SA.7.2. urn problem of Laplace. MG5SA.7.3 A Tennis Match (parity, arithmetic) MG5SA.7.4 The Colored Bowling Pins MG5SA.7.4. triangle of ten spots, no two-color avoids equilateral triangles MG5SA.7.5 The Problem of the Six Matches MG5SA.7.5. plane distant one graphs, edges: 1..7 = 1,1,3,5,10,19,39 MG5SA.7.6 Two Chess Problems: Minimum and Maximum Attacks MG5SA.7.7 How Far Did the Smiths Travel (arithmetic, distance) MG5SA.7.8 Predicting a Finger Count MG5SA.8 A Matchbox Game-Learning Machine MG5SA.8. computerchess, hexapawn, minicheckers, minichess MG5SA.9 Spirals MG5SA.10 Rotations and Reflections MG5SA.11 Peg Solitaire MG5SA.11. move one peg far into the plane (Conway) -> MG13SA.19 MG5SA.11.a Montreal solitaire, JoCT A 60 (1992) 50-66 MG5SA.11.b UMAP Journal 16:2 (Summer 1995) Special Section on Math. Appl. Games MG5SA.12 Flatland MG5SA.12. greates cross section of the unitcube (area=Sqrt(2)) MG5SA.13 Chicago Magic Convention MG5SA.14 Tests of Divisibility MG5SA.14. checks if k in {2,3,4,5,6,8,9,10,11} divides n. MG5SA.14. 9 | n - digitsum(n) as 9 | 100a+10b+c - (a+b+c), nine check MG5SA.14. 11 | n - altdigitsum(n) as 11 | 100a+10b+c - (a-b+c), eleven check MG5SA.14. 21 | 10a+b <=> 21 | a-2b as -2*10 = 1 (mod 21), seven test MG5SA.14. 19 | 10a+b <=> 19 | a+2b as 2*10 = 1 (mod 19), 19-test MG5SA.14. 39 | 10a+b <=> 39 | a+4b as 4*10 = 1 (mod 39), 13-test MG5SA.14. 49 | 10a+b <=> 49 | a+5b as 5*10 = 1 (mod 49), seven test MG5SA.14. 399|100a+b <=> 399| a+4b as 4*100= 1 (mod399), 7*19 test MG5SA.14. Polynomial proof: 9 | P(10)-P(1) and 11 | P(10)-P(-1) MG5SA.14. and 7*11*13 | P(1000) - P(-1) for every polynomial P. MG5SA.14. casting out nines, digital roots -> MG2SA.4 MG5SA.14.a L E Dickson, "Criteria of Divisibility by a Given Number", MG5SA.14.a History_of_the_Theory_of_Numbers (New York: Chelsea Publishing, MG5SA.14.a 1952), Vol. I, Chapter 12, pp. 337-46. MG5SA.14.b E. A. Maxwell, Division by 7 or 13, Math Gaz, 49 (Feb 1965) 84 MG5SA.14.c Die Neunerprobe des Adam Ries und andere Reste, PM 39:6 (1997) 242-6 MG5SA.14.d Adam Ries und die Neunerprobe. Eine historische Studie MG5SA.14.d H Deubner, Mathematik in der Schule 8:7 (1970) 481-492 MG5SA.14.e B. A. Kordemsky, The Moscow Puzzles, 1972, Chap. 11: Divisibility MG5SA.14.x out of any set of 2N-1 integers, there is a subset of size MG5SA.14.x exactly N whose sum is divisible by N. (Erdos-Ginzburg-Ziv theorem) MG5SA.15 Nine Problems MG5SA.15.1 The Seven File Cards MG5SA.15.2 A Blue-Empty Graph (Ramsey) MG5SA.15.3 Two Games in a Row MG5SA.15.4 A Pair of Cryptarithms MG5SA.15.5 Dissecting a Square MG5SA.15.6 Traffic Flow in Floyd's Knob MG5SA.15.6.a Braess's Paradox: A Puzzler from Apllied Network Analysis MG5SA.15.6.a UMAP 13:4 (1992) 303-312 MG5SA.15.7 Littlewood's Foodnotes MG5SA.15.8 Nine to One Equal 100 (123456789) MG5SA.15.9 The Crossed Cylinders (volume, geometry) MG5SA.16 The Eight Queens and Other Chessboard Diversions MG5SA.17 A Loop of String MG5SA.17. Leopard Cat's Cradle, a loop-release trick, a ring-release trick, MG5SA.17. a scissors-release puzzle, Jacob's ladder = Osage Diamonds MG5SA.17.a Caroline F. Jayne, String Figures, Dover Publ. 1962 MG5SA.17.b Rotation of a String Figure, JoRM 8 (1976) 177-181 MG5SA.17.b a three lozenge (diamond) figure MG5SA.17. king paths in a letter rectangle (boggle) MG5SA.18 Curves of Constant Width MG5SA.19 Rep-Tiles: Replicating Figures on the Plane MG5SA.19 rep-tiles - reptiles - replicating figures MG5SA.19.a A Puzzling Journey to the Reptiles and Related Animals MG5SA.19.a http://www.kiwi.gen.nz/~karl/ MG5SA.20 Thirty-Seven Catch Questions (quickies) MG5SA.20.1 pythagoren joke MG5SA.20.2 clock puzzle (time) MG5SA.20.3 probability (X2 > X1) = 5/12, with dice X1, X2 MG5SA.20.4 what is the exact opposit of 'not in'? MG5SA.20.5 underdeterminded geometric question (crossed ladders) MG5SA.20.5.a biquadratic solution (Diophantine), Euclides 68 (1992/93) 228-233 MG5SA.20.6 what was the customer buying? (logarithmic costs, digit costs) MG5SA.20.7 area of the triangle, 13, 18, 31 MG5SA.20.8 wrong pronouncation MG5SA.20.9 remarkable coincidence of two sums -> MG4 MG5SA.20.10 angle of two diagonals at the cube (equilateral triangle) MG5SA.20.11 one word anagram of NEW DOOR MG5SA.20.12 Thales theorem (inversion) & Pythagoras MG5SA.20.13 statistiacal correlations between foot size and math. tests MG5SA.20.14 what is this familiar continuum, (Roy G. Biv. of Rainbow, Oregon) MG5SA.20.15 a simple formular which gives primes only MG5SA.20.16 equilateral triangle, point: minimal sum of distances to the sides MG5SA.20.17 arithmetic: divide 50 by 1/3 and add 3. MG5SA.20.18 word problem: cross out six letters, BSAINXLEATNTEARS MG5SA.20.19 arithmetic: is a topologist a doughnut? MG5SA.20.20 the bookkeeper's question: words with 3 double letters in a row MG5SA.20.21 Can two bisectors of a triangle intersect at right angles? MG5SA.20.22 How many month have 30 days? (at least or exactly) MG5SA.20.23 the last cigarettes, three butts give a new one. MG5SA.20.24 limerick: 1264853971.2758463 MG5SA.20.25 how long will three pills last, taken one every half-hour? MG5SA.20.26 knock-out turnament with 137 players, how many play are necessary? MG5SA.20.27 a ten letter word using only the top row of a typewriter MG5SA.20.27.a Typewriter Words, in Language on Vacation, Scribner's 1965,171-3 MG5SA.20.28 two US coins = 55 cents, one is not a nickel. What are the coins? MG5SA.20.29 arithmetic: a fish weigths 20 pounds plus half its own weight. MG5SA.20.30 a remarkable telegram (palindrome) MG5SA.20.31 three interpretations of III: x = III/III = III III (alphametic) MG5SA.20.32 rearrange six full and empty glases FFFEEE -> FEFEFE (one move) MG5SA.20.33 arithmetic: how many spaces does a wheel with ten spokes have? MG5SA.20.34 logic, semantic: NOT"the number of words in this sentence is nine" MG5SA.20.35 two sisters with the same birthday, but are not twins. why? MG5SA.20.36 how to lose $4. an unfair bet. "I'll bet you $1 that if you give MG5SA.20.36. me $5 I'll give you $100 in exchange". MG5SA.20.37 one dollar and 87 cents, 60 cents of if was in pennies (coinage) MG6SA: M. Gardner's Sixth Book of Mathematical Games from Scientific American MG6SA: Freeman (1971) San Francisco MG6SA.1 The Helix MG6SA.2 Klein Bottles and Other Surfaces MG6SA.2. folding a Klein bottle (a cross-cap, a projective plane, a torus, a MG6SA.2. M"obius surface) from a square, topolagical invariants of seven basic MG6SA.2. surfaces (chromatic number, Betti number), torus-cutting problem MG6SA.3 Combinatorial Theory MG6SA.3. Lo Shu magic square, folding stamps, magic hexagon MG6SA.4 Bouncing Balls in Polygons and Polyhedrons MG6SA.4. Billiard, liquid-pouring problem MG6SA.5 Four Unusual Board Games MG6SA.5. French Military Game, William L. Black's game (analysed), tablut, MG6SA.5. Sidney Sackson's game of Focus MG6SA.6 The Rigid Square and Eight Other Problems MG6SA.6.1 The Rigid Square MG6SA.6.2 A Penny Bet (n coins versus n+1 coins, probability 1/2) MG6SA.6.2.a A Card Game with a Positive Payoff, AMM 98 (1991) 760-762, E3330 MG6SA.6.3 Three-dimensional Maze MG6SA.6.4 Gold Links (n-link chain) MG6SA.6.5 Word Squares (Rotas Square) MG6SA.6.6 The Three Watch Hands (the three hands [clock] meets at 12 only) MG6SA.6.7 Three Cryptarithms MG6SA.6.8 Maximizing Chess Moves (minimum -> MG5SA) MG6SA.6.9 Folding a M"obius Strip (smallest ratio) MG6SA.7 Sliding-Block Puzzles MG6SA.7. Boss, Dad's, L'Ane Rouge, Line up the Quinties, Ma's, Stotts's Tiger MG6SA.7.a permutation puzzles (Topspin, Binary Arts) -> S_20 MG6SA.7.a College Math. J. 24:2 (1993) 163-5 -> (sio: simmilay to my Inver) MG6SA.7.b Dead pigs fly Town -> Dead pigs wont fly (45 moves opt. (Boss)) MG6SA.7.b Euclides 68 (1992/93) 246 Ex 642 MG6SA.7.c The Knight's Tour on the 15-Puzzle, Math Mag 66 (1993) 159-166 MG6SA.8 Parity Checks MG6SA.8. square root of 2 is irrational, two-color map theorem, MG6SA.8. the glass trick, reversing pairs of coins - the head parity is fixed MG6SA.8. R. Sprague's rolling cubes, color opposite faces different, the MG6SA.8. parity of the 3 visible faces of one color changes every move MG6SA.8. the enormous shoe box - for graph #{vertices with odd degree} is even MG6SA.8.-> rolling a tetrahedron (MG6SA.19) MG6SA.8.a On rolling a cube and a tetrahedron, AMM 90 (1983) 711-712, E6388 MG6SA.8.b Some things never change (invariants, parity), MG6SA.8.b Quantum 4:1 (1993) 35-37, 60, M91 MG6SA.8.b kernel of "Genius" (4*4 matrix), { (0+-0, -00+, +00-, 0-+0) } MG6SA.8.b moves: add (mod 1), rows, collomns, parallel of the diagonals. MG6SA.9 Patterns and Primes MG6SA.9. Sieve of Eratosthenes, Ulam's square spiral, prime magic square MG6SA.9. Euler's polynomial x^2+x+41, Mersenne, Fermat, pepunit, gear problem MG6SA.9.a Lucas-Lehmer Test, AMM 100 (1993) 370-371, simple proof, Mersenne P. MG6SA.9.b Fermat Numbers (factorizations) Math. Comp. 61 (1993) MG6SA.9.b 319-350: F9, 463-474: F6 (history) MG6SA.9.c Lenstra, The development of the number field sieve, LNiM 1554 (1993) MG6SA.9.c Fermat number factorizations, e. g. F7 MG6SA.9.d A hundred years of prime numbers, AMM 103 (1996) 729-741 MG6SA.9.e Newman's short proof of the Prime Number Theorem, D. Zagier MG6SA.9.e AMM 104:8 (Oct. 1997) 705-708 MG6SA.10 Graph Theory MG6SA.10. planar graphs, nomplanar K_3,3, K_5, imbedding in a grid, MG6SA.10.a A link between the Jordan curve theorem and Kuratowski planarity MG6SA.10.a criterion, AMM 97 (1990) 216-218; Zbl 736.05036 MG6SA.10.A Discrete Jordan curve theorems, JoCT B 47 (1989) 251-61 MR 90m:05118 MG6SA.10.b Open problems in grid labeling, AMM 97 (1990) 133-135, Zbl741:05058 MG6SA.10.c Cyclic Perfect One Factorizations of K_2n, 259-272 MG6SA.10.c Some Perfect One-Factorizations of K_14, 419-436 MG6SA.10.c Combin. Design Theory, Annals of Discrete Math. 34 (1987) MG6SA.10.d strictly rectangluar representations, Eureka 52 (Mar 1993) 30-44 MG6SA.10.e Football Pool Problem, JoCT A 67 (1994) 161-168 MG6SA.10.e FPP(11) <= 9477, FPP(12) <= 27702 MG6SA.10. nointersecting Euler path (black-white coloring by T. H. O'Beirne), MG6SA.10.f Pairwise compatible Euler Tours, JoCT B 53 (1991) 80-92 MG6SA.10.g Double Euler Tours, JoCT B 50 (1990) 198-207 MG6SA.10. Hamiltonian path, impossible rhombic dodecahedron, knight's tours MG6SA.10. -> KnT: Knight Tours MG6SA.10. -> MG8SA.14 Knights of the Square Table MG6SA.10. -> Hamiltonische Linien, MU 24:3 (1978) 5-40 MG6SA.10. - dodecahedron unique circle, Petersen, knight tours MG6SA.10. - Sachs = Sci Am 10:1992 118-20 (I. Stewart) planar graph criteria MG6SA.10. -> Hamiltonian Checkerboards, Math. Mag. 57 (1984) 291-294 MG6SA.10. - ham. circles in C_n * C_m. When are left or up moves sufficient? MG6SA.10. -> Circuits in Directed Grids, Math Intell. 13:3 (1991) 40-43 MG6SA.10. - ham. circles in C_n * C_m. When are left or up moves sufficient? MG6SA.10. -> Figured Tours (Knight, Rook), Math. Spectrum 25:1 (1992) 16-20 MG6SA.10. - special pattern: e.g. 6*6 knight tour with 1 and 4 in the same row MG6SA.10. -> Hamiltonian Paths in Graphs of Linear Extensions for Unions of MG6SA.10. - Posets, SIAM Disc. Math., 5:2 (1992) 199-206 (G. Stachowiak) MG6SA.10. -> A Hamiltonian path in the transposition graph for multiset MG6SA.10. - permutations, Congr. Numer. 67 (1988) 27-34, MR 90k:05099 MG6SA.10. -> Leaper graphs (r,s-knights), D. E. Knuth, is the 2*(r+s) square MG6SA.10. - Hamiltonian (for (r,s)=1)? Checked for r+s<=15. MG6SA.10. - The Math. Gazette 78 (1994) 274-296 MG6SA.10. -> The Gordon Game of a Finite Group, AMM 99 (1992) 567-569 MG6SA.10. - group-tours x1=e, x2=a1*x1, x3=a2*x2, ... s.t. all moves a_n are MG6SA.10. - different. A finite abelian group G has a tour iff [G:2G]=2. MG6SA.10. - So if G is cyclic then ord(G) must be even (all interval series) MG6SA.10. -> A Fast Method for Sequencing Low Order Non-Abelian Groups, 27-42 MG6SA.10. - Combin. Design Theory, Annals of Discrete Math. 34 (1987) MG6SA.11 The Ternary System MG6SA.11. the counterfeit coin problem, sorting procedure (DEMOCRAT cards) MG6SA.11.a detection of a defective coin with partial weight information, MG6SA.11.a AMM 91 (1984) 173-179, (history, beam scale, spring scale) MG6SA.11.b a tale of two coins, AMM 94 (1987) 121-129 MG6SA.11.c how to find many counterfeit coins? Graphs Combin. 2 (1986) 173-7 MG6SA.11.d Updating a Tale of 2 Coins, Ann. NY Acad. Sci. 576 (1989) 259-265 MG6SA.11.e MR 89a:90153, MR 89g:90121 MG6SA.12 The Trip around the Moon and Seven Other Problems MG6SA.12.1 The Trip around the Moon (deep in the dessert) MG6SA.12.1.a the jeep once more or jeeper by the dozen, AMM 77 (1970) 493-501 MG6SA.12.1.b a new approach to the jeep problem, Bul. EATCS 38 (1989) 145-154 MG6SA.12.1.b only a discrete amont (one tank) may be deposited MG6SA.12.1.c Gale's round-trip jeep problem, AMM 102:2 (Apr 1995) 299-309 MG6SA.12.1.c open case solved. Dewdney's variation is solved too. MG6SA.12.2 The Rectangle and the Oil Well (elementary geometry) MG6SA.12.3 Wild Ticktacktoe (S. W. Golomb) MG6SA.12.4 Coins of the Realm (R. Sprague: efficient coin system of 16 coins) MG6SA.12.5 Bills and Two Hats (maximizing probability) MG6SA.12.6 Dudeney's Word Square (forword, backword, diagonal) MG6SA.12.7 Ranking Weights (rank five weights with seven weightings (balance)) MG6SA.12.7.a weighting seven coins in five weightings, AMM (1989) 254-8,E3023 MG6SA.12.7.b Optimal sorting n=12, MR 92g:05007; Knuth, TAOCP 5.1.3 MG6SA.12.8 Queen's Tours (five chess-tour problems) MG6SA.13 The Cycloid: Helen of Geometry MG6SA.14 Mathematical Magic Tricks MG6SA.15 Word Play MG6SA.15. pangram (sentences with 26 different letters) MG6SA.15. Claude E. Shannon: Squdgy fez, blank jimp crwth vox! MG6SA.16 The Pythagorean Theorem MG6SA.16. chinease proof (dissection of (a+b)^2), shearing proofs, Euclids MG6SA.16. Thm -> Pyth, Pappus theorem, pythagorean tripples MG6SA.16.a The Pythagorean Proposition, DC NCTM, 1968, contains 367 proofs MG6SA.16.b Themenkomplex Pythagoras, MG6SA.16.b Lehrb"ucher und Monographien zur Didaktik der Mathematik, 39, 1995 MG6SA.16.c Some recent discoveries in elementary geometry, MG6SA.16.c Math Gaz 81:492 (Nov 1997) 391-397 MG6SA.16.c Faulhaber's generalization of the Pythagorean Thm. to 3-dim. MG6SA.16.d Note on the Ptolemy Theorem, C. Popescu MG6SA.16.d Nieuw Archief voor Wiskunde, 15:3 (Nov 1997) 193-197 MG6SA.16.e Zum Satz von Ptolem"aus, MNU 44:8 (1991) 464-466, F. Bodnar MG6SA.16.e Ptolemy, the convex tetragon (quadrangle), trigonometric proof MG6SA.16.f Pythagoras und kein Ende? (P Baptiste) MG6SA.16.f Klett Verlag, Leipzig 1997, 152p MG6SA.17 Limits of Infinite Series MG6SA.17. Zeno's Paradox, harmonic series (infinite-offset paradox) MG6SA.17.a W. Stadje, convergence of parts of the harmonic series (deutsch), MG6SA.17.a Elem. Math. 46 (1991) 51-54; #MR 92h:11009 MG6SA.17.b partial sums of the harmonic series, AMM 78 (1971) 864-870 MG6SA.17.c Concrete Math. Ex 9.67: Let Q_n be the least integer m such that MG6SA.17.c H_m > n. Find the smallest integer n such that MG6SA.17.c Q_n <> [exp(n-gamma)+1/2], or prove that no such n exists. (-> d) MG6SA.17.d The Form of Comtet Functions of Divergent Series (-> c) MG6SA.17.d Utilitas Math. 42 (1992) 241-245 MG6SA.18 Polyiamonds MG6SA.18. hexiamonds, three twins, unique star MG6SA.19 Tetrahedrons MG6SA.19. tetrahedron-octahedron space tesselation, pentatope, MG6SA.19. rolling tetraeder trick, nine problems, impossible magic tetrahedron MG6SA.19.a Topsy-turvy pyramids (rollings tetrahedra), Quantum 4:1 (1993) 63-64 MG6SA.19.b Cannonball pyramids, Triads, Tumbleweed (rolling tetrahedra) MG6SA.19.b Quantum (Nov Dec 1993) 63-64 MG6SA.20 Coleridge's Apples and Eight Other Problems MG6SA.20.1 Coleridge's Apples (linear Diophant problem) MG6SA.20.2 Reversed Trousers (topology) MG6SA.20.3 Coin Game (parity) MG6SA.20.4 Truthers, Liars, and Randomizers (logic) MG6SA.20.5 Gear Paradox (mechanical puzzle) MG6SA.20.6 Form a Swastika (Nazi cross) MG6SA.20.7 Blades of Grass Game (probability for fortune-telling in russia) MG6SA.20.7. the ends are tied in pairs. two entwined rings indicate marriage. MG6SA.20.7. 2 perfect matchings form a circle on 2n vertices with MG6SA.20.7. prob=(2/3)(4/5)*...*((2n-2)/(2n-1)) MG6SA.20.8 Casey at the Bat MG6SA.20.9 The Eight-Block Puzzle (3*3 Boss puzzles) MG6SA.21 The Lattice of Integers MG6SA.21. continued fraction of 2, billiard-ball paths, area of polygons MG6SA.21.a Das Dreikruegeproblem #ZBl 634.10001 (three-pitchers problem) MG6SA.21.b Sprague, Zur Theorie der Umfuell-Aufgaben, Jber DMV 49 (1940) 65-73 MG6SA.21.c Musical Scale Constructions: The Continued Fraction Compromise MG6SA.21.c Utilitas Math. 42 (1992) 97-113 (with historical remarks) MG6SA.21.d continued fraction [1,2,3,4,5,..] = I_0(2) / I_1(2) MG6SA.21.d (Hyperbolic Bessel functions) E3264, AMM (Feb 1990) MG6SA.21.e The Euclidean Algorithm Strikes Again, AMM 97 (1990) 125-129 MG6SA.21.e representations for p: a^2 + b^2 = p = 4k+1 (p prime) MG6SA.21.f A One-sentence Proof that Every Prime p = 1 (mod 4) is the Sum MG6SA.21.f of two squares, AMM 97 (1990) 144 (fix point proof) MG6SA.22 Infinite Regress MG6SA.22. impossible cubed cube, snowflake curve, Escher's Drawing Hands MG6SA.23 O'Gara, the Mathematical Mailman MG6SA.23. philately, path problems, minimum (maximum) turn problems MG6SA.24 Op Art MG6SA.24. n-bug problem, length of a logarithm spiral -> MMR.8.1 MG6SA.24. heptiamonds (tiling), tesselation of convex heptagons MG6SA.24.a The patterns of the isonemal two-color two-way two-fold fabrics MG6SA.24.a (weaving), Zbl 736.05022 MG6SA.24.b Random Dot Stereogram, AMM 101 (1994) 715-724 MG6SA.25 Extraterrestrial Communication MG6SA.25. decrypt I. Bell's interplanetary message MG7SA: Mathematical Carnival MG7SA: A. Knopf (1975) MG7SA.1 Sprouts and Brussels Sprouts MG7SA.1. a topological pencil and paper graph game. There is a fairly thorough MG7SA.1. description of it in the Peirs Anthony novel "Macroscope". MG7SA.1.a Graph Theory and the Game of Sprouts, AMM 100 (1993) 478-482 MG7SA.2 Penny Puzzles (Coins) MG7SA.2. doubling problem, close-packing problem (rhomboid <-> circular), MG7SA.2.a Stacking/doubling Coins (Double Five) in a circle at even positions, MG7SA.2.a (possible n=10k, n=4k), Crux Math. 19:6 (1993) 185-186, Ex. 1769 MG7SA.2. Triangle-inversion problem, Triangular solitaire, rotation problem, MG7SA.2. surprising invariance theorem, tree-planting problems, 3penny tricks MG7SA.3 Aleph-null and Aleph-one MG7SA.4 Hypercubes MG7SA.4. Charles Howard Hinton, Hinton cubes, visualizing the tesseract MG7SA.4. largest square in a cube, gives the largest octahedron in a cube MG7SA.4. largest cube in a hypercube, Prince Rupert problem MG7SA.4. 11 hexominoes fold into cubes, 261 ways for the tesseract (hypercube) MG7SA.4. the number of points, lines, cubes, ... of the n-cube; GF = (1+2x)^n MG7SA.4.a Symmeties of the Cube and Outer Automorphisms of S6, MG7SA.4.a AMM 100 (1993) 377-380 MG7SA.4.b Keller's cube-tiling conjecture is false in high dimensions (n>=10) MG7SA.4.b Bull. AMS 27 (1992) 279-283, MR 93e:52040 MG7SA.4.b Keller: If R^n is tiled by congruent parallel unit cubes, then some MG7SA.4.b pair shares a complete (n-1) dimensional face. MG7SA.4.c A new distance metric on strings computable in linear time, MG7SA.4.c Disc. Appl. Math. 20 (1988) 191-203 (sequence space, edit distance) MG7SA.4.d two characterizations of the general. hypercube (Hamming sheme) MG7SA.4.d Disc. Math. 93 (1991) 63-74 MG7SA.4.e Unfolding the tesseract, JoRM 17 (1984-85) 1-16 (261 octacubes) MG7SA.5 Magic Stars and Polyhedrons MG7SA.5. impossible magic pentagram (K5), magic hexagram (septagram,octagram) MG7SA.5. the hexagram is equivalent to octahedron and cube, impossible prism MG7SA.5.a perimeter-magic polygons, JoRM 7:1 (1974) 14-20 MG7SA.5.b magic graphs, a characterization, Europ. J Comb. 9 (1988) 363-8 MG7SA.5.b MR 89f:05138 MG7SA.6 Calculating Prodigies MG7SA.6. mental calculaction tricks MG7SA.7 Tricks of Lightning Calculators MG7SA.7. mental calculaction tricks MG7SA.8 The Art of M. C. Escher MG7SA.8.a Napoleon, Escher, and Tessellation, M. Mag. 64 (1991) 242-246 MG7SA.8.b Tiling Survey, Micro Math 9:2 (1993) 3-24, Book, Poster, Program MG7SA.9 The Red-Faced Cube and Other Problems MG7SA.9.1 The Red-Faced Cube (Harris' cube rolling problems) MG7SA.9.1a Single Vacancy Rolling Cube Problem, JoRM 7 (1974) 220-224 MG7SA.9.2 The Three Cards (logic) MG7SA.9.3 The Key and the Keyhole (topology) MG7SA.9.4 Anagram Dictionary MG7SA.9.5 A Million Points (half the number of points) MG7SA.9.6 Lady on the Lake (hunting game) MG7SA.9.7 Killing Squares and Rectangles (combinatorial geometry) MG7SA.9.8 Cocircular Points (elementary geometry, search in a picture) MG7SA.9.9 The Poisoned Glass (binary search, expected number of tests) MG7SA.10 Card Shuffles MG7SA.10. maximal order of a permutation of 52, MG7SA.10.a the expected order of a random permutation, Bull LMS 23 (1991) 34-42 MG7SA.10.a Zbl 735.11044 MG7SA.10. Groups of Perfect Shuffles -> Math. Magazine 60 (1987) 3-14 MG7SA.10.b Shuffling Cards, AMM (1986) 333-348 MG7SA.11 Mrs. Perkins' Quilt and Other Square-Packing Problems MG7SA.11. squared squares, minimum prime dissections for squares, MG7SA.11. pyramidal = square number (only nontrivial solution n=24, m=70), MG7SA.11. 1..24 (without 7) squares can be packed into a 70 square MG7SA.11. open problem: 1..n square packing of a rectangle (other than n=1) MG7SA.11. table of smallest squares which can be packed by 1..n squares (n<=18) MG7SA.11. 1/2, 1/3, 1/4, ... squares can be packed into a 5/6 square. MG7SA.11. squares with total area 1 can be packed into a square of area 2. MG7SA.11.a Concrete Math. Ex 2.37: can 1/k * 1/(k+1) rectangles be packed into MG7SA.11.a the unite square? (k=1, 2, ... ) open question MG7SA.11.b Tile a rectangle with rectangles of the form 1*2, 2*3, ... n*(n+1) MG7SA.11.b using one tile of each (n<6 solvable), JoRM 24:1 (1992) 57-9 Ex1942 MG7SA.11.c On Some New Simple Perfect Squared Squares, MG7SA.11.c Disc. Math. 106/7 (1992) 67-75 MG7SA.11.d On Packing Unequal Rectangles in the Unit Square (1/k * 1/(k+1)) MG7SA.11.d JoCT 68 (1994) 465-469, they fit into a 133/132 square. MG7SA.11.d 1/k * 1/k (k>=2) fit into a rectangle of area=47/72. MG7SA.11.e Covering a square by e. circles, Elemente d. Math. 50 (1995) 167-170 MG7SA.11.f Dichteste Packungen von gleichen Kreisen in einem Quadrat (densest MG7SA.11.f circle packing of a square), Elemente d. Math. 49 (1994) 16-26 MG7SA.11.g Tiling a rectangle with the fewest squares, JoCT A 76 (1996) 272-291 MG7SA.11.h Tiling a rectangle with the fewest squares, Math Gaz 82 (1998) 134-5 MG7SA.12 The Numerology of Dr. Flies MG7SA.12. biorhythm, 23, 28, 33 day cycles, the largest positive interger that MG7SA.12. can't be expressed as a sum of multiples of two nonnegative integers MG7SA.12. a and b (relatively prime) is (a-1)(b-1)-1. R Sprague (Problem 26) MG7SA.13 Random Numbers MG7SA.14 The Rising Hourglass and Other Physics Puzzles MG7SA.14.1 Two Hundred Pigeons MG7SA.14.2 The Rising Hourglass MG7SA.14.3 Iron Torus MG7SA.14.4 Suspended Horsehoe MG7SA.14.5 Center the Cork MG7SA.14.6 Oil and Vinegar MG7SA.14.7 Carroll's Carriage MG7SA.14.8 Magnet Testing MG7SA.14.9 Melting Ice Cube MG7SA.14.10 Stealing Bell Rops MG7SA.14.11 Moving Shadow MG7SA.14.12 The Coiled Hose MG7SA.14.13 Egg in Bottle MG7SA.14.14 Bathtub Boat MG7SA.14.15 Balloon in Car MG7SA.14.16 Hollow Moon MG7SA.14.17 Lunar Bird MG7SA.14.18 The Compton Tube MG7SA.14.19 Fishy Problem MG7SA.14.20 Bicycle Paradox (mechanics) MG7SA.14.21 Inertial Drive MG7SA.14.22 Worth of Gold MG7SA.14.23 Switching Paradox MG7SA.15 Pascal's Triangle MG7SA.15.a even Multinomial Coefficients, Math. Mag. 64 (1991) 115-122 MG7SA.15.a Kummer's theorem for multinomial coefficients. MG7SA.15.a If g is the number of carries in the p-ary addition of MG7SA.15.a e1, e2, ... et then g is the order of p in Multinom(e1,e2,...et). MG7SA.15.b Zaphod Beeblebrox's Brain & Pascal Triangle, AMM 99 (1992) 318-331 MG7SA.15.b the number of integers p (mod 2^q) in any row of Pascal's triangle MG7SA.15.c Only finitly many rows in Pascal's triangel consits exclusively MG7SA.15.c of rth-power-free integers, AMM 99 (1992) 579-580 MG7SA.15.d Cube Slices, Pictorial Triangles, Prob., M. Mag. 64 (1991) 219-241 MG7SA.15.e A General. of a Congruential Property of Lucas, AMM 99(1992)231-38 MG7SA.15.f The distribution of the Binomial Coefficients modulo P, (Wilf) MG7SA.15.f J. Number Theory 41 (1992) 1-5 MG7SA.15.g Pascal triangle, Zbl 735.05003 MG7SA.15.h Pascal's T. and (1+x+x^2..x^t)^m, MR 93i:11021 MG7SA.16 Jam, Hot, and Other Games MG7SA.16. Jam, Hot are equivalent to ticktacktoe, nim, kayles, Henon's string MG7SA.16. game (kayles), Isaacs' hamstrung squad car game (differential games) MG7SA.16.a Daisies, Kayles, and the Sibert-Conway decomposiotion in misere MG7SA.16. octal games, Theor. Comp. Science 96:2 (1992) 361-388 MG7SA.16.b Mate with bishop and Knight in Kriegspiel, TCS 96, 389-403 MG7SA.17 Cooks and Quibble-Cooks MG7SA.17. wrong solutions, chess opening, Loyd - Dudeney square dissection, MG7SA.17. Dudeney's rook's-tour, Dudeney's clock-puzzle dissection, MG7SA.18 Piet Hein's Superellipse MG7SA.19 How to Trisect an Angle MG7SA.19. Archimedes' method, Kempe's linkage, the tomahawk trisector. MG8SA: Mathematical Magic Show MG8SA: Alfred A. Knopf (1977) New York MG8SA.1 Nothing MG8SA.2 More Ado About Nothing MG8SA.3 Game Theory, Guess It, Foxholes MG8SA.4 Factorial Oddities MG8SA.5 The Cocktail Cherry and Other Problems MG8SA.5.1 The Cocktail Cherry (matches) MG8SA.5.2 The Papered Cube (maximal cube) MG8SA.5.3 Lunch At The TL Club (truther, liar, logic) MG8SA.5.4 A Fair Division (conguence) MG8SA.5.5 Tri-Hex (game, ticktactoe, geometry of incidence, golden ratio) MG8SA.5.6 Langford's Problem MG8SA.5.6. langford sequences: 312132, 41312432; no of solutions for MG8SA.5.6. 3,4,7,8,11,12 = 1,1,26,150,17792,108144; seq. exists iff n=0,3 mod 4 MG8SA.5.7 Overlap Squares (elementary geometry quicky) MG8SA.5.8 Families in Fertilia (probability, statistical fallacy) MG8SA.5.8. expected size of a family, geometric distribution MG8SA.5.9 Christmas and Halloween MG8SA.5.9. Halloween = oct. 31 = dec. 25 = Christmas MG8SA.5.9. octal - decimal, alphametic: 675*31 = 837*25 = 20925 unique MG8SA.5.10 Knot The Rope (party trick) MG8SA.6 Double Acrostics MG8SA.7 Playing Cards MG8SA.7. poker patience: five poker hands which are are a straight or better MG8SA.7. are impossible MG8SA.8 Finger Arithmetic MG8SA.9 M"obius Bands (Mobius, Moebius Bands) MG8SA.9.a The Dark Side of the M"obius Strip, AMM 97 (1990) 890-897 MG8SA.9.a concrete embedding of the M"obius strip in R^3, minimal energy MG8SA.10 Ridiculous Questions MG8SA.11 Polyhexes and Polyaboloes MG8SA.11.a Rhombiominoes, JoRM 24:2 (1992) 144-146, Prob 1961 MG8SA.12 Perfect, Amicable, Sociable MG8SA.12.a Perfect Numbers, Quantum 3:3 (1993) 18-23, 59 (update, history) MG8SA.12.b new sociable numbers, Math. of Comp. 56 (1991) 871-873 MG8SA.12.c Favorable conditions for amicability, JoRM 24 (1992) 245-249 MG8SA.13 Polyominoes and Rectification MG8SA.13.a Polyominoes of Order 3 do not exists, JoCT A 61 (1992) 130-136 MG8SA.13.b 15 L3 + N4 -> 7*7 square, Euclides 67 (1991/92) 93 Ex 628 MG8SA.13. pattern for the 7 tetrahexes and 22 pentahexes MG8SA.13. set of 3 trihexes and 7 tetrahexes sold by Coffin (Snowflake) MG8SA.13. pattern for the 14 tetrabolos MG8SA.13. replication problems: tetrabolo, hexomino, octomino MG8SA.14 Knights of the Square Table MG8SA.14.a the n*n Knight cover problem (n<=20) JoRM 23 (1991) 255-267 MG8SA.15 The Dragon Curve and Other Problems MG8SA.15.a reflections on the emergence of space-filling curves, Zbl 736.01002 MG8SA.15.1 Interrupted Bridge Game MG8SA.15.2 Nora L. Aron (cryptarithm) MG8SA.15.2.a On the Reversing of Digits, Math. Mag. 42 (1969) 208-210 MG8SA.15.3 Polyomino Four-Color Problem MG8SA.15.3. 11 monominoes require 4 colors, 6 cubes each pair shares a surface MG8SA.15.4 How Many Spots? MG8SA.15.5 The Three Coins MG8SA.15.6 The 25 Knights (parity) MG8SA.15.7 The Dragon Curve (paper strip folding) MG8SA.15.8 The Ten Soldiers (longest increasing (decreasing) subsequences) MG8SA.15.9 A Curious Set Of Integers MG8SA.16 Colored Triangles and Cubes (MacMahon) MG8SA.17 Trees MG8SA.17.a Tree Isomorphism Algorithm, Speed Clarity, M. Mag. 64 (1991) 252-61 MG8SA.17.b Gen. binary trees (transpose) J. Algorithm 11 (1990) 68-84 (Ruskey) MG8SA.17.c New Tricks for Old Trees: Maps and the Pigeonhole Principle MG8SA.17.c AMM 101 (1994) 664-667 (graph-spanner) MG8SA.17.c Spanning tree of the hypercube-graph has diameter >= 2n+1 MG8SA.18 Dice MG8SA.18.a Weldon Dice Data Revisted, MG8SA.18.a Amer. Stat. 45:3 (1991) 216-222, 46:3 (1992) 239-240 MG8SA.19 Everything MG9SA: Mathematical Circus MG9SA: Alfred A. Knopf (1979) New York MG9SA.1 Optical Illusions MG9SA.2 Matches MG9SA.3 Spheres and Hyperspheres MG9SA.3. Soddy's fourth circle poem (with hyperspheres) MG9SA.3. sphere packings, kissing number, Leech lattice, -> MG3SA.7 MG9SA.3.a On spherical codes generating the kissing number in dim 8 and 24. MG9SA.3.a Disc. Math. 106/7 (1992) 199-207 MG9SA.4 Patterns of Induction MG9SA.5 Elegant Triangles MG9SA.5. Napoleon thm: the triangle of the centers of exterior equilateral MG9SA.5. triangles is equilateral. (Geometric Transformation, I M Jaglom) MG9SA.5. 5-con triangles: (8,12,18) and (12,18,27) have 5 common sides+angles MG9SA.5.a Simultaneous gener. of the theorems of Ceva and Menelaus MG9SA.5.a Math. Mag. 65:1 (1992) 48-52 Zbl. 756.51016 MG9SA.5. the cross ladders, finding integral parameters MG9SA.5.c Math Quickies Q 201: Segments determine an equilateral triangle MG9SA.6 Random Walks and Gambling MG9SA.6.a letter chains (Kettenbriefe), Stochastik in der Schule 12:3 (1992)37 MG9SA.6.b Probability models of pyramid or chain letter systems demonstrating MG9SA.6.b that their promotional claims are unreliable, MG9SA.6.b Operations Research 32 (1984) 527-536 MG9SA.6.c Gleichverteilung-Entropie, Expositiones Math. II:1 (1993) 3-46 MG9SA.6.c Ehrenfest game, urn model, Maxwell demon MG9SA.7 Random Walks on the Plane and in Space MG9SA.7. Markov chains MG9SA.7.a Back to Square One, Eureka 49 (March 1989) 67-70 MG9SA.7.a E(return) = #vertices, for doublestochastic irreduceable matrices MG9SA.7.b Chip-Firing games on directed graphs, J o Alg. Comb. 1 (1992) 308-328 MG9SA.7.b probabilistic abacus of Engel, undirected -> polynomial run-time MG9SA.7.c Mr. Markov Plays Chutes and Ladders, UMAP 14:1 (1993) 31-38 MG9SA.8 Boolean Algebra MG9SA.9 Can Machines Think? MG9SA.9. Turing machines (addition), Eliza, Turing test, MG9SA.9.a Noncomputability and the Busy Beaver Problem, UMAP 14:1 (1993) 41-73 MG9SA.9. 2001: a space odyssey (A. C. Clarke), HAL - IBM (Ceasar chipher 1) MG9SA.10 Cyclic Numbers MG9SA.11 Eccentric Chess and Other Problems MG9SA.11.1 Eccentric Chess MG9SA.11.1.a Martin Gardner's "Royal Problem", Quantum 4:1 (1993) 45-46 MG9SA.11.2 Talkative Eve MG9SA.11.2. cryptarithm eve/did = .talktalktalk... MG9SA.11.3 Three Squares (elementary geometry) MG9SA.11.3. an angle addition problem, elementary solutions with no trigs MG9SA.11.3. arctan(1/3)+arctan(1/2)=arctan(1), (3+i)(2+i)=5+5i MG9SA.11.3.a 54 different proofs, JoRM 4 (Apr 1971) 90-99 (Ch Trigg) MG9SA.11.3.b Math. Gaz. (Dec 1973) 334-336 + (Oct 1974) 212-215 (R. Narth) MG9SA.11.3.c geometrical proof of a result of Lehmer's, Fib. Quar. 11(1973)539 MG9SA.11.3.c a generalization for n squares in a row. (Ch Trigg) MG9SA.11.3.sio triangle (a,b,c)=(3,4,5) has (s,s_a,s_b,s_c)=(6,3,2,1), rho=1 MG9SA.11.4 Pohl's Proposition (binary magic trick) MG9SA.11.5 Escott's Sliding Blocks MG9SA.11.5. sliding-block puzzle, 4 squares & 6 l-shapes, 30*24 board MG9SA.11.6 Red, White, and Blue Weights MG9SA.11.6. six weights, three light and three heavy, three light-heavy pairs MG9SA.11.6. are colored red, white, blue; two weightings on a balance scale MG9SA.11.6. three weightings without colors, two different solutions MG9SA.11.7 The 10-Digit Numbers MG9SA.11.7.a tally numbers, JoRM 11 (1978-79) 76-77 (F. Rubin) MG9SA.11.7.b Math.Kabinet.3.1.7 selfdescribing sequences MG9SA.11.7. Base4: 1210, 2020, Base5: 521200, BaseR>6: (R-4)2100..001000 MG9SA.11.7.c Concrete Math. Ex 2.36, 9.63: Golomb's selfdescribing sequence, MG9SA.11.7.c f(k) = #{n|f(n)=k}, f is nondescending, 1,2,2,3,3,4,4,4,5,... MG9SA.11.7.d self-ref-sentences, (attractor, cycles), PM 35:6 (1993) 241-244 MG9SA.11.7.e cyclic counting trios (self-ref), Fib Quart. 25 (1987) 11-20 MG9SA.11.7.f selfdescribing sequences, Math. Mag. 66:4 (1993) 276-277 (refs) MG9SA.11.8 Bowling-Ball Pennies MG9SA.11.8. remove minimum number of points, avoiding equilateral triangles MG9SA.11.9 Knockout Geography MG9SA.11.9. isola-game on a graph (states of the USA, two states are connected MG9SA.11.9. if the first letter of one is the last of the other), 2..3 players MG9SA.11.9.a On Ringeisen's Isolation Game, Disc Math 80 (1990) 297-312 MG9SA.11.9.b The vertex picking game and a variation of the game of MG9SA.11.9.b dots and boxes, Disc Math 70 (1988) 311-313, MR 89f:05111 MG9SA.12 Dominoes MG9SA.12.a Tiling with 28 Dominoes, JoRM 24:2 (1992) 157-158 Ex 1881 MG9SA.12.a placing n dominoes in a (2n+1)*(2n+2) rectangle, can MG9SA.12.a determine the entire tiling. MG9SA.12.b 23 hexominoes (4 tetrominoes) are tilable with dominoes, MG9SA.12.b alpha 92:5, p29 + p35 MG9SA.12.c Alternating-Sign Matrices and Domino Tilings, (Elkies,..) MG9SA.12.c J. of Algebraic Combinatorics 1 (19??) MG9SA.12.d counting perfect mathchings in hexagonal systems MG9SA.12.d Graphs, hypergraphs and appl., Proc. Conf. Graph Theory, Eyba, 1984 MG9SA.12.d Teubner-Texte zur Mathematik 73, Leipzig, 72-79 MG9SA.12.e Calcualting the Number of Perfect Matchings and of Spanning Trees, MG9SA.12.e Pauling's Orders, the Characteristic Polynomial, and the MG9SA.12.e Eigenvectors of a Benzenoid System (P. John, H. Sachs) MG9SA.12.e Topics in Current Chemistry 153, Springer, 1990, 145-179 MG9SA.12. 4*4 magic squares with dominoes MG9SA.13 Fibonacci and Lucas Numbers MG9SA.13. the only Fibonacci Squares are 1 and 144. MG9SA.13. the only Fibonacci triangular numbers are 1, 3, and 55. MG9SA.13.a a Fibonacci Based pseudo-random number generator, z(n)=floor(n*p) MG9SA.13.a Zbl 735.65001 MG9SA.13.b Fibonacci numbers and Fermat Last Theorem, Acta. Arith. 60(1992)371 MG9SA.13.c Lagarias: Pseudorandom Number Generators, In: Cryptology and Comp. MG9SA.13.c Number Theory (Proc of Symp. in Appl. Math. AMS 42 (1989)) MG9SA.13.d A Fibonacci version of Kraft's inequality applied to discrete MG9SA.13.d unimodal search, SIAM J. Computing 22:4 (1993) 751-777 MG9SA.13.e A fast algorithm of the Chinese remainder theorem and its applic. MG9SA.13.e to Fibonacci numbers, MR 93i:11004 MG9SA.13.f Wie erkennt man eine Fibonacci Zahl? (Fibonacci number detection) MG9SA.13.f MSem. 45 (1998) 243-246, M M"obius MG9SA.13.f z is a Fibonacci iff [tau z - 1/z, tau z + 1/z] contains an integer. MG9SA.14 Simplicity MG9SA.14. von Aubel's Thm: segments between opposite centers of squares on the MG9SA.14. side of a quadrilateral have equal length and are perpendicular MG9SA.14.a Von Aubel's Quadrilateral Thm. (P.J.Kelly), MG9SA.14.a Math. Mag. (Jan. 1966) 35-37, vector proof and generalizations MG9SA.14. Leo Moser's Circle Dissection Problem, (1,2,4,8,16,31,...) -> DME1.9 MG9SA.15 The Rotating Round Table and Other Problems MG9SA.15.1 Rotating Round Table MG9SA.15.1. no matches on a round table, semiqueen problem on a cylinder MG9SA.15.2 Single-Check Chess MG9SA.15.2. one check wins, presto chess, white wins with only 5 knights moves MG9SA.15.2. first check with a figure that can't be taken wins (open problem) MG9SA.15.3 Word Guessing Game MG9SA.15.4 Triple Beer Rings MG9SA.15.5 Two-Cube Calendar MG9SA.15.6 Uncrossed Knight's Tours MG9SA.15.6.a longest uncrossed Knigth's tours n*m, JoRM 2 (1969) 154-157 MG9SA.15.7 Two Urn Problems (probability, Carroll's Pillow Problem 5) MG9SA.15.8 Ten Quickies MG9SA.16 Solar System Oddities MG9SA.17 Mascheroni Constructions MG9SA.18 The Abacus MG9SA.18.a the Algorists vs. the Abacists, TCMJ 24:3 (1993) 218-223 MG9SA.19 Palindromes: Words and Numbers MG9SA.19.a Palindrome squares for various bases, Zbl 755.11004 MG9SA.19.a open problem: is the number of base 2 solutions infinite? MG9SA.20 Dollar Bills MG10SA: Wheels, Life, and Other Mathematical Amusements MG10SA: Freeman (1983) New York MG10SA.1 Wheels MG10SA.2 Diophantine Analysis and Fermat's Last Theorem (FLT) MG10SA.2.a On A^4 + B^4 + C^4 = D^4, Math. of Comp. 51 (1988) 825-35 MG10SA.2.b (3+Sqrt(93))^3 + (3-Sqrt(93))^3 = 12^3, Math. Mag. 63 (1990) 55 MG10SA.2. near misses: 10^3 + 9^3 = 12^3 + 1, 6^3 + 8^3 = 9^3 - 1 (t=1, t=-1) MG10SA.2. (9t^3)^3 + (9t^4)^3 + (-9t^4-3t)^3 = 1 (t=1 Ramanujan, t=-1 Euler) MG10SA.3 The Knotted Molecule and Other Problems MG10SA.3.1 The Knotted Molecule (trefoil knot in Z^3) MG10SA.3.2 Pied Numbers MG10SA.3.2. represent integer n with the minimum number of Pi. one is allowed MG10SA.3.2. to use \Pi, + , -, *, \, Sqrt, Floor, [monadic -] which gives MG10SA.3.2. 4 = -floor(-Pi). n=1..100 computed. -> Four Fours MG4SA.5 MG10SA.3.3 The Five Congruent Polygons (dissection, fake) MG10SA.3.4 Starting a Chess Game (permutation) MG10SA.3.5 The Twenty Bank Deposits (linear Diophantine equation) MG10SA.3.6 The First Black Ace (probability) MG10SA.3.7 A Dodecahedron-Quintomino Puzzle MG10SA.3.7. quintominoes: pentagon colored with 5 different colors, there are MG10SA.3.7. 12 different not counting rotations and reflections. If 11 tile MG10SA.3.7. the dodecahedron the 12th fits automatically. MG10SA.3.8 Scrambled Quotation MG10SA.3.9 The Blank Column MG10SA.3.10 The Child with the Wart MG10SA.3.10. sum product problem, x*y*z=36, x+y+z=A (x, y, z positive integer) MG10SA.3.10. not enough information sum-product problem MG10SA.3.10.a The Census-Taker Problem, Math. Mag. 63 (1990) 86-88 MG10SA.3.10.a census-taker numbers: products with exactly one pair of sums A MG10SA.4 Alephs and Supertasks MG10SA.4. the power set of a set is bigger than the set, proof there is no MG10SA.4. bijection even in the infinite case (Cantor's proof from 1890). MG10SA.4. continuum hypothesis, bijection segment - line - square - R^omega MG10SA.4. examples for 2^continuum: set of all real one-valued functions MG10SA.4. as there is no highest integer, there are some paradoxes MG10SA.4. H. Weyl suggested supertasks, Thomson lamp paradox, MG10SA.4.a Are 'Infinite Machines' Paradoxial?, Science 159 (Jan 1968) 396-406 MG10SA.4.b Zeno, Aristotle, Weyl and Shuard: Two-and-a-half millenia of MG10SA.4.b worries over number, Math Gazette 64 (Oct 1980) 149-158 MG10SA.4.c On Some Paradoxes of the Infinite. (V. Allis and T. Koetsier) MG10SA.4.c Brit. J. Phil. Sci. 42 (1991) 187-194. -> Art. 228 of rec.puzzles MG10SA.4. false proof, bijection integers-reals, reversing digits, pi - e joke MG10SA.5 Nontransitive Dice and Other Probability Paradoxes MG10SA.5. Efron nontransitive dices, four dices, each beats another 2:1. MG10SA.5.a Non-Transitive Dominance (3 dice), Math. Mag. 49 (1976) 115-120 MG10SA.5. wrong usage of the principle of indifference in probalility MG10SA.5. Pascal's wager, deciding for god, but there are many relegions MG10SA.5. n-card monte: R red and B black cards are given pick a pair MG10SA.5. prob(same color) = (C(R,2)+C(B,2))/C(R+B,2) MG10SA.6 Geometric Fallacies MG10SA.7 The Combinatorics of Paper Folding MG10SA.8 A Set of Quickies MG10SA.8. 36 quick questions, Austin's Dog. MG10SA.8.7 two closed curved curves have an even number of common intersections MG10SA.8.10 which base maximizes the area of a isoscelene triangle? MG10SA.8.16 probability of seeing two sides of a regular pentagon (symmetry) MG10SA.8.19 superqueens, unique solution on the n=10 board, chess problem MG10SA.8.24 number of abigous dates mmddyy versus ddmmyy - 12*11 MG10SA.8.30 you are x years old in year x*x MG10SA.8.33 24 cubes can face-touch, kiss, a central one of the same size MG10SA.8.33 8 nonoverlapping squares can face-touch a square of the same size MG10SA.9 Ticktacktoe Games MG10SA.9. positions, legal, retro analysis. MG10SA.9. reverse game - making 3 in a line loses. MG10SA.9. 3d game is won by the first player MG10SA.9. 4cube game qubic is won by the first player - big game tree MG10SA.9. go-moku has been analyzed by Victor Allis MG10SA.9. pairing scheme avoiding 9 in a row or diagonal MG10SA.10 Plaiting Polyhedrons MG10SA.10. build the platonic polyhedra with colored strips MG10SA.10.a Asymptotic Euclidean type constructions without Euclidean tools, MG10SA.10.a Fib. Quart. 9 (Apr 1971) 199-216 (J P Pedersen) plaiting MG10SA.10 is the icosahedro or the dodecahedron more round? MG10SA.11 The Game of Halma MG10SA.12 Advertising Premiums MG10SA.12. puzzles of Sam Loyd, the T-puzzle, the Pythagorean-square puzzle, MG10SA.12. T-puzzle figures: the alternative T, the 'Teezer' puzzle, MG10SA.12. the Trick Donkeys - three pieces, two jockeys ride two hourses MG10SA.12. the Pony puzzle (inversion, background) MG10SA.12. pencil with a loop: buttonholer, magic pencil, Knopflochstab MG10SA.12.p Wei Zhang, Exploring Math Through Puzzles, 1996, Key Curriculum Pr. MG10SA.12. Get off the Earth paradox (Chinese warriors on a circle) MG10SA.12.a The Disappearing Man and Other Vanishing Paradoxes, Games MG10SA.12.a Nov-Dec 1980, 14-18 (Mel Stover) MG10SA.12.b A Centennial Tribute to Sam Loyd, College Math J.23 (1992) 402-404 MG10SA.12. mental addition: 1000+40+1000+30+1000+20+1000+10 MG10SA.13 Salmon on Austin's Dog MG10SA.13 geometric series added backwards, space-time-diagram, Zeno's paradox MG10SA.13.a Austins's paradox, MM 44 (Jan 1971) Q503, (Sep 1971) Coment 238-239 MG10SA.14 Nim and Hackenbush MG10SA.15 Golomb's Graceful Graphs MG10SA.15. harmonious: label the vertices with 0..e-1 such that the MG10SA.15. f(x)+f(y) (mod e) are distinct for all edges xy. MG10SA.15.a Graceful and harmonious Labelling of Prism Related Graphs, MG10SA.15.a Ars Combin. 34 (1992) 213-222, the cube is not harmonious MG10SA.15.b on edge-graceful regular graphs and trees, MG10SA.15.b Ars Combin. 34 (1992) 129-142 MG10SA.15.c Labeling Grids (graceful), Ars Combin. 34 (1992) 167-182 MG10SA.16 Charles Addams' Skier and other Problems MG10SA.16.1 The Flexible Band (topological puzzle) MG10SA.16.2 The Rotating Disk MG10SA.16.3 Frieze Pattern MG10SA.16.4 The Can of Beer (center of gravity) MG10SA.16.5 The Three Coins MG10SA.16.6 Kobon Triangles MG10SA.16.7 A Nine-Digit Problem (alphametic) MG10SA.16.7. abc*de = fg*hi = P, maximize P, No. 81 of Amusements in Math. MG10SA.16.7. abc*de = fgh*ij, ab*cde = fghi, a*bcde = fghi, ab*cde = fghij MG10SA.16.7. ab*c = de+fg = hi, ab*c = de*f = ghi, a*bc = d*ef = g*hi MG10SA.16.8 Crowning The Checkers MG10SA.16.9 Charles Addams' Skier MG10SA.16.9. plausible explanation of a cartoon, ski tracks arround a tree MG10SA.17 Chess Tasks MG10SA.17. extremal chess problmes MG10SA.17. multicolor nonatacking queens, MG10SA.17. 5-square: 3w+5b queens <=> 3 queens and 5 nonatacked squares (uniq) MG10SA.18 Slither, 3x+1 and other curious Questions MG10SA.18. 3x+1 the Collatz problem -> Col MG10SA.18. chromatic number of the Euclidian plane (distance one graph) <= 7 MG10SA.18.a irrational distance graph (galactic no=3),M Mag. 64 (1991) 141-2 MG10SA.18. triangles dissectable into 5 similar triangles: right, (30, 120, 30) MG10SA.18. 8 points in the plane, the mid orthogonal of each pair match a pair MG10SA.19 Mathematical Tricks With Cards MG10SA.20 The Game of Life, Part I MG10SA.21 The Game of Life, Part II MG10SA.22 The Game of Life, Part III MG10SA.22.a stable, multi-state, time-reversible cellular automata with rich MG10SA.22.a particle content, Questiones Math. 15 (1992) 325-343 MG11SA: Knotted Doughnuts and Other Mathematical Entertainments MG11SA: Freeman (1986) New York MG11SA.1 Coincidence MG11SA.2 The Binary Gray Code MG11SA.2. Chinese Ring Puzzle, Tower of Hanoi, The Brain (Mag-Nif) MG11SA.2. Loony Loop (ternary Gray code), Hammilton Path on the n-Cube (n<6) MG11SA.2.a Efficient gen. of the bin. reflected Gray code and its appl. MG11SA.2.a (Gen. combinations (transpose)), CACM 19 (1976) 517-521 MG11SA.2.b Gen. combinations (transpose), MR 90d:05008 MG11SA.2.c The Gray code function g(n) = n xor (n/2); Zbl 764.11011 MG11SA.2.d Toeplitz sequences, paperfolding, Tower of Hanoi and progression MG11SA.2.d free sequences of integers, MR93j:11017 (unified concept) MG11SA.2.e A Gray code for necklaces of fixed Density MG11SA.2.e SIAM J o Disc Math 9:4 (1996) 654-673 MG11SA.2.f Pascal's Triangle and the Tower of Hanoi, AMM 99 (1992) 538-544 MG11SA.2.g Shortest Paths Between Regular States of the Tower of Hanoi, MG11SA.2.g Information Sciences 63 (1992) 173-181 (A. M. Hinz) MG11SA.2.h Variation on the Tower of Hanoi, Math Mag. 64 (1991) 199-203 MG11SA.2.h start with the odd and even numbered disks on different pegs MG11SA.2.i A optimal algorithm for the twin-tower problem, #MR 92a:05001 MG11SA.2.j A Survey of Combinatorial Gray Codes, SIAM Review 39:4 (1997)605-629 MG11SA.2.j generating combinations, variations, permutations, partitions, ... MG11SA.3 Polycubes MG11SA.3. Soma figures (9 animals, 3 structures with holes, impossible wall) MG11SA.3. Soma Cube, Diabolic cube, Mikusinski cube, Tetracubes, Pentacubes MG11SA.3. Lesk cube, Qube, Dorian cube (subset of 3 units wide Pentacubes) MG11SA.3. Putzl (two Players: 2*Tetracubes in two colors to build a 4-cube) MG11SA.3. tricube (9 Triminoes), Solid Pentominoes, 13-hole pentomino problem MG11SA.3.a Solid Polyomino Constructions, Math. Mag. 49 (1976) 137-139 MG11SA.3.a 3*3*3 Cube == one tricube & six tetracubes, one set is impossible MG11SA.3.a 4*4*4 Cube == one tetracube & twelve pentacubes or MG11SA.3.a six tetracubes & eight pentacubes MG11SA.3.a 2*3*31 Brick from all polycubes of orders one to five MG11SA.4 Bacon's Cipher MG11SA.4.a de Bruijn sequences, Math Mag 55 (1982) 131-143 MG11SA.4.b A new look at the de Bruijn graph, (Fredricksen) MG11SA.4.b Disc. Appl. Math. 37/38 (1993) 193-203 MG11SA.4.c Methods for constructing de Bruijn sequences (Russian) Zbl 764.05006 MG11SA.4.d "Periods" of de Bruijn sequences, (Golomb) MG11SA.4.d Adv. Appl. Math. 13 (1992) 152-159, Zbl 766.11014 MG11SA.4.e On the de Bruijn Torus Problem, JoCT A 64:1 (1993) 50-62 MG11SA.5 Doughnuts: Linked and Knotted MG11SA.5. Whether two seperate knots on a closed rope can cancel each other? MG11SA.5. reversible cloth torus, knotted torus MG11SA.5.a TWisted Tubes (functions for knots) Mathematica J. 3:1 (1993) MG11SA.5.b Wente's twisting constant-mean-curvature torus, AMM 95 (1988)570 MG11SA.5.c K Scherer, Rubber Wrapper, JoRM 12 (1979/80) 60, all box knot != ring MG11SA.5.d S. Moran, The Math. Theory of Knots and Braids (1983) MG11SA.5.e L. Siebenmann, New geometric splittings of classical knots, LMS75 MG11SA.5.f Torangles and Torboards, Quantum 4:4 (1994) 63-65 (torus chess) MG11SA.5.g B. M. Steward, Adventures Among the Toroids (Polyhedra)10QC450S849 MG11SA.5.h Jurisic, Aleksandar, The Mercedes knot problem, AMM 103(1996)756-770 MG11SA.6 The Tour of the Arrows and Other Problems MG11SA.6.1 The Tour of the Arrows (hammiltonian circle) MG11SA.6.2 Five Couples (handshaking) MG11SA.6.3 Square-Triangle Polygons (convex polygons) MG11SA.6.4 Ten Statements (logic) MG11SA.6.5 Pentomino Farms (fence problems) MG11SA.6.6 The Uneven Floor (continous argument) MG11SA.6.7 The Chicken-Wire Trick (paper folding) MG11SA.6.8 Where was the King? (R. Smullyan chess problem) MG11SA.6.9 Polypowers (ladders, Infinite Exponetials) MG11SA.7 Napier's Bones MG11SA.7. Genaills's Rods, The Mathematica J. 3:2 (1993) 60-62 MG11SA.8 Napier's Abacus MG11SA.9 Sim, Chomp and Racetrack MG11SA.9. Chomp is isomorphic to Fred Schuh's divisor game MG11SA.9.a Chomp, Math Intelligenzer 15:3 (1993) 59-60 MG11SA.10 Elevators MG11SA.10. The Gamov-Stern elevator problem MG11SA.11 Crossing Numbers MG11SA.12 Point Sets an the Sphere MG11SA.12. Cover the Sphere with arcs of a great circle MG11SA.12. cromatic number of the plane (distance one graph) MG11SA.12.a K. V. Mardia, Statistics of Directional Data MG11SA.12.b Geoffrey Watson, Statistics on Spheres MG11SA.13 Newcomb's Paradox MG11SA.14 Refections on Newcomb's Paradox MG11SA.15 Reverse the Fish and Other Problems MG11SA.15.1 The Gunport Problem (dominoes on rectangle with many holes) MG11SA.15.1.a Clumsey Packing of Dominoes, Disc. Math. 71 (1988) 33-46 MG11SA.15.2 Figures Never Lie (number joke) MG11SA.15.3 Functional Fixedness (the stand problem, the string problem) MG11SA.15.4 Monochromatic Chess (R. Smullyan chess problem) MG11SA.15.5 The Two Bookcases (reverse two bookcases) MG11SA.15.6 Irrational Probabilities (generated with a coin (P Diaconis)) MG11SA.15.6.a Monte Carlo Simulation of Infinite Series,M Mag. 64 (1991)188-96 MG11SA.15.6.b searching for losers (random subsets, not bit optimal) MG11SA.15.6.b Random Structures & Algorithms 4:1 (1993) 99-110 MG11SA.15.6.c Unbiased coin tossing with a biased coin (random walk, Pascal MG11SA.15.6.c triangle), The Annals of Math. Statistics 41:2 (1970) 341-352 MG11SA.15.7 Who's Behind the Mad Hatter? (logic, word problem) MG11SA.15.8 Reverse the Fish (a toothpick (match) puzzle) MG11SA.15.9 The Intersecting Circles (Elem. Geometry Theorem) MG11SA.15.9. -> 3 or 4 equal circles, Quantum (May 90) M6, (Sep 91) M33 MG11SA.16 Look-See Proofs MG11SA.16. figurative numbers, triangular, square, sum of powers MG11SA.16. sum(i^3) = (sum(i^1))^2, sum(i^5) + sum(i^7) = 2 (sum(i^3))^2 MG11SA.16. 3*3 + 4*4 = 5*5 a four piece dissections MG11SA.16. 2*2 + 3*3 + 6*6 = 7*7 a five piece dissections MG11SA.16. 3*3*3 + 4*4*4 + 5*5*5 = 6*6*6 a nine block dissections MG11SA.17 Worm Paths MG11SA.17. Spirolaterals, LOGO (Logo) MG11SA.17.a Serial Isogons of 90 Degrees, Math. Mag. 64 (1991) 315-324 MG11SA.18 Waring's Problems MG11SA.19 Cram, Bynum and Quadraphage MG11SA.19. cram (Cogito), crosscram, Bynum, linear cram (.007 = James Bond) MG11SA.19. Quadraphage: trap a king, bishop, rook, or knight (Silverman) MG11SA.20 The I Ching MG11SA.20. probabilities provided by the stick and coin procedures MG11SA.21 The Laffer Curve MG11SA.21. the Phillips curve, supply-side defended versus attacked MG12SA: Time Travel and Other Mathematical Bewilderments MG12SA: Freeman (1988) New York MG12SA.1 Time Travel MG12SA.1. science fiction, tachyons paradoxes, G"odel's cosmos, MG12SA.2 Hexes and Stars MG12SA.2. figurative numbers, triangular, square, hex, star, cube MG12SA.2. hex = cube (only 1), triangular = square = hex (only 1) MG12SA.2. square = square-pryramid (only 1, 70*70) MG12SA.2. tetrahedral = square-pryramid (only 1) -> MG12SA.2.i MG12SA.2.a (Moessner's process) -> Adding up to powers, AMM 97 (1990) 139-143 MG12SA.2.b trapezoidal numbers -> Math. Mag. 58 (1985) 108-110 MG12SA.2.c Tetrahedral Numbers as Sums of Square Numbers, M. Mag. 64 (1991) 104 MG12SA.2.d Squares expressible as a sum of n consecutive squares -> MG12SA.2.d AMM 96 (1989), 622-625 (Solution 6552); MG12SA.2.e Khare, Indian J. of Math. 30 (1988) 219-225 MG12SA.2.f Anglin, AMM 97 (1990), 120-124 (Square Pyramid Puzzle) MG12SA.2.g J. Rung, Praxis der Mathematik, 32 (1990) 102-106 MG12SA.2.h J. Rung, Praxis der Mathematik, 33:5 (1991) 230 MG12SA.2.i Beukers, tetrahedral = pryramid, Nieuw Arch. Wisk. 6 (1988) 203-210 MG12SA.2.j A Halmos problem and a related problem, AMM 101 (Dec 1994) 993-996 MG12SA.2.j sums of consecutive integers: n+(n+1)+...+(n+k) = not a power of 2 MG12SA.3 Tangrams, Part 1 MG12SA.3. history of Sam Loyd's pseudohistory, Tangram paradoxes MG12SA.4 Tangrams, Part 2 MG12SA.4. 13 convex Tangrams, 53 pentagons, Tangram Heuristics, farm problem MG12SA.4.a Three-Triangle-Tangram, BIT 24 (1984) 380-382 MG12SA.4.a 3 similar right-angled triangles, (1,a,aa), (a,aa,a^3), (aa,a^3,a^4) MG12SA.4.a with a^4+a^2=1 -> a=0.78615. 16 different convex figures MG12SA.4.b Triangle decompositions (germ), Beitr. Algebra Geom. 32 (1991) 87-93 MG12SA.4.b Zbl 761:51014, decomp. triangles into triangles similar to it MG12SA.5 Nontransitive Paradoxes MG12SA.5. Arrow's voting paradox, tournament paradox (magic square 3*3) MG12SA.5. nontransitive sucker bet, bingo cards, W. Penney's penny game MG12SA.5. triplet and quadruplet probabilities, Conway's algorithm MG12SA.5. -> "Lucifer at Las Vegas" MG12SA.5. -> "Nontransitive Dice and Other Probability Problems" MG10SA MG12SA.5.a How many random digits are required until given sequences are MG12SA.5.a obtained, J. Appl. Prob. 19 (1982) 518-531, (Blom; Thorburn) MG12SA.5.b The occurrence of sequence pattern in ergodic Markov chains MG12SA.5.b Stochastic Proc. Appl. 17 (1984) 369-373, (Benveneto) MG12SA.6 Combinatorial Card Problems MG12SA.6. permutation genaration (H. Steinhaus = Johnson-Trotter order) MG12SA.6. permutation (reflected Gray code with mixed bases) MG12SA.6. motel problem, traveling-burglar problem (Lehmer) MG12SA.6. k-swaps, k-drops (Conway), upper for topswops 2^n (Wilf) MG12SA.6.a randomized adaptive sorting, (skip-sort, skip-list) MG12SA.6.a Random Structure & Algorithms 4:1 (1993) 37-57 MG12SA.6.b Records: The Mathematica J. 2:4 (1992) 10-12 (Boston Competition) MG12SA.6.c Gen. alternating permutations (transpose) Order 6 (1989) 227-233 MG12SA.6.d Gen. alternating permutations (lexicogarphic) BIT 30 (1990) 17-26 MG12SA.6.e Card-shuffling can create chaos, Math. Intellig. 14:1 (1992) 54-56 MG12SA.6.e 123456.., 213456.., 324156.., 135264.., (shuffle 1, 2, 3, ...) MG12SA.6.f The permutational power of a priority queue, BIT 33 (1993) 2-6 MG12SA.6.f a priority queue transforms permutation s -> t. MG12SA.6.f The number of possible pairs (s, t) is (n+1)^(n-1). MG12SA.6.g Euler numbers and Skew-Hook, Math Mag 66 (1993) 181-188, MG12SA.6.g up-down permutations, tangent numbers, tan(x)+sec(x) MG12SA.6.h the smallest length of a string containing all k-element MG12SA.6.h permutations (n letter), MR 93k:05008 MG12SA.6.i metrics on permutations, a survey, J Combinatorics, Information & MG12SA.6.i System Sciences, 23 (1998) 173-185 (M Deza; T Huang) MG12SA.6. Langford problem, Silverman problem, Ransom problem MG12SA.6.A Seltsame Zahlreihen - Ketten, die sich selbst abzaehlen, MG12SA.6.A Math.Kabinet.3.1.7 (selfdescribing sequences, Langford (error)) MG12SA.6.B Game of Cards, Dynamical Systems, and a Characterization of MG12SA.6.B Floor and Ceiling Functions, AMM 97 (1990) MG12SA.6.B f(x)=a+ceiling(x/b), (f^k)(x)=ceiling(w+(x-w)b^(-k)), w=ab/(b-1), MG12SA.6.B f^3 = f(f(f)), w = ab/(b-1), b<>1, x \in Z MG12SA.6.C Langford seq. perfect and hooked, Disc. Math. 44 (1983) 97-104 MG12SA.6.D selfdescribing programs (Selbstreproduzierende Programme) MG12SA.6.D J. Kraus, Forschungsbericht 110/1981 Abt. Informatik Uni Do. MG12SA.6.D examples: Simula, Pascal, Assembler, loop-program MG12SA.6.E selfdescribing programs (Pascal) Wurzel 27:12 (1993) 278-282 MG12SA.7 Melody-Making Machines MG12SA.8 Anamorphic Art MG12SA.9 The Rubber Rope and Other Problems MG12SA.9.1 The Rubber Rope (harmonic series) MG12SA.9.1.a The beetle and the rubber band, Quantum 4:4 (1994) 42-45 MG12SA.9.2 The Sigil of Scoteia MG12SA.9.3 Integer-Choice Game (lower number wins except for predecessors) MG12SA.9.4 Three Circles (conics) MG12SA.9.5 The Multilated Score Sheet (chess problem) MG12SA.9.6 Self-Numbers (Kaprekar) MG12SA.9.6.a Schaaf, Bibliography of Rec. Math. IV.2.8 Kaprekar's Number MG12SA.9.6.b Conway's RATS and Other Reversals, AMM 96 (1989) 425-428 MG12SA.9.6.c Length of the n-number game, Fib. Quart. 28 (1990) 259-265 MG12SA.9.6.c (s1,..,sn) -> (|s2-s1|,|s3-s2|,..,|sn-s1|), MR 91i:11026 MG12SA.9.6.d Palindromisierungsprozesse, Wurzel 29 (1995) 50-53 MG12SA.9.6.e On non-palindromic pattern in palindromic processes, MG12SA.9.6.e "additive palindromisation", Math. Gazette 80:489 (Nov 1996) 577-9 MG12SA.9.6.e Base g=2^n: 10(g-1)_r(g-2)(g-1)0_r r>=2 (non-palindromial) MG12SA.9.7 The Colored Poker Chips (map-coloring with discs) MG12SA.9.8 Rolling Cubes (John Harris) MG12SA.10 Six Sensational Discoveries MG12SA.10. april joke, four-color-map theorem, Ramanujan, chess-program MacHic MG12SA.10.a Churchhouse, RF & Muir, TE, J. Inst. Math. Appl. 5, 318-328, 1969 MG12SA.10.a Continued Fractions, Algebraic Numbers and Modular Invariants MG12SA.10.a exp(pi*sqrt(163)) differs from an integer by less than 10^-12. MG12SA.10.a Why and when does exp(pi*sqrt(x)) approximate an integer? MG12SA.10. special relativity (Gedankenexperiment), Leonardo da Vinci MG12SA.10. parapsychology MG12SA.11 The Csaszar Polyhedron MG12SA.11. seven vertex polyhedron (torus), (v,e,f) = (7,21,14), polyhedron MG12SA.11. without diagonals, Steiner tripple systems, Room (Hadamard) Squares MG12SA.11. regular polyhedra can form a ring (impossible tetrahedron) MG12SA.11.a A new polyhedron of genus 3 with 10 vertices, MG12SA.11.a TH Darmstadt Preprint 914 (1985) MG12SA.11.b The search for Hadamard Matrices, (Golomb,...) AMM 70 (1963) 12-17 MG12SA.12 Dodgem and Other Simple Games MG12SA.12. Star nim, Lewthwaite's counter game (5x5 square, pairing strategy) MG12SA.12. Meander (5x5 square, sliding of unit squares), Dodgem (NxN square) MG12SA.12. N=3 is first player win, Rex = reverse Hex, Ulam's triplet game MG12SA.12.a Dodgem (dodge-ausweichen; autoscooter), Winning Ways II, p685 MG12SA.12.a 3*3 Dodgem analysis (table), there are no drawn positions MG12SA.13 Tiling with Convex Polygons MG12SA.13. classification of monohedral tilings with tri and tetragons MG12SA.13. monohedral tiling with pentagons (last found 1985 by Rolf Stein) MG12SA.13.a Tiling Polygons with Parallelograms, (S. Kannan) MG12SA.13.a Discrete Computational Geometrie 7 (1992) 175-188 MG12SA.13.b Translational Prototiles on the Lattice, M. Mag. 64 (1991) 3-12 MG12SA.13.c The problem of the calissons, AMM 96 (1989) 429-431 (David; Tomei) MG12SA.13.c bijection: hexagonal tilings <-> (n,n,n) plane partitions MG12SA.13.c invariance of the orientations s. a. AMM 97 (1990) 131 (Galvin) MG12SA.13.d Boomerangs cannot tile convex polygons, M. Mag. 60 (1987) 182 MG12SA.13.e Doris Schattschneider, In praise of amateurs, pp 140-166 (1981) MG12SA.13.e in David Klarner, ed., The Mathematical Gardner. MG12SA.13.f a new convex pentagon tiler, M. Mag. 58 (Nov. 1995) 308 (+ cover) MG12SA.13.g Equilateral convex pentagons which tile the plane (Hirschhorn&Hunt) MG12SA.13.g JoCT A 39 (1985) 1-18, MR 86g:52022 MG12SA.13.h Unsolved Prob. in Geom., C14. Which polygons tile the plane? MG12SA.14 Tiling with Polyominoes, Polyiamonds, and Polyhexes MG12SA.14. Karl Scherer: Karl@kiwi.gen.nz MG12SA.14.a Boxes with the U-Pentacube (90), JoRM 24:2 (1992) 146 Prob 1963 MG12SA.14.a prime boxes: 2.3.5, 3.3.10, 3.7.15; 15^3 possible, (complete sio) MG12SA.14.a JoRM 25:4 (1994) 226-229, 3.10.10, 4.4.5 MG12SA.14.b Boxes with the Pentacube (61), JoRM 24:1 (1992) 62-64 Ex 1615 MG12SA.14.b prime boxes: 2.3.5, 2.2.5, 5.5.9; MG12SA.14.c Tilings of Lattice Points in Euklid. n-Space, Disc. Math 29 (1980) MG12SA.14.c 169-174, each |S|=3 tiles Z^n, (Z*Z tileble -> N*Z tileble) MG12SA.14.d Tiling with Sets of Polyominoes, JoCT 9 (1970) 60-71 (Golomb) MG12SA.14.d U5, F5 tiles 10*10. MG12SA.14.e smallest rectangles with FI, ZI, UI, WI, JoRM 25 (1993) 149-150 MG12SA.14.f Packing rectangles with congruent polyominoes, JoCT A 77:2 (1997) MG12SA.14.f 181-192, W. R. Marshall MG12SA.14.g Tiling with Polyominoes, Math Intell 18:2 (1996) 38-47, S W Golomb MG12SA.14.h Packing boxes with N-tetracubes, Crux Math 23:6 (1997) 336-342 MG12SA.14.sio oo_oo tiles Z*Z but not N*Z. MG12SA.15 Curious Maps MG12SA.15. stereographic (cylindrical) projection, Mercator map, Mecca map, MG12SA.15. Gilbert's prob. problem: n points are randomly distributed around MG12SA.15. the globe, the prob. that all lie in one hemisphere is (n*n-n+2)/2^n MG12SA.15.a World Plot (creates maps), Mathematica J. 3:3 (1993) 10-13 MG12SA.15.b The Probability of Covering a Sphere With N Circular Caps, MG12SA.15.b E. N. Gilbert, Biometrika 52, 1965, p323. p = (N*N-N+2)/2^N MG12SA.16 The Sixth Symbol and Other Problems MG12SA.16.1 What Symbol Comes Next? MG12SA.16.2 Which Symbol is Different? MG12SA.16.3 Cutting a Cake (two parameter equality) MG12SA.16.3.a A combinatorial algorithm to establish a fair border MG12SA.16.3.a Europ. J. Comb. 11 (1990) 301-304, #MR 92a:05237 MG12SA.16.4 Two Cryptarithms MG12SA.16.4.a Some Ideas about the Solution of Cryptarithms, JoRM 7,309-14 MG12SA.16.5 Lewis Carrol's "Sonnet" MG12SA.16.6 Third-Man Theme (chess problem) MG12SA.17 Magic Squares and Cubes MG12SA.17. Number of 5x5 magic squares, impossible of 4^3 magic cube MG12SA.17.a Balanced Magic Rectangles, Europ J of Comb. 14:4 (1993) 285-299 MG12SA.17.b Magic squares of order 4, Kathleen Ollerenshaw and Herman Bondi, MG12SA.17.b Philosophical Trans. of the Roy. Soc. London, A 306 (1982) 443-532 MG12SA.17.c W. S. Andrews, Magic Squares and Cubes MG12SA.17.d Magic Squares and Linear Algebra, Christopher J. Henrich, MG12SA.17.d AMM 98:6 (June-July 1991) 481-488. chenrich@monmouth.com MG12SA.17.e Most-perfect pandiagonal magic squares: their construction and MG12SA.17.e and enumeration, ISBN 0 90501 06X, Kathleen Ollerenshaw, David Bree MG12SA.18 Block Packing MG12SA.18. pangram, Brick-packing Puzzles, harmonic brick: a x ab x abc, MG12SA.18.a S. Wagon, fourteen proofs, AMM 94 (1987) 601-617, #Zbl 691.05011 MG12SA.18. Color impossible proof for 27 1x2x4 bricks to form a 6x6x6 cube, MG12SA.18. impossible proofs for 15 1x2x4 bricks to get into a 5x5x5 cube MG12SA.18. 1st: covering the surface of the cube, a mod 4 restriction. MG12SA.18. Color impossible proof for 2x2, 3x3 squares to get a 25x25 square, MG12SA.18. the number of 1x2x4 bricks which can be packed in a NxNxN cube. MG12SA.18. Conway box: 6*1x2x2 and 3*1x1x1 gives a 3-cube. MG12SA.18. Conway box: 6*1x2x4, 6*2x2x3, and 5*1x1x1 gives a 5-cube. MG12SA.18. Conway box: 3*1x1x(2N-1) and 1x2x2 boxes pack a (2N+1)-cube. MG12SA.18. 3*1x1x3 and 12*1x2x4 boxes pack a 3x5x7 box (Klarner) MG12SA.18.e 42*1x2x4 + 7*1x1x1 = 7x7x7 ?, AMM 82 (Mar 1975) E2524 question MG12SA.18.e 42*1x2x4 + 7*1x1x1 != 7x7x7, AMM 83 (Nov 1976) 741-742 E2524 answer MG12SA.18.b How many 1*2*4 Bricks Can You Get into an Odd Box? MG12SA.18.b Disc. Math. 133 (1994) 55-78 (rather complex) MG12SA.18.c Strips on a Board (packing n*m by 1*k) B. Kotlyar MG12SA.18.c Quantum 5:2 (Nov Dec 1994) 63-65 & 61 MG12SA.18.d J. Fricke, Quadratzerlegung eines Rechtecks, MG12SA.18.d Math. Semesterberichte 42 (1995) 53-62 MG12SA.18.d rectange x*y is tiled by n times a*a and m times b*b and p holes MG12SA.19 Induction and Probability MG12SA.19. Goodman's 'grue' paradox, Simpson's reversal paradox, MG12SA.19. Blyth's paradox (A beats B, and A beats C, but A is mot the best) MG12SA.19.a A Mathematical Rating System, UMAP 13:4 (1992) 313-334 MG12SA.20 Catalan Numbers MG12SA.20. seven interpretations for Catalan numbers, bijective proofs. MG12SA.20.a JoCT A 27 (1979) 392-3 (Z^n paths below a hyperplan, trees) MG12SA.20.a let a1, a2 .. am, m=1+Sum ni ki, be an arbitrary sequence of MG12SA.20.a operands and ni operators of arity ki. there is a unique cyclic MG12SA.20.a permutation of the sequence which is well. MG12SA.20.b Bootstrap percolation, the Schr"oder numbers,and the n-King problem MG12SA.20.b SIAM J. Disc. Math. 4 (1991) 275-280; Zbl 736.05008 MG12SA.20.c How to Guess a Generating Function, SIAM J Disc Math 5 (1992)497-9 MG12SA.20.c LDU decomposition of a Hankel matix, Schr"oder numbers MG12SA.20.d Sur les Polynomes de Catalan Simples et Doubles MG12SA.20.d Europ. J. Comb. 12 (1991) 389-396 (Kreveras) MG12SA.20.e Six etudes in generating functions (Catalan, Motzkin, Narayana) MG12SA.20.e Intern. J. Comput. Math. 29 (1989) 201-215 (Zeilberger) MG12SA.20.f The Motzkin family, PU. M. A. Pure Math Appl. A 2:3/4 (1991)249-79 MG12SA.20.f survey: Motzkin, central trinomial, Catalan, Zbl. 756.05003 MG12SA.20.g Catalan Numbers, Their Generalization, and Their Uses, MG12SA.20.g Math. Intelligenzer, 13 (1991) 64-75 (Hilton, Pederson) MG12SA.20.h Lattice path enumeration by formal schema, Adv. in Appl. Math. MG12SA.20.h 13 (1992) 216-251 MR 93i:05007 MG12SA.20.i Lattice path reflections, & dimension changing bijektions MG12SA.20.i Ars Comb. 34 (1992) 3-15, MR 93i:05008 MG12SA.20.i NSEW-path <--> linear NS-path MG12SA.20.j Three combinatorial sequences derivable from lattice path counting MG12SA.20.j Ann. Disc. Math. 52 (1992) 81-92 (45 ref, new), MR 93j:05005 MG12SA.20.k Refinements of the Narayana numbers (6 param.) MR 93j:05008 MG12SA.20.l Super ballot numbers, J. Symb Comp 14 (1992) 170-194 MR 93k:05009 MG12SA.20.m http://www-math.mit.edu/~rstan/ec/catalan.ps.gz R. Stanley MG12SA.20.m 65 realizations of the Catalan numbers, part of a 47p excerpt. MG12SA.20.n JoCT A23(1977) 291, gives 14 manifestations of Motzkin numbers MG12SA.21 Fun with a Pocket Calculator MG12SA.21. nim game (subtraction game) with moves are adjacent on the 9-block MG12SA.22 Tree-Plant Problems MG13SA: Penrose Tiles to Trapdoor Chiphers ... and the return of Dr. Matrix MG13SA: Freeman (1989) New York MG13SA.1 Penrose Tiling MG13SA.2 Penrose Tiling II MG13SA.2.a Will it tile? try the Conway criterion, M. Mag. 53 (1980) 224-232 MG13SA.3 Mandelbrot's Fractals MG13SA.3.a Generalized Mandelbrot Rule for Fractal Sections MG13SA.3.a Physical Rev. A 45 (1992) 654-656 MG13SA.3.b Can we see the Mandelbrot Set? The College Math J 26:2 (1995) 90-99 MG13SA.3.c Area of the Mandelbrot Set, Numerische Math 61 (1992) 59-72 (<-b) MG13SA.4 Conway's Surreal Numbers MG13SA.4. picture of John 'Horned' Conway, the Alexander horned sphere, there MG13SA.4. is a four-horned mechanical puzzle sold as Loony Loop with a loop to MG13SA.4. removed. surreal numbers mentioned in MG8SA.1 Nothing. MG13SA.4. Crosscram -> MG11SA.19 which Conway calls Domineering. MG13SA.4. the games Crosscram, Col, Snort, Silver Dollar Game without the MG13SA.4. Dollar, Silver Dollar Game with the Dollar, Rims (Conway's name: MG13SA.4. Ralyes) (rule: take (with optional split)), Prim, Dim, Cutcake MG13SA.4.a Surreal Numbers, (D. E. Knuth), 1974 MG13SA.5 Back from the Klondike and Other Problems MG13SA.5. Loyd's Klondike, Chinese checkers, no-three-in-line MG13SA.5.a no-three-in-line with restricted slopes, ax+by=c, |a|<=2, |b|<=2, MG13SA.5.a Math. Semesterberichte 39 (1992) 202-203, problem 48, part solution MG13SA.6 The Oulipo MG13SA.6. wordplay, palindrom MG13SA.7 The Oulipo II MG13SA.8 Wythoff's Nim MG13SA.8. queen (I. P. Rufus), king, rook, bishop, and combined [reverse] Nim MG13SA.8.a Berge, Graphs and Hypergraphs, p319-20 (Withoff) Grundy [0..10]^2 MG13SA.8.b From Wythoffs NIM to Chebyshev Inequality, AMM 98 (1991) 889-900 MG13SA.8.c Wythoff Pairs as Semigroup Invariants, Adv in Math 85 (1991) 69-82 MG13SA.8.d (AS-Fraenkel) Nimhoff Games, JoCT A 58 (1991) 1-25 MG13SA.8.e Disjoint Covering Systems of Rational Beatty Sequences, MG13SA.8.e Discrete Math. 92 (1991) 361-369 MG13SA.8.f Recent Problems and Results About Kernels in Directed-Graphs, MG13SA.8.f Discrete Math. 86 (1990) 27-31 (C-Berge, P-Duchet) MG13SA.8.g (king) Wythoff ((0,1),(1,0),(1,1),(1,-1),(-1,1)), PM 35:1 (1993) MG13SA.8.g 42-43 A641, G(n,m)=0 <=> n and m even, (otherwise G=infinity) MG13SA.9 Pool-Ball Triangles and Other Problems MG13SA.9.1 Pool-Ball Tiangles MG13SA.9.1. absolute-difference triangles of consecutive numbers must have 1 MG13SA.9.1. as its lowest number (C. Trigg). Only order 1..5 are possible. MG13SA.9.1. No. of solutions: 1..5; 1, 2, 4, 4, 1. MG13SA.9.1. A triangular array (even) has always an even-odd sum pattern with MG13SA.9.1. an equal number of even and odd ones (H. Harborth). MG13SA.9.1. modulo-m-sum triangle of 0..m-1 (order 4 ok, order 5, 6 no) MG13SA.9.2 Toroidal Cannibalism (topology) MG13SA.9.2. two linked toruses; one with a hole. This can swallow the other. MG13SA.9.3 Exploring Tetrads MG13SA.9.3. four congruent tiles, each pair of which shares a finite portion MG13SA.9.3. of a common boundary. (polyhex, polyamond, polyomino solutions) MG13SA.9.4 Knights and Knaves (logic) MG13SA.9.4. 4 logicals, R. Smullyan, truth teller and liars. MG13SA.9.5 Lost-King's Tours MG13SA.9.5. hamiltonian path of a king, which change of direction after each MG13SA.9.5. move, with minimal number of crossings. MG13SA.9.5.a A King's Tour of the Chessboard, Math. Mag. 58 (1985) 285-286 MG13SA.9.5.a the king (after the first move) can only move to a square which MG13SA.9.5.a touches an even number of squares which have already been MG13SA.9.5.a visited (impossible for rectangles, except 1*1 and 1*2) MG13SA.9.6 Steiner Ellipses MG13SA.9.6. find for a triangle the minimal area of a circumscribed ellipse. MG13SA.9.7 Different Distances MG13SA.9.7. place n counters on an n*n grid so that the pairwise distences are MG13SA.9.7. different. This is possible only for n<=7. MG13SA.9.7. No. of solutions: 1..7; 1, 2, 5, 16, 28, 2, 1. MG13SA.9.7.a Erd"os, Guy; distinct distances, Elem. Math. 25 (1970) 121-133 MG13SA.9.7.b distinct slopes or lengths, Combinatorica 12 (1992) 39-44 MG13SA.9.7.c distinct slopes (lower bound), Combinatorica 13 (1993) 127-128 MG13SA.9.8 A Limerick Paradox MG13SA.10 Mathematical Induction and Colored Hats MG13SA.11 Negative Numbers MG13SA.12 Cutting Shapes into N Congruent Parts MG13SA.12.a The L-shaped Dissection Problem, JoRM 24:1 (1992) 64-69 Ex 1771 MG13SA.12.a for which n is a dissection into n congruent pieces possible? MG13SA.12.b Tiling with n congruent pieces (L), JoRM 22 (1990) 185-191 MG13SA.12.c Dissection of a triangle in 3 similar pieces (Scherer), Quantum 4:5 (1994) 26-27 MG13SA.12.d Alexander Soifer, How does one cut a triangle? (1990) MG13SA.12.d Center for the Excellence in Math. Education. MG13SA.12.d Find all n, such that every triangle can be cut into n parts similar (congruent) MG13SA.12.d to each other. (p13). Solution: I) N\{2,3,5} II) Square numbers MG13SA.13 Trapdoor Ciphers MG13SA.13.a Sharing a Secret, UMAP 13:4 (1992) 335-350 MG13SA.14 Trapdoor Ciphers II MG13SA.15 Hyperbolas MG13SA.16 The New Eleusis MG13SA.17 Ramsey Theory MG13SA.17.b An Extremal Problem for Triangle-Free Graphs, AMM (1989) E3284 MG13SA.17.c Boltyanskij, Geometric etudes in combinatorial math., 1991 MG13SA.18 From Burrs to Berrocal MG13SA.18. there are 119979 possible pieces for the burr puzzle MG13SA.18.a the Six-Piece Burr (W. Cutler), JoRM 10 (1977-78) 241-250 MG13SA.18.b Computer Recreations (Dewdney), SA 10/85 16-22 MG13SA.18.c Computer Recreations (Dewdney), SA 01/86 (2-dimensional) MG13SA.19 Sicherman Dice, the Kruskal Count and Other Curiosities MG13SA.19.a Renumbering of the Faces of Dice, Math. Mag. 52 (1979) 312-315 MG13SA.19.b Cyclotomic Polynomials and Nonstandard Dice, Disc Math 27 MG13SA.19.b (1979) 245-259; -> Dice with Fair Sums, AMM 95 (1988) 316-328 MG13SA.19.c J. C. Lagarias and R. J. Vanderbei, 1988, The Kruskal Count, MG13SA.19.c AT&T Bell Laboratories, Murray Hill, New Jersey 07974 MG13SA.19. Visible number of points of a dice: 1 to 15 without 13. Renumber MG13SA.19.s a dice so that the visible number of points is 1 to 26 (= faces+ MG13SA.19.s egdes+vertices). Set x1,x2,x3,x4,x5,x6 := 1,3,9,18,6,2. (unique) MG13SA.19.s GF-solution:(1+z^x1+z^x6)(1+z^x2+z^x5)(1+z^x3+z^x4)=(1-z^27)/(1-z) MG13SA.19. Dice rolling - top even: roll right; top odd: roll up. You end in MG13SA.19. the 6-cycle: 1,4,5,6,3,2 after at most three rolls. MG13SA.19. Cube of 8 dice with the minimal=40 (maximal=306) sum of the MG13SA.19. product of the 12 pair of touching faces. MG13SA.19. Cube of 27 dice with the minimal=294 (maximal=1028) sum of the MG13SA.19. product of the 54 pair of touching faces. MG13SA.19. equal-sized squares in the plane with every vertex is corner of at MG13SA.19. least two squares, Hypercube projection (Kim's knight problem) MG13SA.19. 16 knights on a chessboard so that each knight attacks just 4 other MG13SA.19. Kruskal's principle or Krusckal count (confluence, probability) MG13SA.19. counterintuitive: the prob is ca 5/6 that any two arbitrarily MG13SA.19. started chains of cards will intersect (tree) MG13SA.19.f Card Corner, The Linking Ring (Dec 1976) 82-87 MG13SA.19.f same column (Dec 1957) & (Mar 1978) related trick: Kraus principle MG13SA.19. a pairing strategy for the amazon game of D. L. Silverman, this is MG13SA.19. isola-game, old or attacked positions are forbidden. MG13SA.19.a Checker Jumping in Three Dimensions, Math. Mag. 52 (1979) 227-231 MG13SA.19.b Scouts in Space, JoRM 21:3 (1989) 195-202 MG13SA.19.c Scouts in Hyperspace, JoRM 24:2 (1992) 116-120 MG13SA.20 Raymond Smullyan's Logic Puzzles MG13SA.21 The Return of Dr. Matrix MG13SA.21. 3*3 magic square named "lo shu", {{2,7,6},{9,5,1},{4,3,8}} = M3 MG13SA.21. 15 can be partitioned into a triplet of distict integers 1..9 in MG13SA.21. exactly 8 ways. -> "lo shu" is unique -> MG12SA.17 MG13SA.21.a A replicaton property for magic sq., Math Mag 65 (1992) 175-181 MG13SA.21. Vector space: M = {{a-c,a-b+c,a+b},{a+b+c,a,a-b-c},{a-b,a+b-c,a+c}} MG13SA.21. magic constant = 3a, GF = x^a (1 + x^b + x^{-b}) (1 + x^c + x^{-c}) MG13SA.21. as M^t M is symmetric according to both diagonals, we have MG13SA.21. (100,10,1) M3^t M3 (100,10,1) = 276^2+951^2+438^2=672^2+159^2+834^2 MG13SA.21. alphametic squares->Sallows, Abacus 4 (1986) 28-45, (1987) 20-29,43 MG13SA.21. M with one egde 8 and constant 15, -> b-c = 3 -> (a,b,c)=(5,4,1). MG13SA.21. prime 3*3 magic square, smalest constant is 177 MG13SA.21. 3*3 magic square with consecutive primes (a,b,c)=(1480028171,12,30) MG13SA.21.b Characteristic Polynomials of Magic Squares, M.Mag. 57(1984)220-1 MG13SA.21.b the inverse of a 3*3 magic square is also magic MG13SA.21. Smith numbers: composite integers with sum of digits is equal MG13SA.21. the sum of digits of the prime factorisation MG13SA.21. Smith brothers (n, n+1), palindromic Smiths, 3*3 Smith magic square, MG13SA.21. Conjecture: the density of Smith numbers is 3%. MG13SA.21. From every repunit whose prime factors are known one can construct MG13SA.21. a Smith number -> the set of Smith numbers is infinite. MG13SA.21. Feynman's fraction 1/243 in decimal notation (Los Alamos joke) MG14SA: Fractal Music, Hypercards and More Math. Recreations from SA Magazin MG14SA: Freeman (1991) New York MG14SA.1 White, Brown, and Fractal Music MG14SA.2 The Tinkly Temple Bells MG14SA.2. Bell and Stirling 2nd numbers, set partition, rhyme shemes MG14SA.2.a Asymt. Estimates of Stirling Numbers, Studies Appl Math 89 (1993)233 MG14SA.2.b Apropos, Two Notes on Notations (Stirling 1st, 2nd), D. E. Knuth MG14SA.2.b AMM 101 (1994) 771-778 MG14SA.3 Mathematical Zoo MG14SA.4 Charles Sanders Peirce MG14SA.5 Twisted Prismatic Rings MG14SA.6 The Thirty Color Cubes (Mac Mahon) MG14SA.6.a Der Keplersche Koerper und andere Bauplaene (Kowalewski) MG14SA.6.a JFM 64 (1938) 643-644 MG14SA.7 Egyptian Fractions MG14SA.8 Minimal Sculpture MG14SA.9 Minimal Sculpture II MG14SA.10 Tangent Circles MG14SA.11 The Rotating Table and Other Problems MG14SA.11.1 The Rotating Table MG14SA.11.1.a Rotating-table games and derivatives of words MG14SA.11.1.a TCS 108 (1993) 311-329 (Yehuda, Etzion, and Moran) MG14SA.11.1.a coins with S states, soluble iff S and N are power of the same prime MG14SA.11.2 Turnablock MG14SA.11.2. Game of ONaG, nim-multiplication MG14SA.11.3 Persistences of Numbers MG14SA.11.4 Nevermore MG14SA.11.5 Rectangling the Rectangle MG14SA.11.6 Three Geometric Puzzles MG14SA.12 Does Time Ever Stop? Can the Past Be Altered? MG14SA.13 Generalized Ticktacktoe MG14SA.13.a tic-tac-toe using invariant subsets JoRM 25 (1993) 128-135 MG14SA.14 Psychic Wonders and Probability MG14SA.15 Mathematical Chess Problems MG14SA.15. n-queens problem (reflected, modular), partition of n*n MG14SA.15.a A gener. of the n-queen problem, MR 92k:05042 MG14SA.15.b On the Queen Domination Problem, Disc. Math. 86 (1990) 21-26,cover MG14SA.15.c Queen Attacks, JoRM 12 (1979) 53 (k=1) MG14SA.15.d A Problem of Chess Queens, JoRM 24 (1992) 264-271 (k<=4) MG14SA.15.d greatest number of queens, s. t. each attacks precisely k others MG14SA.15.e On the 8-Queens-problem, Proc. Edin. Math. Soc. 17 () 43-68 (n<12) MG14SA.16 Douglas Hofstadter's G"odel, Escher, Bach MG14SA.17 Imaginary Numbers MG14SA.18 Pi and Poetry: Some Accidental Patterns MG14SA.18. e^pi > pi^e as x^(1/x) has a maximum value for x=e MG14SA.18. positive solutions x^y = y^x are x=(1+1/t)^(t+1) and y=(1+1/t)^t MG14SA.19 More on Poetry MG14SA.20 Packing Squares MG14SA.20.a On Tiling an m*m Square with m Squares, MG14SA.20.a Crux Math. 19:7 (1993) 189-191, cases 11-33 solved MG14SA.20.a 1^3 + 2^3 .. + n^3 = (n(n+1)/2)^2 MG14SA.20.b (update of a) Cuttler determined the minimum 8. MG14SA.21 Chaitin's Omega MG14SA.21.a Information-Theoretic Incompletness (Chaitin) MG14SA.21.a Appl. Math. and Comp. 52 (1992) 83-101 MG15SA: The Last Recreations, Hydras, Eggs, and Other Math. Mystifications MG15SA: Springer (1997) New York MG15SA.1 The Wonders of a Planivers MG15SA.2 Bulgarian Solitaire and Other Seemingly Endless Tasks MG15SA.3 Fun with Eggs, Part I MG15SA.4 Fun with Eggs, Part II MG15SA.5 The Topology of Knots MG15SA.6 M-Pire Maps MG15SA.7 Directed Graphs and Cannibals MG15SA.7. river crossing problems: 3 missionaries and 3 cannibals MG15SA.7.a The jealous husbands and the missionaries and cannibals, MG15SA.7.a I. Pressman, D. Singmaster, Math Gazette 73 (Jun 1989) 73-81 MG15SA.8 Dinner Guests, Schoolgirls and Handcuffed Prisoners MG15SA.8. Kirkman's schoolgirl problem, Steiner triple systems, designs MG15SA.9 The Monster and Other Sporadic Groups MG15SA.9. the classification of finite simple groups. MG15SA.10 Taxicab Geometry MG15SA.10. geometry with the 1-norm. MG15SA.11 The Power of the Pigeonhole MG15SA.11.a Pigeonhole principle (problem book, russian) (Letchikov) MG15SA.11.a Zbl 749.00004 MG15SA.11.b three into two won't go, Math Gazette 61:415 (Mar. 77) 25-31 MG15SA.11.c the pigeonhole principle, TYCMJ mock issue (Jan 1979) 4-12 MG15SA.11.d Das Schubfachprinzip (Pigeonhole, Ramsey, Kronecker, Schur) MG15SA.11.d MU 25:1 (1979) 23-37; 61 Aufgaben, 14 Beispiele MG15SA.11.f pigeons in every Pigeonhole, Quantum, (Jan 1990) 25-26, 32 MG15SA.11.e Das Schubfachprinzip I, alpha 29:5 (1995) 30-33 (10 problems) MG15SA.11.e Das Schubfachprinzip II, alpha 29:9 (1995) 30-34 (30 contest prob.) MG15SA.11.g Appl of the pigeonhole principle, Math Gazette 79 (1995) 286-292 MG15SA.12 Strong Laws of Small Primes MG15SA.12. -> SEQ: sequences, strong law of small numbers MG15SA.13 Checker Recreations Part I MG15SA.14 Checker Recreations Part II MG15SA.15 Modulo Arithmetic and Hummer's Wicked Witch MG15SA.16 Lavinia Seeks a Room and Other Problems MG15SA.16.1 Lavinia Seeks a Room MG15SA.16.2 Mirror-Symmetric Solids MG15SA.16.3 The Damaged Patchwork Quilt MG15SA.16.3. 2 part dissection: 9*12 - 1*8 = 10*10, (checkered 3 parts) MG15SA.16.4 Acute and Isosceles Triangles MG15SA.16.5 Measuring with Yen MG15SA.16.6 A New Map-Coloring Game MG15SA.16.7 Whim MG15SA.16.7. The Nim game plus the whim move (selects the type: normal-misere) MG15SA.17 The Symmetry Creations of Scott Kim MG15SA.17.a Inversions, Scott Kim, Byte 1981, Key Curriculum Press 1996 MG15SA.18 Parabolas MG15SA.19 Non-Euclidean Geometry MG15SA.19.a The Trigonometriy of Escher's Woodcut "Circle Limit III", MG15SA.19.a Math Intell. 18:4 (1996) 42-46 MG15SA.20 Voting Mathematics MG15SA.20.a approval voting: a best buy method for multi-candidate elections MG15SA.20.a S. Merrill, MM 52 (Mar 1979) 98-102 MG15SA.20.b Democraty and math. (Problems and paradoxes in free elections) MG15SA.20.b Quantum 3:3 (1993) 4-9, 58 (Arrow's Theorem) MG15SA.21 A Toroidal Paradox and Other Problems MG15SA.21.1 A Poker Problem MG15SA.21.2 The Indian Chess Mystery MG15SA.21.2. chess logic, retro analysis, which color corresponds to White? MG15SA.21.2.a The Chess Mysteries of Sherlock Holmes, R. M. Smullyan, 1979 MG15SA.21.3 Redistribution in Oilaria MG15SA.21.4 Fifty Miles an Hour MG15SA.21.4. averaging a continous function MG15SA.21.5 A Counter-Jump "Aha!" MG15SA.21.5. parity in peg jumping MG15SA.21.6 A Toroidal Paradox MG15SA.22 Minimal Steiner Trees MG15SA.23 Trivalent Graphs, Snarks, and Boojums MMR: Madachy's Mathematical Recreations MMR: Joseph S. Madachy, Dover Publ., 2nd 1979 MMR.1 Geometric Dissections MMR.2 Chessboard Placement Problems MMR.3 Fun With Paper MMR.3.1 geometric constructions with paper only MMR.3.2 flexagons MMR.3.3 other flexagon diversions MMR.3.4 solid flexagons MMR.4 Magic and Antimagic Squares MMR.4.a Unsolved problems on magic squares, Disc. Math. 127 (1994) 3-13, G. Abe MMR.4.b J L Fults, Magic Squares, 1974, Open Court Publ. Co, La Salle Illinois MMR.4.1 magic squares MMR.4.2 miscellaneous magic configurations MMR.4.3 antimagic squares MMR.4.4 talisman squares MMR.5 Puzzles and Problems MMR.5.1 moonshine sharing MMR.5.2 number toughies MMR.5.2. 10^9 = a*b with a, b zerofree (for which p are 2^p, 5^p zerofree?) MMR.5.2. 2^p zerofree: 1..9, 13..16, 18, 19, 24, 25, 27, 28, 31..37, 39, MMR.5.2. 49, 51, 67, 72, 76, 77, 81, and 86. (Check upto p<46000000) MMR.5.2. 2^33250486 -- the last 176 digits (out of 10009394) are non-zero MMR.5.2. 2^18894561 -- the last 174 digits (out of 5687830) are non-zero MMR.5.2. 2^4400728 -- the last 164 digits (out of 1324752) are non-zero MMR.5.3 nine-coin move MMR.5.3. NPDNPDNPD -> DDDPPPNNN MMR.5.4 scotch and water MMR.5.5 geometric construction MMR.5.5. smalest (largest) equilateral triangle incribable in a square MMR.5.5.a An old Max-Min Problem Revisted, AMM 96 (1989) 421 MMR.5.5.a inscribing a max area rectangle into a triangler. MMR.5.6 the mad hatted! (logic) -> MG13SA.10 MMR.5.7 high stakes (binary) MMR.5.8 death in the decanter (logic) MMR.5.9 problems in probabilities (different random digits) MMR.5.9.a The first digit problem, AMM 83 (1976) 521-538 (Benford's Law) MMR.5.9.b Sequential Partitioning, AMM 99 (1992) 846-855 MMR.5.9.b leapfrog sequence, logarithmic sequence, optimal cutting procedure MMR.5.9.b Steinhaus' Three Gap Theorem (points on the circle), Benford's Law MMR.5.9.c Optimal spacing of points on a circle, Fib. Quart. 27 (1989) 18-24 MMR.5.9.d The distribution of leading digits and uniform distribution mod 1, MMR.5.9.d Diaconis, Ann. Probability 5 (1977) 72--81, MR 54 #10178 MMR.5.10 the golden spheres MMR.5.11 a paper-covering problem MMR.5.12 the commoner's dilemma MMR.5.12. how to get white from an urn with black marbles MMR.5.13 did the butler do it? (congruence) MMR.5.14 an airport problem MMR.5.15 the seven fortunes (Diophantine) MMR.5.16 occupational mix-up (logic) MMR.5.17 a traveling man (arithmetic) MMR.5.18 a problem in confusion (arithmetic) MMR.5.19 cigarette selling (Diophantine) MMR.5.19. Frobenius Problem, (Sylvester) MMR.5.19. largest number not representable n*a+m*b with (a,b)=1, n>=0, m>=0. MMR.5.19.a ax+by+cz = N, MR 93e:11033 MMR.5.19.b Math. Scand. 58 (1986) 161-175, ZBl 607.10038 MMR.5.19.c The coin exchange problem for arithmetic progressions MMR.5.19.c AMM 101 (1994) 779-781 MMR.5.19.d http://www.cs.cmu.edu/~kannan/Papers/pubs.html gives an algorithm MMR.5.20 eight stamps (logic) -> MMR.5.6, MG13SA.10 MMR.5.21 squared eggs (Diophantine, Pell) MMR.5.22 the oracle of the three gods (logic) MMR.5.23 cube formation MMR.5.23. What is the shortest strip of paper 1'' wide and black on one side MMR.5.23. that can be folded to form a 1'' cube that is black on all sides? MMR.5.24 balancing cubes (Diophantine) MMR.5.25 major Perkins (logic) MMR.5.26 the chicken yolks (arithmetic) MMR.6 Number Recreations MMR.6.1 numbers and their divisors MMR.6.1.a $\sigma_7(n)=\sigma_3(n)+120\sum_{m=1}^{n-1}\sigma_3(m)\sigma_3(n-m)$ MMR.6.1.a where for each $d$ we denote by $\sigma_d(k)$ the sum of the $d$-th MMR.6.1.a powers of the divisors of $k$. MMR.6.1.a N.P. Skoruppa, A quick combinatorial proof of Eisenstein series MMR.6.1.a idenities, J. Number Theory, vol. 43, 68-73, 1993. MMR.6.2 perfect and multiperfect numbers MMR.6.3 prime numbers MMR.6.4 amicable numbers MMR.6.5 0123456789, or digital diversions MMR.6.5. 123 - 45 - 67 + 89 = 100 // uses +, -, conc MMR.6.5. 1+2+3+4+5+6+7+8*9 = 100 // uses +, *, conc MMR.6.5. (1 + 2 - 3 - 4)(5 - 6 - 7 - 8 - 9) = 100 // uses +, -, * MMR.6.5. (1!)(2!)(3! + 4!) - 5! + 6! - ((7!)(8!)/9!) = 100 MMR.6.5. 98 - 76 + 54 + 3 + 21 = 100 MMR.6.5.sio impossible with +, conc only as the sum is equal zero mod 9. MMR.6.5.sio but possible with repeating decimal fractions: 98.'765+1.'234 = 100 MMR.6.5. squares: 139854276=11826^2, 923187456=30384^2 MMR.6.6 narcissistic numbers MMR.6.6. 153=1^3+5^3+3^3, numbers which are the sum of the cubes of there digits MMR.6.6.a Armstrong numbers: 153=1^3+5^3+3^3, Fib. Quart. 30 (1992) 221-224 MMR.6.6.a MR 93e:11014; Zbl 759.11001 -> (b) Zbl 98.262, (c) Zbl 524.10007 MMR.6.6.b Can. J. Math. 12 (1960) 374-389 MMR.6.6.c Nieuw Arch. Wiskd. IV (1) (1983) 345-360 MMR.6.6. Digital Invariants MMR.6.6. PDI (perfect digital invariant) MMR.6.6. RDI (recurring digital invariant) 55: 5^3+5^3 -> 250 -> 133 -> 55 MMR.6.6. ADI (amicable digital invariant) 136: 1^3+3^3+6^3 -> 244 -> 136 MMR.6.6. VR (Visible Representation) 5882353 = 588^2 + 2353^2 MMR.6.6. sum of factorials: 40585 = 4! + 0! + 5! + 8! + 5! MMR.6.6. sum of powers: 34012224 = (3+4+0+1+2+2+2+4)^6 MMR.6.6. Multigrades: 1^n+5^n+8^n+12^n=2^n+3^n+10^n+11^n (n = 1, 2, 3) MMR.6.6. 792 has 11 representations as sum of three cubes MMR.6.7 factorial products MMR.6.8 printer's "errors" MMR.6.9 automorphic numbers MMR.6.10 two rather large numbers MMR.7 Alphametics MMR.7.a A/DE + B/FG + C/HI = 1 (Digits 1..9) unique, REC 8:5 (1993) 5-6 (Nob) MMR.8 Conglomerate MMR.8.1 the problem of the mice MMR.8.1.a a computer assisted study of pursuit in a plane, AMM 82 (1975) 804-12 MMR.8.1.b Ant Trails, Math. Intell. 15:2 (1993) 59-62 (pursuit problems) MMR.8.2 deployment MMR.8.2. a tick-tack-toe like game played on a 5*5 square with 4 symbols. MMR.8.3 domino recreations MMR.8.4 lost cords MMR.8.5 bouncing billiard balls DG1MI: Tracking the Automatic Ant, DG1MI: David Gale, Springer, New York, 1998 DG1MI.1 Simple Sequences with Puzzling Properties DG1MI.1. Somos sequences DG1MI.1. A Theorem Joke: Perfect squares don't exist DG1MI.2 Probability Paradoxes DG1MI.3 Historic Conjectures: More Sequence Mysteries DG1MI.4 Privacy-Preserving Protocols DG1MI.4. Somos sequences update DG1MI.5 Surprising Shuffles DG1MI.5. A Re-view of some Review: Erd"os - Integral distances DG1MI.6 Hundrets of New Theorems in a 2000 Year-Old Subject: Where will it End? DG1MI.6. Clark Kimberling explores the special points of a triangle DG1MI.7 Pop Math ans Protocols DG1MI.8 Six Variations on the Variational Method DG1MI.8. gcd, Sylvester's problem, Billiard balls DG1MI.9 Tiling a Torus: Cutting the Cake DG1MI.10 The Automatic Ant: Compassless Constructions DG1MI.11 Games Real, Complex Imaginary DG1MI.12 Coin Weighing: Square Squaring DG1MI.13 The Return of the Ant and the Jeep DG1MI.14 Go DG1MI.15 More Paradoxes. Knowledge Games DG1MI.16 Triangles and Computers DG1MI.16. The dance of the Simson Lines DG1MI.16. Configurations with Rational Angles DG1MI.17 Packing Tripods DG1MI.18 Further Travels with My Ant DG1MI.19 The Showlace Problem DG1MI.20 Triangles and Proofs DG1MI.20. The Morley Triangle (Newman's proof) DG1MI.20. Fermat point, Kiepert's hyperbola, projective generalization DG1MI.21 Polyominoes (Golomb) DG1MI.22 A Pattern Problem, A Probability Paradox, and A Pretty Proof DG1MI.23 The Sun, the Moon, and Mathematics DG1MI.24 In Praise of Numberlessness DG1MI.a1 A Curious Nim-Type Game DG1MI.a2 The Jeep Once More or Jeeper by the Dozen DG1MI.a3 Nineteen Problems in Elementary Geometry (Armando Machado) DG1MI.a4 The Truth and Nothing But the Truth DME1: Ross Honsberger DME1: Mathematical Gems 1 DME1: The Dolciane Mathematical Exposition No. 1, MAA, 1973 DME1: german: Mathematische Edelsteine DME1.1 an old chinease theorem and Pierre de Fermat DME1.1.a A generalization of Euler's theorem, Math Gaz 82:493 (1998) 80 DME1.1.a let (a, m_i) = 1 for all i then (with [...] is the lcm) we have DME1.1.a a ** [phi(m1), phi(m2), ..., phi(mk)] = 1 (mod [m1, m2, ..., mk]) DME1.1.b Combinatorial Proof of Fermat's little Theorem, AMM 63 (1956) 718 DME1.1.c A maximal generalization of Fermat's (little) Theorem, MM 39 (1966) DME1.1.c 103-107 - Fermat, Euler, Carmichael, Gauss, Redei. DME1.2 Louis Posa DME1.3 equilateral triangles DME1.3. Sum of the distances to the sides is constant (Viviani) DME1.3. Fermat point, Torricelli problem, Steiner's proof with ellipses DME1.3. Napoleon thm: the triangle of the centers of exterior equilateral DME1.3. triangles is equilateral. DME1.3.a Napoleon thm made simpler, 3 proofs, extendable DME1.3.a Complex Numbers & Geometry, Liang-shin Hahn, MAA (1994), 60-62, 180-81 DME1.3.b G. Pickert, Bemerkungen zum Satz von Napoleon, MSem 39 (1992) 37-41 DME1.3.c L. Gerber, Napoleon's Thm and the Parallelogram Inequality for DME1.3.c Affine-Regular Polygons, AMM 87:8 (Oct 1980) 644-648 DME1.3 a set of diameter 1 can be covered with a equilateral triangle s=sqrt(3) DME1.4 the ochart problem DME1.5 delta curves DME1.6 its combinatorics that counts! DME1.7 Hamiltonial circles DME1.8 the theorem of Morley DME1.9 a combinatorial problem DME1.9. diagonals in a convex polygon, counting crossings and regions DME1.10 multiperfect, superabundant and practical numbers DME1.11 circles, squares, lattice points DME1.12 recursion DME1.13 Poulet's, super Poulet's and other numbers DME1.13. pseudoprime numbers of base 2, Mersenne numbers GG: The Grazing Goat in n Dimensions (Weidende Ziege) GG: A goat is tethered to the circumference of a circular field radius R. GG: How long does the goat's tether need to be so that it can eat half the GG: grass in the field? GG- Geometry Problem 30, AMM 1 (1894) 395-396 GG- M. Fraser, A tale of two goats, Math. Mag., 55:4 (1982) 221-227 GG- M. D. Meyerson, Return of the grazing goat in n dim., CMJ 15 (1984) 430-432 GG Correction for the infinit dim. case, CMJ 15:2 (1984) 126-134, 0-1 law GG In odd dimensions the problem reduces to a polynomial equation GG- D. Treiber, Zum Problem der "Weidenden Ziege", PM 33:3 (1991) 97-100 GG- Bild der Wissenschaft 1970, S. 928 u. 1050 GG- Problem 711 aus PM 21 (1979) 282 GG- Bemerkung zur "Weidenden Ziege", PM 33:6 (1991) 277-278 (G. Bach) GG- another grazing problem (P 710), JoRM 12 (1979/80) 74-75 GG- FAQ for the Swarthmore Forum's "Ask Dr. Math." GG http://forum.swarthmore.edu/dr.math/faq/faq.grazing.html GG- http://mathworld.wolfram.com/GoatProblem.html GG- rec.puzzles FAQ ==> analysis/goat <== GG- The Bull and the Silo: An Application of Curvature, AMM 105:1 (1998) 55-58 GG the bull is eating at the outer side of a convex curve. M. E. Hoffman LB: Ladder-Box Problem (Leiter und Kiste Problem) LB: |\ LB: 1/a | \ c given b and c, calculate a LB: |__\ LB: 1/b | |\ x = a + b LB: |__|_\ y = 1/a + 1/b LB: b a c_min = (b^(2/3) + b^(-2/3))^(3/2) LB: Solution-type for (b=1): T(rigonometric), C(onic section), LB: L(adder-symmetry), S(ymmetry a + 1/a), P(olynomSym in a), LB: N(umeric), D(iophantic Solutions) LB: rel: similar triangles: x/y = a/(1/b) = b/(1/a) LB: rel: 2 * triangle area: x*y = (1/b)*x + b*y LB: rel: Pythagorean thm: (a+b)^2 + (1/a+1/b)^2 = c^2 LB- Arthur Cyril Pearson, The twentieth century standard puzzle book, LB George Routledge & Sons, London 1907, Part II, no. 102, p. 103 LB box = (15, 12), ladder = 52, -> (x, y) = (20, 48) (<- Euclides Ex 623) LB- G. Mott-Smith, Math. Puzzles, Dover Publ. (1954, 1946 repr.) Problem 102 LB The Bay Window, box = (3,9), ladder = 20, -> (x, y) = (12, 16) [D] LB- A. Sutcliffe, A ladder and wall problem, AMM 74 (1967) 325-326 [D] LB E1832 [1965, 1021] <- Schaaf, Bibliography of Rec. Math. II.3.12 LB- A. Dunn, Mathematical Bafflers, Dover Publ. (1980, 1964 repr.), p14 [S] LB- Bild der Wissenschaft (Math. Kabinet) 10:1970 p1050 [T] LB- Bild der Wissenschaft (Math. Kabinet) 1:1972 p89 [S] LB- Bild der Wissenschaft (Math. Kabinet) 9:1978 p147-147 [C] LB- Bild der Wissenschaft (Math. Kabinet) 1:1979 p112 [S] LB- H. Apsimon, Math. Byways in Ayling, Beeling, and Ceiling, LB Oxford Univ. Press 1984, (Chap. 1 [S?] & Chap. 2 [D]) LB- M. Zerger, The ladder problem, Math. Magazine 60:4 (1987) 239-242 [T,L,(C)] LB- letter to the editor, Math. Magazine 61:1 (1988) 63 (->Zerger) [L] LB- section of circle and hyperbola, mathematiklehren 1 (1983) 50-54 [C?] LB- Euclides 66 (1990/91) 222 Ex 623 (2 refs) [???] LB- rec.puzzles FAQ ==> geometry/ladder.and.box <== [???] LB- sio: general solution for b=1 and special problem parameter otherwise. LB b=2, c=10/3: (x,y) = (8/3, 2) or (2.949163, 1.553560) cubic root LB b=2, c=7/2: (x,y) = (3.249, 1.300) or (2.5178, 2.4312) with LB a1 = (-2 - Sqrt[5] + Sqrt[23 + 10 Sqrt[5]])/2 LB a2 = (-2 + Sqrt[5] + Sqrt[23 - 10 Sqrt[5]])/2 LB b=1: standard case LB a + 1/a = z = - 1 + sqrt( 1 + c^2 ) LB a1 = (z + sqrt(z^2 - 4))/2, a2 = (z - sqrt(z^2 - 4))/2 LB x1 = (z + 2 + sqrt(z^2 - 4))/2, x2 = (z + 2 - sqrt(z^2 - 4))/2 LB b=1: special case c = 2*r + 1/r for some rational r. LB a1 = (Sqrt[c^2 + 1] + Sqrt[4*r^2 + 1] - Sqrt[r^-2 + 1] - 1)/2 LB a2 = (Sqrt[c^2 + 1] - Sqrt[4*r^2 + 1] + Sqrt[r^-2 + 1] - 1)/2 LB x1 = (Sqrt[c^2 + 1] + Sqrt[4*r^2 + 1] - Sqrt[r^-2 + 1] + 1)/2 LB x2 = (Sqrt[c^2 + 1] - Sqrt[4*r^2 + 1] + Sqrt[r^-2 + 1] + 1)/2 LB b=1, c=3 (r=1): (x,y) = (1.670, 2.492) or (2.492, 1.670) LB a1 = (Sqrt[10] + Sqrt[5] - Sqrt[2] - 1)/2 LB a2 = (Sqrt[10] - Sqrt[5] + Sqrt[2] - 1)/2 LB x1 = (Sqrt[10] + Sqrt[5] - Sqrt[2] + 1)/2 LB x2 = (Sqrt[10] - Sqrt[5] + Sqrt[2] + 1)/2 LB b=1, c=9/2 (r=2): (x,y) = (1.302, 4.307) or (4.307, 1.302) LB a1 = (Sqrt[85] + 2*Sqrt[17] - Sqrt[5] - 2)/4 LB a2 = (Sqrt[85] - 2*Sqrt[17] + Sqrt[5] - 2)/4 LB x1 = (Sqrt[85] + 2*Sqrt[17] - Sqrt[5] + 2)/4 LB x2 = (Sqrt[85] - 2*Sqrt[17] + Sqrt[5] + 2)/4 LB b=1, c=11/3 (r=3/2): (x,y) = (1.420, 3.381) or (3.381, 1.420) LB a1 = (Sqrt[130] + 3*Sqrt[10] - Sqrt[13] - 3)/6 LB a2 = (Sqrt[130] - 3*Sqrt[10] + Sqrt[13] - 3)/6 LB x1 = (Sqrt[130] + 3*Sqrt[10] - Sqrt[13] + 3)/6 LB x2 = (Sqrt[130] - 3*Sqrt[10] + Sqrt[13] + 3)/6 LB- sio[S]: you can shear the box (rombic box) without disturbing symmetry (b=1) LB a + 1/a = z = cos(phi) - 1 + sqrt( (cos(phi)+1)^2 + c^2 ) LB a1 = (z + sqrt(z^2 - 4))/2 a2 = (z - sqrt(z^2 - 4))/2 CCh: Conway's Recursive Sequence (series): 1, 11, 21, 1211, 111221, 312211, ... CCh- J. H. Conway, Eureka 46 (1986) 5-16, reprinted in: CCh Open Problems in Communications and Computations, Springer, 1987, 173-188 CCh The Weird and Wonderful Chemistry of Audioactive Decay CCh (Correction of the final asymptotic formula: Ilan Vardi) CCh- Ilan Vardi, Computational Recreations in Mathematica, Chapter 1. (1991) CCh ilan@leland.Stanford.EDU (ilan vardi) CCh Organization: DSG, Stanford University, CA 94305, USA CCh- Endless self-description (Hilgemeier's "likeness sequence", CCh [Die Gleichniszahlen-Reihe]) Quantum 4:1 (1993) 17 CCh- Hilgemeier, M., Die Gleichniszahlen-Reihe, BdW (Dec. 1986) 19 CCh- "One Metaphor Fits All" : A Fractal Voyage With Conway's Audioactive Decay CCh http://www.is-bremen.de/~mhi/frahor00.htm CCh- P 958 R"atselhafter Folgenanfang? Praxis der Mathematik, 33:3 (1991) 136 CCh- Lsg P 958: Praxis der Mathematik, 34:1 (1992) 42-43 -> C. Stoll. CCh Folge: 1, 11, 21, 1211, 111221 = a1, a2, a3, a4, a5 gegeben. CCh Lsg I: V(n) = N(n-1) V(n-1) : 1, 1, 2, 12, ... (n >= 2 auÞer n=3) CCh N(n) = V(n-2) N(n-2) : -, 1, 1, 11, ... (n >= 3) CCh a(n) = V(n) N(n) CCh Lsg II: Fibonacci Lsg von Paasche ist Quatsch. CCh Lsg III: a(n) sei in Trinaersystem dargestellt, dies gibt im CCh Dezimalsystem 1, 4, 7, 49, 376, ... CCh Interpoliere durch ein Polynom 4ten Grades. CCh Lsg IV: Ersetze: 11->21, 1->11, 2->12 indem man immer die erste CCh moegliche Ersetzung (links nach rechts) macht. CCh- Clifford Stoll, Kuckucksei, Fischer Verlag, Kap. 48, S. 358 CCh- 1, 11, 21, 1211, 1231, 131221, ..., REC 7:4 (Sep 1992) 4-5 CCh- On a Curious Property of Counting Sequences, AMM 101 (1994) 560-563 CCh- XXX math.CO/9808077 Shalosh B. Ekhad, Doron Zeilberger CCh Proof of Conway's Lost Cosmological Theorem Col: Collatz sequence, syrakuse-algorithm, 3n+1 problem, Col- E16, R. K. Guy, Unsolved problems in number theory, 1981, 10 Refs <= 1978 Col- MG10SA.18 Slither, 3x+1 and other curious Questions Col- Lagarias, the 3x+1 problem and its general., AMM 92 (1985) 3-23 (70 refs) Col http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/node1.html Col- G. Venturini, Iterates of number theoretic functions with periodic Col rational coefficients, Studies in Appl. Math., 86 (Apr. 1992) 185-218 Col- P. Filipponi, On the 3n+1 problem: Something old, something new, (1991) Col Zbl 735.11010 uncomplete proof. Col- Adv in Appl. Math. 10 (1989) 344-347 Col- Jeffrey Lagarias, The Set of Rational Cycles for the 3x+1 problem, Acta Col Arithmetica. LVI 1990, 33-53; #MR 91i:11024. Lagarias considers the same Col problem on the set of rationals that have an odd denominator [local ring Col of fractions of the integers at the ideal (2)]. In this ring the situation Col differs from the one conjectured for Z, and has many interesting features. Col He proves several theorems, formulates other interesting conjectures and Col povides heuristic arguments for them. Col On a different side, John H.Conway showed in _Unpredictable Operations_ Col (Proc. of the 1972 Number Theory Conference. Boulder-Colorado. pp. 49-52) Col that for the following generalized problem: Col ======================================================================== Col g(n) = a_i n + b_i (n congruent to i mod p) with the Col a_i's and b_i's rationals chosen so that g(n) is always an integer Col ======================================================================== Col the question of whether there is a k-th iterate of g(n) which equals one, Col for a given n, is undecidable; and that is still true even if b_i = 0. Col This doesnt mean that Collatz's problem is undecidable, but it gives an Col idea of how hard it probably is. Col- new lower bounds on nontrivial cycle length, Disc Math 118 (1993) 45-56 Col- Lagarias also maintains an online annotated bibliography that covers work Col since his AMM 92 (1985) 3-23 article. Col http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/links.html Col- R. Guy, Conway's Prime Producing Machine, Math. Mag., 56:1 (1983) 26-33 Col contains a construction of Conway to prove unsolvable the convergence Col of a certain class of congruential iteration problems. Col- S. Burckel, Functional equations associated with congruential functions Col Theor. Comp. Sci. 123 (1994) 397-406 (generalize Conway's construction) CPS: Conway's permutation sequence, N -> N (bijective) (-> "Col") CPS 2n -> 3n, 4n-1 -> 3n-1, 4n+1 -> 3n+1 CPS- E17, R. K. Guy, Unsolved problems in number theory, 1981, 1 Ref CPS- J. H. Conway, Unpredictable iterations, Proc Number Theory Conf., Boulder CPS 1972, 49-52 HOS: Hofstadter Sequence: HOS a(1)=a(2)=1, a(n) = a(n-a(n-1)) + a(n-a(n-2)) HOS- E31, R. K. Guy, Unsolved problems in number theory, 1981, 1 Ref HOS- S. M. Tanny, A well-behaved cousin of the Hofstadter sequence, HOS Disc. Math. 105 (1992) 227-239 HOS- Conway's Challenge Sequence, AMM 98 (1991) 5-20, 99 (1992) 563-564 HOS- Conway's Challange, AMM 100 (1993) 396-397 HOS- -Tal Kubo kubo@math.harvard.edu or kubo@zariski.harvard.edu HOS A draft of the article is available (in AMSLaTeX format). Interested HOS parties should contact me by e-mail. <- Conway's Challenge Sequence NCS: Newman Conway Sequence: NCS p(1) = p(2) = 1, p(n) = p(p(n-1)) + p(n-p(n-1)) NCS- Math Mag 68:5 (1995) 400-401 Solution 1459 NCS (*) 0 <= p(n+1) - p(n) <= 1 (**) p(2n) <= 2*p(n) MNQ: modular n-queen problem (n Damen Problem) MNQ- T. Klove, Disc. Math. 19 (1977) 289-291, 36 (1981) 33-48 MNQ part II contains solutions with rotational sym.: n=49 and n=77 MNQ open: are there always solutions if n contains prime factors 3 mod 4? MNQ- O. Heden, Disc. Math. 102 (1992) 155-161, on the modular N-Queen Problem MNQ place n-2 queens for all n (solved), Addendum: Disc Math 118 (1993) 293 MNQ -> AMM 1989 (E 3162) MNQ- O. Heden, Disc. Math. 120 (1993) 75-91, Maximal partial spreads and MNQ the modular N-Queen Problem MNQ- Sprague, Mathematische Unterhaltungen... , Aufgabe 16 (zwei Tafelrunden) MNQ- G. Polya, Uber die "doppelt-periodischen" Losungen des n-Damen Problems MNQ (in: W. Ahrens, Mathematische Unterhaltungen und Spiele, Bd II MNQ (1918), 374-374), Collected Papers IV, 237-247 MNQ- ..., I. Vardi, The n-queens problem, AMM 101 (1994) 629-639 MNQ exponential lower bound 2^(n/5) if for a prime-factor p = 1 mod 4. MNQ- infinite queens, Annals New York Acad. Sci. 555, p133 MNQ iteration of the (13524) solution MNQ- A Neural Network solves n-Queens, Biolog. Cybern. 66 (1992) 375 MNQ- Fast Search Algor. n Queens, IEEE Trans. on Syst. Man and Cybern. MNQ 21:6 (1991) 1572-76 MHP: Monty Hall Paradox (let's make a deal) MHP- AMM 99:1 (Jan 1992) 3-7 MHP- Math. Gazette 75 (Okt 1991) 275-277 MHP- Math. Teacher 85:4 (Apr 1992) 250-252-256, 85:6 (Sep 1992) 409 MHP- Math. & Comp. Education 26:2 (1992) 153-156 MHP- E. Engel, A. Ventoulias, Chance 4:2 (1991) 6-9 MHP- Der Spiegel 34/1991 212-213 Drei Tueren Problem MHP- Der Spiegel 36/1991 12-13 (Leserbriefe zum Drei Tueren Problem) MHP- Spektrum der Wissenschaft, 11/1991 12-16 MHP- Die Zeit 34/1991 58 -> Die Zeit 30/1991 und 33/1991 (Leserbriefe) MHP- Stochastik in der Schule, 11:3 (1991) 46-51 Problemecke MHP- Math. Magazine 64 (1991) 359 [Reviews], -> New York Times, Focus MHP- H. Winter, JMD 13:1 (1992) 23-53, intuitive Aufkl"arung prob. Paradoxien MHP- von Randow, Das Ziegenproblem, rororo 1992 MHP- The 5-Step Probability Solver, Pi Mu Epsilon J. 9 (1992) 445-447 MHP- Simulation..., Stochastik in der Schule 12:3 (1992) 2-25 MHP- The Problem of the Car and Goats (survey) MHP The College Math. J. 24:2 (1993) 163-165 MHP- Probability Problems References (Update) MHP The College Math. J. 26:2 (1995) 132-134 MHP- Generalizing Monty's Dilemma MHP Quantum 5:4 (Mar Apr 1995) 17-21 & 59-60 and 40-41 MHP- http://www.cartalk.msn.com/About/Monty/ MHP- http://www.cut-the-knot.com/hall.html PFE: Patological Functions PFE- Gelbaum, Olmstedt; Counterexamples in analysis, 1964 PFE- Everywhere differentiable nowhere monoton function, AMM 81 (1974) 349-354 PFE- On functions that are monoton on no interval, AMM 88 (1981) 754-755 PFE- Functions with a proper local maximum in each interval, AMM 90 (1983) 281-2 PFE- A universal entire fuction, AMM 90 (1983) 331-332 PFE- On van der Waerden's nowhere differentiable function, AMM 91 (1984) 307-8 PFE- Holes in Graphs, Quantum 2:1 (Sep/Oct 1991) ..-14, 19, 63 PFE function continuous only on the integers; bijection irrational -> real PFE- White and Brown Musik, Fractal Curves and 1/f Fluctuations, SA 4/1978, 16 PFE- Nowhere-Differential Functions (Kieswetter Curves), AMM 99 (1992) 565-566 PFE- Noncentral Difference Quotients and the Derivative, AMM 95 (1988) 551-553 PFE (f(h)-f(ah))/((1-a)h) -> f'(0) for a <> +/- 1 PFE- Derivatives, Why They Elude Classification, Math. Mag. 49 (1976) 5-11 PFE- A. Bruckner, J. Leonard, Derivatives, AMM 73 Part II (1966) 24-56; 216 Refs PFE a necessary and sufficient condition that a set E subset of [a,b] be the PFE set of discontinuities of a derivative, is that E be an F_sigma of the PFE first category (meager). (theorem p27). E may have mesure one PFE- Critical Points of Gateaux Differential Functions, AMM 95 (1988) 566-567 PFE f'(x)=0, f''(x)>0 (Gateaux) =/=> x is local minimum PFE- holomorphic functions C -> C, with Q -> Q are more than polynomials(Achim) PFE f(x) = Sum[ c_i Product[ x-x_i, {i,n} ], {n, Infinity} ] (Newton form) PFE N -> Q: i -> x_i (surjektive), c_i rational and small enough. PFE- there is no differential metric for R^n, AMM 86 (1979) 585-586 PFE- Continuous Functions and Connected Graphs, AMM 97 (1990) 337-339 PFE- The Shortest Planar Arc of Width 1, AMM 96 (1989) 309-327 PFE- Every Smooth Map of Euclidean Space into Itself is an Expansion Followed PFE by a Contraction, AMM 95 (1988) 713-716; continuity is not sufficient PFE- Composition of differential functions f(f(x))=exp(x), M. Mag. 64(1991)354-5 PFE- Cantorset C: C+C = [0,2]. Math. Mag. 64 (1991) 357, A785 PFE- real functions, Bull. LMS 19 (1987) 396-398 (review LNiM 1170) PFE- Gegenbeispiele zum Infimum und Supremum bei Ableitungen, PFE Math. Semesterberichte 39 (1992) 137-142 (G. Pickert) PFE- On the definition of 0^0, FU-Berlin Preprint A-92-5 (W. Koepf) PFE |y(t)| < M (x(t))^a, with M>0, a>0, then lim_{t->0} x(t)^y(t) = 1 PFE- The Exotic World of Invertible Polynomial Maps (Jacobi Conjecture) PFE Nieuw Archief voor Wiskunde 11 (1993) 21-31 PFE- G. Harris and C. Martin, The roots of a polynomial vary continuously PFE as a function of the coefficients, Proc. Amer. Math. Soc. 100 (1987), PFE 390-392. (SHORTER NOTES) PFE- T. Sauer, Ein algorithmischer Zugang zu Polynomen und Splines, PFE Math. Semesterberichte, 43 (1996) 169-189. Bezier, Casteljau, de Boor AGM: arithmetic-geometric mean AGM- Gauss, Landen, Ramanujan, the a-g mean, ellipses, and the ladies diary, AGM AMM 95 (1988) 585-608 AGM approximations for the perimeter of the ellipse (table, history) AGM in each formula is the error increasing with eps; max error, err<10^(-6) AGM Pi (a+b) <= P [Kepler] 21.46% eps<.0894 AGM Pi Sqrt[2(a^2+b^2)] >= P [Euler] 11.07% eps<.0894 AGM 2 Pi ( (a+b)/(Sqrt[a]+Sqrt[b]) )^2 >= P [Sipos] 57.08% eps<.4446 AGM Pi ( 3/2 (a+b) - Sqrt[ab] ) >= P [Peano] 17.81% eps<.4883 AGM 2 Pi ((a^(3/2) + b^(3/2))/2)^(2/3) <= P [Muir] 1.046% eps<.5492 AGM Pi ( 3(a+b) - Sqrt[(a+3b)(3a+b)] ) <= P [Ramanujan] 0.416% eps<.8280 AGM Pi (a+b) (1+3z^2/(10+Sqrt[4-3z^2]))<= P [Ramanujan] .0402% eps<.9811 AGM Pi a (3 z^4 - 128)^2 / (2048 (4 - z^2)(1 + z )) + a*O(z^9) [Eric Blom] AGM z = (a-b)/(a+b) AGM- the Gauss-Slamin algorithm, Math. Gazette 76 (July 1992) 231-242 AGM- elementary approximations to the area of n-dimensional ellipsoids, AGM AMM 78 (1971) 280-283; Pi(a+b) <= P <= Pi Sqrt[2(a^2+b^2)]; AGM 4 Pi (bc+ca+ab)/3 <= A <= 4 Pi Sqrt[((bc)^2+(ca)^2+(ab)^2)/3] AGM false: 4 Pi (a^2+b^2+c^2)/3 <= A <= 4 Pi Sqrt[(a^4+b^4+c^4)/3] (Polya) AGM- bccarlson@decst2.ams.ameslab.gov AGM B. C. Carlson, elliptic integrals, Math of Comp 59 (1992) 165-180 AGM- Area of an Ellipsoid: a=b, c AGM A = 2 Pi a^2 (1 + 1/Sqrt[1-(a/c)^2] Arctan[Sqrt[(c/a)^2-1]]) for c>a, AGM A = 2 Pi a^2 (1 + 1/Sqrt[(a/c)^2-1] Arctanh[Sqrt[1-(c/a)^2]]) for c<a. AGM- Approximations of the Area (Achim Flammenkamp) AGM 2 Pi (ab + bc + ac - 3abc/(a+b+c)) max error 10% AGM Pi ( (1-1/Sqrt[3]) (ab + bc + ac) + (1+1/Sqrt[3]) Sqrt[aabb+bbcc+aacc] ) AGM max error 2.09% AGM- An Overlooked Series for the Elliptic Perimeter, M Mag. 68:3 (1995) 216-220 AGM- http://mathworld.wolfram.com/E/Ellipse.html AGM- http://mathworld.wolfram.com/E/Ellipsoid.html AGM-J. M. Borwein, P. B. Borwein, and D. H. Bailey, "Ramanujan, AGM Modular Equations, and Approximations to Pi", American Mathematical AGM Monthly, vol. 96, no. 3 (March 1989), p. 201 - 220. AGM-P. Beckman AGM A history of pi AGM Golem Press, CO, 1971 (fourth edition 1977) AGM-J.M. Borwein and P.B. Borwein AGM The arithmetic-geometric mean and fast computation of elementary AGM functions AGM SIAM Review, Vol. 26, 1984, pp. 351-366 AGM-J.M. Borwein and P.B. Borwein AGM Pi and the AGM - a study in analytic number theory and AGM computational complexity AGM Wiley, New York, 1987 AGM-David Chudnovsky and Gregory Chudnovsky AGM The computation of classical constants, Columbia University, AGM Proc. Natl. Acad. Sci. USA, Vol. 86, 1989. AGM-E. Salamin AGM Computation of pi using arithmetic-geometric mean AGM Mathematics of Computation, Vol. 30, 1976, pp. 565-570 ISO: Isoperimetric Inequalities ISO- R. Osserman, Bonnesen-Style Isoperimetric Inequalities, AMM 86 (1979) 1-29 ISO (77 Refs) PIA: Piano, Chair Moving Problem, Sofa or Ladder moving PIA- Mountain Climbing, Letter Moving, and the Ring-Width of a Polygon, PIA AMM 96 (1989) 494-510 PIA- The sofa problem, AMM 83 (1976) 188-189 (refs) SEQ: sequences, strong law of small numbers SEQ- W. L. Schaaf, Bibliography of Rec. Math. III.1.13 Number Sequences SEQ- Guy, The Strong Law of Small Numbers, AMM 95 (1988) 697-712 SEQ- Guy, The Second Strong Law of Small Numbers, Math. Mag. 63 (1990) 3-20 SEQ- N. J. A. Sloane, A Handbook of Integer Sequences SEQ- Curious Recursion: x(n+3) = (1+x(n+1)+x(n+2))/x(n), M. Mag. 64(1991)133-4 SEQ- 51th Putnam: T(0)=2, T(1)=3, T(2)=6, T(n)=(n+4)T(n-1)-4nT(n-2)+(4n-8)T(n-3) SEQ T(n) = n! + 2^n, Math. Mag. 64 (1991) 141 SEQ- u(1)=1, u(n+1)=floor(Sqrt(2)(u(n)+(1/2)) -> u(2n+1)-2u(2n-1) is the nth SEQ digit in the binary expansion of Sqrt[2]. Math. Mag. 64 (1991) 168-171 SEQ- Periodicity of Somos Sequences, Proc AMS 116 (1992) 613-619 SEQ- The strange and surprising saga of the Somos sequences, SEQ Math. Intelligenzer 13:1 (1991) 40-42 (Gale) SEQ-> v(n+1) = v(n)**2 - c SEQ- Aho & Sloane, Some doubly exponential sequences, Fib. Quart. SEQ 11 (1973), 429-437; SEQ- Franklin & Golomb, A function-theoretic approach to the study SEQ of nonlinear recurring sequences, Pacific J. Math. 56 (1975), 455-468. SER: Euler and the series \Sum n! (-x)^n SER- Hardy, Infinite Series, II.A, Euler and the series \Sum n! (-x)^n SER- Knopp, Unendliche Reihen, Par. 66 SER- Bromwich, Infinite Series, 2nd Ed, Pra. 104, 105, 109(2) SER- Carlson, Special Functions, Chap. .. 2F0 CAL: Calender, Easter, Day of Week, Friday 13th CAL- Calendrical Calculations, (N. Dershowitz, E. M. Reingold), CAL Cambridge Univ. Press, 1997 CAL URL http://emr.cs.uiuc.edu/home/reingold/calendar-book/index.html CAL URL http://emr.cs.uiuc.edu/~reingold/calendar.C (C++ source) CAL- calendrical calculations, (N. Dershowitz, E. M. Reingold), CAL Software Practice and Exp. 20:9 (Sep 1990) 899-929 CAL Julian, Gregorian, Islam, Hebrew, Conversions, Gnu Lisp program CAL- calendrical calculations II, (E. Reingold, N. M. Dershowitz, S. Clamen), CAL Software Practice and Exp. 23:4 (Apr 1993) 383-404, CAL three historical calendars CAL- Chr. Zeller, Kalender-Formeln, Acta Mathematica, 9 (1887) 131-136 CAL- Hatcher, D. A., Simple Formulae for Julian Day Numbers and Calendar Dates, CAL Quarterly Journal of the Royal Astronomical Society, 25 (1984) 53-55. CAL- Hatcher, D. A., Generalized Equations for Julian Day Numbers and Calendar CAL Dates, Quarterly Journal of the Royal Astronomical Society, 26 (1985) CAL 151-155. Includes coefficients for various calendar systems, CAL including the Egyptian, Alexandrian, Roman, Gregorian, and Islamic. CAL- Keith & Craver; The ultimate perpetual calendar?, JoRM 22:4 (1990) 280-282 CAL day of the week as a 44 character expression in C. (illegal use of --) CAL The following 45 character C expression by Keith is correct. CAL dow(y,m,d) { return (d+=m<3?y--:y-2,23*m/9+d+4+y/4-y/100+y/400)%7; } CAL- Mathematics of the Gregorian Calendar (V F Rickey) CAL The Math. Intelligenzer, 7:1 (1985) 53-56, leap year rules, history CAL- Kalender-Formeln (C. Zeller), Acta Mathematica, 9 (1887) 131-136 CAL- The Day of the Week for Gregorian Calendars (A. D. Bradley) ?? 82-87 CAL- W. S. B. Wodhouse, Calendar, Encyclopaedia Britannica, CAL 11th and 13th ed., 4, 987-1004 CAL- Feast or Famine of Friday the 13th, AMM 98 (1991) 646-649 CAL- B. H. Brown - Friday the 13th -, AMM 40 (1933) 607 CAL the 13th is most likely to be a Friday CAL- S. R. Baxter - Friday the 13th -, Math. Gazette 53 (1969) 127-129 CAL the 13th is most likely to be a Friday CAL- J. O. Irwin, Friday 13th, Math. Gazette, 55 (1971) 412-415 CAL The number of Friday 13th per year is at least one and at most three. CAL- A. W. Butkewitsch, M. S. Selikson - Ewige Kalender, Teubner (Leipzig) 1974 CAL Kleine Naturwissenschaftliche Bibliothek, Bd. 23 CAL- Ilan Vardi, Computational Recrations in Mathematica, (1991) CAL Chap 4: The Calender, p35-55 PhG: Physical Games PhG- Superball (Flummy) MNU 45:8 (1992) 477-483 MaM: Mastermind MaM: The origin of mastermind is 'bulls and cows'. MaM- The Computer as Mastermind, JoRM 9:1 (1976) 1-6 (D. E. Knuth) MaM- Towards an Optimum Matermind Strategy, JoRM 11:2 (1978) 81-87 (Irving) MaM- Some Stategies for Mastermind, ZOR 26 (1982) 257-278 (Neuwirth) MaM- Mastermind Strategies, JoRM 18:3 (1985) 194-202 (Flood) MaM- Sequential Search Strategies with Mastermind Variants-Part 1&2 MaM JoRM 20:2 (1988) 105-126, 20:3 (1988) 168-181 (Flood) MaM- A Prolog Matermind Program, JoRM 23:2 (1991) 81-93 MaM- Mastermind as a Test-Bed for Search Algorithms, Chance 6 (1993) 31-37 MaM- An optimal Mastermind Strategy, (4 pos, 6 colors) MaM JoRM 25:4 (1994) 251-256 PRN: Pseudo Random Numbers PRN- random numbers (Tech Correspondence) CACM 36:7 (1993) 105-110 PRN- LNiEaMS 374 (1992) Part III, random numbers, transformation, package. PRN- Karloff, Raghavan; Randomized Algorithms and Pseudorandom Numbers, PRN J ACM 40:3 (1993) 454-476 PRN quicksort can be bad with pseudorandom numbers (lin. congruence) PRN- Eduardo M. Engel, A Road to Randomness in Physical Systems (1992) (10QB...) PRN- A McGrail, Randomness Properties of Two Chaotic Mappings, p265-295 PRN in: Cryptography and Coding III (Ed. M. J. Ganley) Clarendon Press (1993) PRN some new references for testing PRNs. PRN- G. Marsaglia, A current view of random number generators PRN Computer Science and Statistics (Ed. L Billard) (1985) 3-10 PRN- B A Cipra, AAAS'94: Random Numbers, Art and Math, SIAM News 27:7(1994)24,18 PRN- Rehashing Pearson's string hash (Letters), C++ Report (Feb. 1995) 6-15 PRN bad use of rand() to generate a good mixing permutaion PRN- S. v Hoerner; Herstellung von Zufallszahlen auf Rechenautomaten, PRN Z. Angew. Math. Physik 8 (1957) 26-52 (v. Neuman mid-square-method) PRN- Halton; SIAM Review 12 (1970) 1-63 PRN- Random HP42S, CACM 38:1 (1995) 121-124 DRS: Descartes Rule of Signs DRS- A Pedersen, A Refinement of Descartes Rule of Signs, AMM 98 (1991) 862-865 DRS if v in S with: a(v-1) a(v) < 0, a(v) a(v+1) < 0 and DRS a(v)^2 <= a(v-1) a(v+1) DRS the intersection on all v in S: [ -a(v)/a(v+1), -a(v-1)/a(v) ] DRS is not empty then : #positive roots <= #sign changes - 2 #S DRS- The Sign Rule from Descates Formulation (1637) to Gauss Proof (1828) DRS Archive for Hist. of Exact Sci. 45:4 (1993) 335-374 DRS- Fekete, Polya; Ueber ein Problem von Lagerre, DRS Rend. Circolo Mat. Palermo 34 (1912) 1-32 TOP: finite Topologies, finite Posets (T0 Topologies) TOP- Posets Pu(n) : 1, 2, 5, 16, 63, 318, 2045, 16999, 183231, 2567284, TOP 46749427, 1104891746, 33823827452, ... Sloane 588 TOP- Topologies Tu(n): 1, 3, 9, 33, 139, 718, 4535, ... Sloane 1133 TOP- Posets (labeled) P(n) : 1, 3, 19, 219, 4231, 130023, 6129859, TOP 431723379, ... TOP- Topologies (labeled) T(n) : 1, 4, 29, 355, 6942, 209527, 9535341, TOP 642779354, 63260289423, ... TOP- N. J. A. Sloane, A Handbook of Integer Sequences, p14-16 TOP posets (labeled): series 588 (1244) TOP topologies (labeled): series 1133 (1476) TOP connected topologies (labeled): series 648 (1245) TOP- N. J. A. Sloane & S. Plouffe, Academic Press, ISBN 0-12-558630-2. TOP The Encyclopedia of Integer Sequences TOP- Chaunier & Lygeros, Proges dans l'enumeration des posets, TOP C. R. Acad. Sci. Paris 314 serie I (1992) 691-694 TOP- Chaunier & Lygeros, The Number of Orders with Thirteen Elements, TOP Order 9:3 (1992) 203-204 (Table: P(n) n=1..13) TOP- Culberson & Rawlins, New results from an algorithm for counting posets, TOP Order 7 (1991) 361-374 TOP- H J Proemel, Counting Unlabeled Structures, JoCT A 44 (1987) 83-93 TOP Cor 2.3a: Almost all partial ordes are rigid, i. e. have no TOP nontrivial automorphism. TOP Cor 2.3: Let Pu(n) denote the number of unlabeled partial orders on TOP an n-element set. Then there exists a constant s such that TOP Pu(n) <= P(n) / n! ( 1 + s / 2^(n/4) ). TOP- D Kleitman and B Rothschild, The number of finite topologies, TOP Proc. AMS, 25, 1970, 276-282. TOP- D Kleitman & B Rothschild, Asymptotic enumeration of partial orders on a TOP finite set, Trans. AMS V 205 (1975) 205-220 TOP almost all posets (labeled) consists of three levels. This is a TOP deep result. An explicid formular (for 3 levels) is given (a double sum). TOP- M Erne, The Number of Posets with More Points Than Incomparable Pairs, TOP Disc Math 105 (1992) 49-60 TOP- M Erne & K Stege, Counting finite posets and topologies, TOP Order 8 (1991) 247-265 TOP- M Erne, On the cardinalities of finite topologies and the number of TOP antichains in partially ordered sets, Disc Math 35 (1981) 119-133 TOP table of labeled posets 1..8 TOP- M Erne, Struktur- und Anzahlsformeln fuer Topologien auf endlichen TOP Mengen, Manuscripta Math. 11 (1974) 221-259 TOP- J. W. Evans, F. Harary and M. S. Lynn; On the computer enumeration of TOP finite topologies; Comm. Assoc. Computing Mach. 10 (1967), 295--298. TOP- L Comtet, Advanced Combinatorics, (1974), p229 Ex. TOP table of labeled topologies 1..9 TOP- L Comtet, Recouvrements, bases de filtre et topologies d'un ensemble fini, TOP C. R. Acad. Sci. Paris 262 A (1966) 1091-1094 TOP tables of labeled posets and topologies 1..6 TOP- Das, Shawpawn Kumar: Jour. ACM 24 (1977) 676-692 TOP- Rodionov. V.I. MR#83k:05010 T(12) and T0(12) calculated (Russ) TOP- Borevich, Z.I. MR#83K:05004b k==l (mod p-1) => T0(k)==T0(l) (mod p) (Russ) TOP- Davison, Asymptotic Enumeration of Partial Orders, TOP Congressus Numerantium 53 (1986) 277-286 (labeled partial orders) TOP He evaluates the double sum of Kleitman and Rothschild. TOP P(n) = (1+O(1/n)) Sqrt[2/(Pi n)] 2^(n^2/4 + 3n/2) c(n) TOP c(n) = sum_{k in Z} 2^(-(k+1/2)^2) for n even TOP c(n) = sum_{k in Z} 2^(-k^2) for n odd TOP for the unlabeled case see Proemel. TOP- Congressus Numerantium, 8 (1973) 180, TOP Table T(n): n=1..7 TOP- R. Bumby, R. Fisher, H. Levinson and R. Silverman; Topologies on TOP Finite Sets; Proc. 9th S-E Conf. Combinatorics, Graph Theory, and TOP Computing (1978), 163--170 TOP- P. Renteln, ``On the enumeration of finite topologies'', Journal of TOP Combinatorics, Information, and System Sciences, 19 (1994) 201-206 TOP- P. Renteln, ``Geometrical Approaches to the Enumeration of Finite Posets: TOP An Introductory Survey'', Nieuw Archief voor Wiskunde, 14 (1996) 349-371 TOP- R W Robinson, Countng labeled acyclic digraphs, 239-273, TOP in: New directions in the theory of graphs, F Haray, Academic Press 1973 TOP- Dave Rusin (rusin@math.niu.edu) TOP http://www.math.niu.edu/~rusin/known-math/point-set.top/finite.top PAC: Trickpack PAC- Euclides 69 (1993) 92 [646] PAC 2 different trick packings for the box: (Euclides and Nob) PAC- P. van Delft and J. Botermans, PAC Creative Puzzles of the World, 1977 PAC (German: Denkspiele der Welt, Deutsche Bearbeitung Eugen Oker, 1977 PAC Section: The dissected block (the T. H. O'Beirne box) PAC- V(3a,3b,3c) + V(a,b,c) = V(2b,2c,7a) = 28abc, V-Increment = 1/27 PAC [ -3, 2, 0 ]-1 [ 9, 6, 4 ] PAC [ 0, -3, 2 ] = [ 14, 9, 6 ] = M PAC [ 7, 0, -3 ] [ 21, 14, 9 ] PAC (a,b,c) = M * (d_a,d_b,d_c). All vectors are column-vectors. PAC + (a,b,c) = (4,6,9) = M * (0,0,1) (Haba) PAC V(12,18,27) + V(4,6,9) = V(12,18,28) PAC several rearrangements for the small box possible as: 3a=2b, 3b=2c. PAC + (a,b,c) = (6,9,14) = M * (0,1,0) (T. H. O'Beirne) PAC V(18,27,42) + V(6,9,14) = V(18,28,42) PAC a small box (18,28,41) with hole (1,14,18) is possible too. PAC + (a,b,c) = (9,14,21) = M * (1,0,0) (sio) PAC V(27,42,63) + V(9,14,21) = V(28,42,63) PAC + (a,b,c) = (10,15,23) = M * (0,1,1) PAC V(30,45,69) + V(10,15,23) = V(30,46,70) PAC + (a,b,c) = (13,20,30) = M * (1,0,1) PAC V(39,60,90) + V(13,20,30) = V(40,60,91) PAC + (a,b,c) = (15,23,35) = M * (1,1,0) PAC V(45,69,105) + V(15,23,35) = V(46,70,105) PAC + (a,b,c) = (19,29,44) = M * (1,1,1) (Euclides) PAC V(57,87,132) + V(19,29,44) = V(58,88,133) PAC + (a,b,c) = (108,164,249) = M * (4,6,9) (sio) PAC maximal increment factor: 1 + 1/81 PAC + (a,b,c) = (164,249,378) = M * (6,9,14) (sio) PAC maximal increment factor: 1 + 1/81 PAC + (a,b,c) = (249,378,574) = M * (9,14,21) (sio) PAC maximal increment factor: 1 + 1/81 PAC - 8+1 Boxes: {a, 2a} * {b, 2b} * {c, 2c} plus (a,b,c). PAC- V(3a,5b,6b) + V(a,2b,3b) = V(4b,4a,6b) = 96abb, V-Increment = 1/15 PAC [ -3, 4 ]-1 [ 5, 4 ] PAC [ 4, -5 ] = [ 4, 3 ] = M PAC (a,b) = M * (d_a,d_b). All vectors are column-vectors. PAC + (a,b) = (4,3) = M * (0,1) PAC V(12,15,18) + V(4,6,9) = V(12,16,18) PAC + (a,b) = (5,4) = M * (1,0) (Nob) PAC V(15,20,24) + V(5,8,12) = V(16,20,24) PAC + (a,b) = (9,7) = M * (1,1) (sio) PAC V(27,35,56) + V(9,14,21) = V(28,36,56) PAC + (a,b) = (31,24) = M * (3,4) (sio) PAC V(93,120,144) + V(31,48,72) = V(96,124,144) PAC maximal increment factor: 1 + 1/30 PAC + (a,b) = (40,31) = M * (4,5) (sio) PAC V(120,155,186) + V(40,62,93) = V(124,160,186) PAC maximal increment factor: 1 + 1/30 PAC + (a,b) = (26,20) : Koffer packen (Hannappel) PAC - 6+1 Boxes: {a, 2a} * ({3b} * {2b, 4b} and {2b} * {6b}) plus (a,2b,3b). PAC- V(3a,b,5c) + V(a,b,c) = V(b,4c,4a) = 16abc, V-Increment = 1/15 PAC [ -3, 1, 0 ]-1 [ 5, 5, 4 ] PAC [ 0, -1, 4 ] = [ 16, 15, 12 ] = M PAC [ 4, 0, -5 ] [ 4, 4, 3 ] PAC + (a,b,c) = (14,43,11) = M * (1,1,1) (sio) PAC V(42,43,55) + V(14,43,11) = V(43,44,56) PAC + (a,b,c) = (46, 141, 36) = M * (3,3,4) PAC maximal increment factor: 1 + 1/45 PAC + (a,b,c) = (60, 184, 47) = M * (4,4,5) PAC maximal increment factor: 1 + 1/45 PAC + (a,b,c) = (235, 720, 184) = M * (15,16,20) PAC maximal increment factor: 1 + 1/45 PAC - 2+1 Boxes: {3a} * {b} * {c, 3c} plus (a,b,c). PAC- V(3a,2b,5c) + 2abc = V(b,4c,8a) = 32abc, V-Increment = 1/15 PAC + (a,b,c) = (92, 282, 144) PAC maximal increment factor: 1 + 1/45 PAC- V(7c,3a,3b) + V(a,b,c) = V(4a,4b,4c) = 64abc, V-Increment = 1/63 PAC [ 4, 0, -7 ]-1 [ 16, 21, 28 ] PAC [ -3, 4, 0 ] = [ 12, 16, 21 ] = M PAC [ 0, -3, 4 ] [ 9, 12, 16 ] PAC + (a,b,c) = (65,49,37) = M * (1,1,1) (sio) PAC V(259,195,147) + V(65,49,37) = V(260,196,148) PAC + (a,b,c) = (760, 573, 432) = M * (16,12,9) (sio) PAC maximal increment factor: 1 + 1/189 PAC + (a,b,c) = (1008, 760, 573) = M * (21,16,12) (sio) PAC maximal increment factor: 1 + 1/189 PAC + (a,b,c) = (1337,1008, 760) = M * (28,21,16) (sio) PAC maximal increment factor: 1 + 1/189 PAC- V(5a,5b,5c) + V(a,b,c) = V(3b,6c,7a) = 126abc, V-Increment = 1/125 PAC [ -5, 3, 0 ]-1 [ 25, 15, 18 ] PAC [ 0, -5, 6 ] = [ 42, 25, 30 ] = M PAC [ 7, 0, -5 ] [ 35, 21, 25 ] PAC + (a,b,c) = (58,97,81) = M * (1,1,1) (sio) PAC V(290,485,405) + V(58,97,81) = V(291,486,406) PAC + (a,b,c) = (1128, 1885, 1575) = M * (15,25,21) (sio) PAC V(5640,9425,7875) + V(1128,1885,1575) = V(5655,9450,7896) PAC maximal increment factor: 1 + 1/375 PAC + (a,b,c) = (1350, 2256, 1885) = M * (18,30,25) (sio) PAC maximal increment factor: 1 + 1/375 PAC- V(5a,5b,5c) + 3abc = V(4b,4c,8a) = 128abc, V-Increment = 3/125 PAC + (a,b,c) = (400, 504, 635) PAC maximal increment factor: 1 + 1/125 EXC: Exchange (Register Swap without tmp-memory) EXC- Dennis Shasha, Codes, Puzzles, and Conspiracy, Freeman (1992), Ex. 37 EXC exchange 2 registers, cyclic permute 7 registers (without minimal. proof) EXC- Peter van der Linden, Expert C Programming - Deep C Secrets, EXC Prenctice Hall (SunSoft Press) (1994), p 287-289 EXC The Limitations of Program Proofs EXC- sio: S_n is a subgroup of GL(n,F2). EXC S_n is generated by transpositions, GL(n,F2) is generated by 'exoring'. EXC Elementary Matrices E_i,j : E_i,j(n,m) = (n == i)(m == j) EXC I_i,j := I + E_i,j with i != j. EXC Generate GL(n,F2) = < {I_i,j | i != j} > EXC Problem: Is #transpositions = 3*#exorings for permutaion-matrices? EXC Problem: How long is a worst case minimal representation? EXC Infomation theoretic bound is: c * n^2 / log(n) EXC trivial upper bound (gauss elimination): n^2 EXC- Kiltinen, How few transpositions suffice? ... you already know EXC M. Mag. 67:1 (Feb 1994) EXC- Letter: M. Mag. 68:1 (Feb 1995) 79 (How few transpositions suffice?) EXC- Jacobson, Lectures in Abstract Algebra I (1951) p 36 (transpositions) EXC- Sury, An Integral Polynomial, M. Mag. 68:2 (Apr 1995) 134-135 EXC Let a1 < a2 < ... < a(n) integers EXC Product_(i>j) (a(i) - a(j))/(i - j) is an integer. EXC Product_(i>j) (x^(a(i)-a(j)) - 1)/(x^(i-j) - 1) is an integral polynom. ROT: roots of a polynomial ROT For any cubic with three different irrational real roots, ROT the roots are NOT elements of the real-radical field. (H"older, Isaacs) ROT- A new approach to solving the cubic, Cardan's solution revealed ROT The Math. Gazette 77 (1993) 354-359 ROT- H\"older. Math. Annalen 38, 307 (1891) ROT- I. M. Isaacs. Am. Math. Monthly 92, 571 (1985) ROT- B. K. Spearman, K. S. Williams; Characterization of Solvable Quintics ROT x**5 + a*x + b, AMM 101 (Dec. 1994) 986-992 ROT- H. B. Griffiths, A. E. Hirst; Cubic equations, or where did the ROT examination question come from?, AMM 101 (Feb. 1994) 151-161 ROT- Polynome 4. Grades mit ganzzahligen Null- und Extremstellen -- ROT pythagoreische Zahlentripel, PM 35 (1993) 39 ROT a^2 + b^2 = c^2 <-> f = (x-a-b)(x-a+b)(x+a-b)(x+a+b), ROT f'= 4x(x-c)(x+c) ROT- College Math. J. 29 (1998), no. 4, 276-277, Michael D. Hirschhorn ROT The Fundamental Theorem of Algebra, (homotopy proof) ROT- Fine & Rosenberg, The fundamental theorem of Algebra, Springer UTM ROT give at least 8 different proofs... KnT: Knight Tours KnT- Uncrossed Knight's Tours (MG9SA.15.6) KnT- Knights of the Square Table (MG8SA.14) KnT- Hamiltonian path, knight's tours (MG6SA.10.) KnT- W. Ahrens, Mathematische Unterhaltungen und Spiele I, 2nd Edition (1910) KnT Chap11: Roesselsprung 319-398, contains many references KnT- H. Schubert, Mathematische Mussestunden, Chap29: Roesselspruenge KnT- A. Kowalewski: Die Buntordnung Heft 1; JFM 48 (1922) 70 KnT see also: JFM 56 (1930) 97-98, JFM 45 356, 46 110, 47 59, 13 153-154 KnT- A. Kowalewski: Topologische Bedeutung von Buntordnungssystemen, KnT Wien Ber., 126 (1917) 963-1007; JFM 46 (1920) 110 KnT- A. De Polignac; Note sur la marche du cavalier dans un echiquier, KnT Bulletin de la Societe Mathematique de France, Paris, IX (1881) 17-24 KnT cyclic Knight's Tours n>=6, n even; JFM 13 (1881) 153-154 KnT- W.W. Rouse Ball, H.S.M. Coxter, Mathematical Recreations and Essays, KnT Chap6: Chess-Board Recreation (re-entrant paths on a chess-board) KnT- I. Stewart, Another fine math you've got me into, Freeman, 1992, KnT Chap7: Knights of the flat Torus KnT- D. E. Knuth, Leaper graphs (r,s-knights), is the 2*(r+s) square KnT Hamiltonian (for (r,s)=1)? Checked for r+s<=15. KnT The Math. Gazette 78 (1994) 274-296 KnT- Figured Tours (Knight, Rook), Math. Spectrum 25:1 (1992) 16-20 KnT special pattern: e.g. 6*6 knight tour with 1 and 4 in the same row KnT- Hamiltonische Linien, MU 24:3 (1978) 5-40 KnT dodecahedron unique circle, Petersen, knight tours KnT Sachs = Sci Am 10:1992 118-20 (I. Stewart) planar graph criteria KnT- roesselspruenge 6*6, Elem. Math. 43 (1988) 1-17, Zbl 734.05011 KnT- (4r+2)^2 roesselspruenge (rot 90 sym), PM 34:4 (1992) 178-180 KnT- geschlossene roesselspuenge (4r)^2, PM 34 (1992) 229-231 KnT- Number Pattern I, JoRM 10 (1977/78) 195-201, knight tour symmetry KnT- das Springerproblem, Informatik Spektrum 15:3 (1992) 169-172 KnT- Hamiltonian Checkerboards, Math. Mag. 57 (1984) 291-294 KnT ham. circles in C_n * C_m. When are left or up moves sufficient? KnT- Circuits in Directed Grids, Math Intell. 13:3 (1991) 40-43 KnT ham. circles in C_n * C_m. When are left or up moves sufficient? KnT- Which Rectangles have a Knight's Tour, M. Mag. 64:5 (1991)325-332 KnT complete inductive solution for Knight's circuits on rectangles KnT- Knight's circuits and tours, Ars Combinatoria 17A (1984) 145-167 KnT the Knight graph of the 4*4 and 3*6 board has crossing number 2. KnT every finite concatenation of 4*4 board (>1) has a knight's tour KnT- Die Rundtour der Narren, Math.Kabinet.3.3.7 KnT- Knight's Tour, mathematik-lehren, 53 (1992) 67-70 KnT- Touring M*N Boards (#hamiltonian tours), JoRM 27:4 (1995) 267-276 KnT- Conrad, Hindrichs, Morsy, Wegener; Solution of the knight's Hamiltonian KnT path problem on chessboards, Disc Appl Math 50 (1994) 125-134 KnT for n>=6 all s-t path exists (s, t have right colors) KnT- Kyek, Parberry, Wegener; Bounds on the number of knight's tours, KnT Tech. Report 555, (1994), Univ. Dortmund, Informatik II KnT 1.3535 <= Tours(n,n)^(1/n^2) <= 4 (lower bound for large n) KnT- M. Loebbing, I. Wegener; The number of Knight's Tours Equals KnT 33439123484294 - Counting with Binary Decission Diagrams, KnT Tech. Report 589, (1995), Univ. Dortmund, Informatik II (13 pages) KnT undirected tours, symmetry group has not been factored out KnT (Number is false, as not a multiple of 4.) KnT- M. Loebbing, I. Wegener; The number of Knight's Tours Equals KnT 33439123484294 - Counting with Binary Decission Diagrams, KnT Electronic J. Combin., 3 (1996) R5 (4 pages) (Number is false) KnT- B. McKay; Knight's Tours of an 8*8 Chessboard, Australian Nat. Univ. 1997 KnT The number of undirected tours is 13267364410532 and the number of KnT equivalence classes under rotation and reflection is 1658420855433. KnT There are 608233 equivalence classes of symmetrical tours. LiG: Little Games LiG- Philip D. Straffin, Graphing the Berry Patch, LiG UMAP J. 17:2 (Sum 1996) 117-122 LiG Berry Path Scramble: B W the game is drawn LiG W o B LiG B|W LiG- Philip D. Straffin, Position Graphs for Pong Hau K'i and Mu Torere LiG Math Mag 68 (1996) 382-386 (Corr fig. 69 (1996) 65) LiG Pong Hau K'i: B | B Mu Torere: W W W both games are drawn LiG (B starts) o W o B LiG W W B B B LiG- Philip D. Straffin - straffin@beloit.edu ABB: ABB- AMM = American Mathematical Monthly ABB- BdW = Bild der Wissenschaft ABB- CACM = Comunications of the ACM ABB- CFF = Cubism For Fun, Dutch Cubus Club ABB- CMJ = The College Mathematical Journal ABB- DMV = Deutsche Mathematische Vereinigung ABB- JFM = Jahrbuch "uber die Fortschritte der Mathematik ABB http://www.emis.de/projects/JFM/ ABB- JMD = Journal f"ur Didaktik der Mathematik ABB- JoRM = Journal of Recreational Mathematics ABB- JoCT = Journal of Combinatorial Theory ABB- LMS = London Mathematical Society ABB- LNiCT = Lecture Notes in Computer Science, Springer ABB- LNiM = Lecture Notes in Mathematics, Springer ABB- MAA = Mathematical Association of America ABB- MG = The Mathematical Gazette ABB- M In = Math Intell = The Mathematical Intelligenzer ABB- MM = M Mag = Mathematical Magazine ABB- MNU = Der mathematisch naturwissenschaftliche Unterricht ABB- MR = Mathematical Reviews ABB- MSem = Mathematische Semesterberichte ABB- MU = MU - Der Mathematikunterricht ABB- PM = Praxis der Mathematik ABB- SciAm = Scientific American ABB- TAOCP = The Art of Computer Programming (Knuth) ABB- TCS = Journal of Theoretical Computer Science ABB- TYCMJ = The Two Year College Mathemetical Journal ABB- Wurzel= (a german Math magazine) ABB- XXX = xxx Math Archive Front at http://front.math.ucdavis.edu/ ABB- ZBl = Zentralblatt der Mathematik ABB- ZOR = Zeitschrift fuer Operations Research