Synopses & Reviews
This classic work offers scores of stimulating, mind-expanding games and puzzles: arithmetical and geometrical problems, chessboard recreations, magic squares, map-coloring problems, cryptography and cryptanalysis, much more. "A must to add to your mathematics library" The Mathematics Teacher. Index. References for Further Study. Includes 150 black-and-white line illustrations.
Synopsis
Classic treasury of arithmetical and geometrical problems, chessboard recreations, cryptography, and much more.
Synopsis
This classic work offers scores of stimulating, mind-expanding games and puzzles: arithmetical and geometrical problems, chessboard recreations, magic squares, map-coloring problems, cryptography and cryptanalysis, much more. Includes 150 black-and-white line illustrations.
About the Author
H. S. M. Coxeter: Through the Looking Glass
Harold Scott MacDonald Coxeter (1907-2003) is one of the greatest geometers of the last century, or of any century, for that matter. Coxeter was associated with the University of Toronto for sixty years, the author of twelve books regarded as classics in their field, a student of Hermann Weyl in the 1930s, and a colleague of the intriguing Dutch artist and printmaker Maurits Escher in the 1950s.
In the Author's Own Words:
"I'm a Platonist — a follower of Plato — who believes that one didn't invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."
"In our times, geometers are still exploring those new Wonderlands, partly for the sake of their applications to cosmology and other branches of science, but much more for the sheer joy of passing through the looking glass into a land where the familiar lines, planes, triangles, circles, and spheres are seen to behave in strange but precisely determined ways."
"Geometry is perhaps the most elementary of the sciences that enable man, by purely intellectual processes, to make predictions (based on observation) about the physical world. The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry."
"Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." — H. S. M. Coxeter
Table of Contents
I ARITHEMETICAL RECREATIONS
To find a number selected by someone
Prediction of the result of certain operations
Problems involving two numbers
Problems depending on the scale of notation
Other problems with numbers in the denary scale
Four fours problems
Problems with a series of numbered things
Arithmetical restorations
Calendar problems
Medieval problems in arithmetic
The Josephus problem. Decimation
Nim and similar games
Moore's game
Kayles
Wythoff's game
Addendum on solutions
II ARITHEMETICAL RECREATIONS (continued)
Arithmetical fallacies
Paradoxical problems
Probability problems
Permutation problems
Bachet's weights problem
The decimal expression for 1/n
Decimals and continued fractions
Rational right-angled triangles
Triangular and pyramidal numbers
Divisibility
The prime number theorem
Mersenne numbers
Perfect numbers
Fermat numbers
Fermat's Last Theorem
Galois fields
III GEOMETRICAL RECREATIONS
Geometrical fallacies
Geometrical paradoxes
Continued fractions and lattice points
Geometrical dissections
Cyclotomy
Compass problems
The five-disc problem
Lebesgue's minimal problem
Kakeya's minimal problem
Addendum on a solution
IV GEOMETRICAL RECREATIONS (continued)
Statical games of position
Three-in-a-row. Extension to p-in-a-row
Tessellation
Anallagmatic pavements
Polyominoes
Colour-cube problem
Squaring the square
Dynamical games of position
Shunting problems
Ferry-boat problems
Geodesic problems
Problems with counters or pawns
Paradromic rings
Addendum on solutions
V POLYHEDRA
Symmetry and symmetries
The five Platonic solids
Kepler's mysticism
"Pappus, on the distribution of vertices"
Compounds
The Archimedean solids
Mrs. Stott's construction
Equilateral zonohedra
The Kepler-Poinsot polyhedra
The 59 icosahedra
Solid tessellations
Ball-piling or close-packing
The sand by the sea-shore
Regular sponges
Rotating rings of tetrahedra
The kaleidoscope
VI CHESS-BOARD RECREATIONS
Relative value of pieces
The eight queens problem
Maximum pieces problem
Minimum pieces problem
Re-entrant paths on a chess-board
Knight's re-entrant path
King's re-entrant path
Rook's re-entrant path
Bishop's re-entrant path
Route's on a chess-board
Guarini's problem
Latin squares
Eulerian squares
Euler's officers problem
Eulerian cubes
VII MAGIC SQUARE
Magic squares of an odd order
Magic squares of a singly-even order
Magic squares of a doubly-even order
Bordered squares
Number of squares of a given order
Symmetrical and pandiagonal squares
Generalization of De la Loubère's rule
Arnoux's method
Margossian's method
Magic squares of non-consecutive numbers
Magic squares of primes
Doubly-magic and trebly-magic squares
Other magic problems
Magic domino squares
Cubic and octahedral dice
Interlocked hexagons
Magic cubes
VIII MAP-COLOURING PROBLEMS
The four-colour conjecture
The Petersen graph
Reduction to a standard map
Minimum number of districts for possible failure
Equivalent problem in the theory of numbers
Unbounded surfaces
Dual maps
Maps on various surfaces
"Pits, peaks, and passes"
Colouring the icosahedron
IX UNICURSAL PROBLEMS
Euler's problem
Number of ways of describing a unicursal figure
Mazes
Trees
The Hamiltonian game
Dragon designs
X COMBINATORIAL DESIGNS
A projective plane
Incidence matrices
An Hadamard matrix
An error-corrrecting code
A block design
Steiner triple systems
Finite geometries
Kirkman's school-girl problem
Latin squares
The cube and the simplex
Hadamard matrices
Picture transmission
Equiangular lines in 3-space
Lines in higher-dimensional space
C-matrices
Projective planes
XI MISCELLANEOUS
The fifteen puzzle
The Tower of Hanoï
Chinese rings
Problems connected with a pack of cards
Shuffling a pack
Arrangements by rows and columns
Bachet's problem with pairs of cards
Gergonne's pile problem
The window reader
The mouse trap. Treize
XII THREE CLASSICAL GEOMETRICAL PROBLEMS
The duplication of the cube
"Solutions by Hippocrates, Archytas, Plato, Menaechmus, Apollonius, and Diocles"
"Solutions by Vieta, Descartes, Gregory of St. Vincent, and Newton"
The trisection of an angle
"Solutions by Pappus, Descartes, Newton, Clairaut, and Chasles"
The quadrature of the circle
Origin of symbo p
Geometrical methods of approximation to the numerical value of p
"Results of Egyptians, Babylonians, Jews"
Results of Archimedes and other Greek writers
"Results of European writers, 1200-1630"
Theorems of Wallis and Brouncker
"Results of European writers, 1699-1873"
Ap