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Martin Gardner's New Mathematical Diversions from Scientific American

Martin Gardner

Gardner, Martin;

Martin Gardner's New Mathematical Diversions from Scientific American

Simon and Schuster, 1966, 253 pages

ISBN 0671209892, 9780671209896

topics: |  math | puzzle

Some puzzles:

  1.  The binary system:
      Leibniz (1646- 1716) felt that binary digits represent 0=nonbeing or
      nothing; and 1 as being or substance.  This is remarkably similar to
      much of the analysis in India, where shunya or kha can mean emptiness
      or the sky or ether.

      Leibniz was incidentally also a proponent of the Scythian theory of
      languages, which is a predecessor to the idea of Indo-European
      languages.  He was also very interested in Chinese studies, and indeed,
      the idea for binary numbers came to him from an exposure to the Chinese
      "Yijing" diagram, which has similar metaphysical attributes, but is not
      used for computation a such.

  2.  Group theory and braids
      Game of "Tangloids" invented by Piet Hein (poet, friend of Bohr). Cut a
      plaque out of heavy paper in a coat-of-arms shape, punch three holes,
      and pass three sash cords through these, tying the other end at a
      chair.  Now by rotating the plaque through the strings one can braid
      the rope in six ways. Doing it again creates further braids.
      Game involves untangling them by weaving, not rotating.  Theorem: can
      always be done if there are an even number of rotations.  Intuition:
      because even rotations sort of undo each other.

  3.  Eight problems
      problem 4: A group of cadets is marching in a 50m x 50m formation.
      Their pet dog starts from the middle of the back, runs to the front,
      turns back instantaneously, and runs back to the middle, at constant
      speed.  In the meantime, the formation has moved 50m.  How far did the
      dog run?

DETAILED CONTENTS  (listing all puzzles, journal references, and links)
1 The binary system
2 Group Theory and Braids
    - A Random Ladder Game: Permutaions, Eigenvalues, and Convergence of
	Markov Chains, [College Math J. 23:5 (1992) 373-385]
    - PSL(2,7) = PSL (3,2), MR 89f:05094 elegant proof
3 Eight Problems
  3.1. Acute Dissection
     Triangle cut into seven acute ones (or eight acute isosceles)
     acute dissection of a square (8), pentagram (5), Greek cross (20)
     NU-Configurations in tiling the square, Math Comp 59 (1992)195-202
     tiling a square with integer triangles
  3.2 How Long is a "Lunar"?
      radius of the sphere, such that surface = volume
  3.3 The Game of Googol (probability)
   3.3. maximising the chance of picking the largest number
   3.3. maximizing the value of the selected object (proposed be Cayley)
     3.3.a On a Problem of Cayley, Scripta Mathematica (1956) 289-292
     3.3.b An Optimal Maintenance Policy of a Discrete-Time Markovian
     3.3.b Deterioration System, Comp. & Math. with Appl 24 (1992) 103-108
     3.3.c A Secretary Problem with Restricted Offering Chances and Random
     3.3.c Number of Applications, Comp. & Math. with Appl 24 (1992) 157-162
     3.3.d On a simple optimal stopping problem, Disc. Math. 5 (1973) 297-312
     3.3.e Stopping time techniques for analysts and probabilits (L. Egghe)
     3.3.e LMS LNS 100
     3.3.f Algebraic Approach to Stopping Variable Problems, JoCT 9 (1970)
     3.3.f 148-161, distributive lattices <-> stopping variable problems
     3.3.g secretary problem, Wurzel 27:12 (1993) 259-264
     3.3.h Ferguson, Who solved the secretary problem?
     3.3.h Statistical Science 4 (1989) 282-296
     3.3.i Freeman, the secretary problem and its extensions: a review
     3.3.i International Statistical Review 51 (1983) 189-206
  3.4 Marching Cadets and a Trotting Dog
  3.5 Barr's Belt
  3.6 White, Black and Brown (logic)
  3.7 The Plane in the Wind
  3.8 What Price Pets? (linear Diophantine equation)
4 The Games and Puzzles of Lewis Carroll
5 Paper Cutting
5. theorem of Pythagoras, dissection proof,
6 Board Games
7 Packing Spheres
  7.a figurative numbers, square, triangular, tetrahedral
8 The Transcendental Number Pi
9 Victor Eigen: Mathemagician
10 The Four-Color Map Theorem
11 Mr. Apollinax Visits New York
12 Nine Problems
  12.1 The Game of Hip
  12.1. two color the 6*6 square, s. t. there is no monochromatic square
  12.1. the number of different squares in the n*n square is n²(n²-1)/12
     12.1.a enumerating 3-, 4-, 6-gons with vertices at lattice points,
     12.1.a Crux Math 19:9 (1993) 249-254
  12.2 A Switching Puzzle: change two cars with a locomotive (circle and tunnel)
  12.3 Beer Signs on the Highway (calculus, speed, time, distance)
  12.4 The Sliced Cube and the Sliced Doughnut (geometry)
       cut the cube (regular hexagon), doughnut (two intersecting cirles)
  12.5 Bisecting Yin and Yang (geometry)
      12.5.a Bisection of Yin and of Yang, Math. Mag. 34 (1960) 107-108
  12.6 The Blue-Eyed Sisters (probability)
  12.7 How old is the Rose-Red City? (linear equations)
  12.8 Tricky Track (logic, reconstruct a table)
  12.9 Termite and 27 Cubes (hamiltonian circle, parity)
13 Polyominoes and Fault-Free Ractangles
  13.a On folyominoes and feudominoes, Fib. Quart. 26 (1988) 205-218
  13.b Rookomino (Kathy Jones) JoRM 23 (1991) 310-313
  13.c Rookomino (K. Jones) JoRM 22 (1990) 309-316 (Problem 1756)
  13.d Polysticks, JoRM 22 (1990) 165-175
  13.e Fault-free Tilings of Rectangles (Graham) The Math. Gardner 120-126
14 Euler's Spoilers: The Discovery of an Order-10 Graeco-Latin Square
   Universal Algebra and Euler's Officer Problem, AMM 86 (1979)466-473
15 The Ellipse
  15.a robust rendering of general ellipses and elliptic arcs,
  15.a ACM Trans. on Graphics, 12:3 (1993) 251-276
16 The 24 Color Squares and the 30 Color Cubes (MacMahon)
   12261 solutions of the 4*6 rectangle, 3*8 is impossible
17 H. S. M. Coxeter
    Coxeter's book Introduction to Geometry 1961
    appl. of the M"obius band, contructions for 257, 65537 gon
    Morley's triangle, equal bisectors - Steiner-Lehmus Thm
  17.a Angle Bisectors and the Steiner-Lehmus Thm, Math. Log 36:3 (1992)1&6
  17.b equal external bisectors, not isoscele, M. Math. 47 (1974) 52-53
  17.c A quick proof of a generalized Steiner-Lehmus Thm,
  17.c Math Gaz. 81:492 (Nov. 1997) 450-451
  17.h Morley's triangle (D.J.Newman's proof), M In 18:1 (1996) 31-32.
       kissing circles, Soddy's formular - Descartes' Circle Theorem
  17.d Circles, Vectors, and Linear Algebra, Math. Mag. 66 (1993) 75-86
       semiregular tilings of the plane, the 17 cristallographic groups
	tilings of Escher: Heaven-Hell, Verbum
  17.e The metamorphosis of the butterfly problem (Bankoff)
  17.e Math. Mag. 60 (1987) 195-210  (47 refs)
  17.f A new proof of the double butterfly theorem, M. Mag. 63 (1990) 256-7
  17.g Schaaf, Bibliography of Rec. Math. II.3.3 The butterfly problem
18 Bridg-it and Other Games
     winning Bridg-it, pairing stategy (Shannon switching game)
     Connections (ASS, 1992) = Bridg-it board: connect or circle
  18.b Directed switching games on graphs and matroids, JoCT B60 (1986)237
  18.c Shannon switching games without terminals,  draft (I), see II, III
  18.c Graphs and Combinatorics 5 (1989) 275-82 (II), 8 (1992) 291-7 (III)
19 Nine More Problems
  19.1 Collating the Coins (coin moving xyxyx -> xxxyy)
  19.2 Time the Toast (optimal shedule)
  19.3 Two Pentomino Posers
  19.3. 6*10 Rectangle with all pentominoes touch the border (unique)
  19.4 A Fixed Point Theorem
  19.5 A Pair of Digit Puzzles (cryptarithms)
  19.6 How did Kant Set His Clock (calculus, time, speed)
  19.7 Playing Twenty Questions when Probability Values are Known
  19.7. Huffman coding, data compression
  19.8 Don't Mate in One (chess)
  19.9 Find the Hexahedrons
  19.9. there are seven varieties of convex hexahedrons (six faces)
20 The Calculus of Finite Differences
  20.d Symmetry Types of Periodic Sequences, Illionois J. of Math.
  20.d 5:4 (Dec 1961) 657-665, appl. to music and switching theory
  20.a generating two color necklaces, Disc. Math. 61 (1986) 181-188
  20.b Generating Necklaces, J. of Algorithms 13:3 (1992) 414

Many of the chapters have many sub-problems; see the index at
for a link of which problem appears in which text.

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This article last updated on : 2014 Jun 15