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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)
from http://www.mathematik.uni-bielefeld.de/~sillke/gardner/lit
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
http://www.ms.uky.edu/~lee/ma502/gardner5/gardner5.html
for a link of which problem appears in which text.

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