topics: |  math | puzzle | history | geometry

Review of chapters from:
J. S. Frame, Bull. Amer. Math. Soc. Volume 46, Number 3 (1940), 211-213.

CONTENTS
 1. ARITHMETICAL RECREATIONS
    includes arithmetical recreations whose interest is mainly
    historical rather than arithmetical. Some of these are of the "think of a
    number" type, others involve digit notations, and still others are tricks
    with cards or games with counters.
    TOC:
        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 problem | 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

 2. ARITHMETICAL RECREATIONS (continued)
    problems of probability derangements and arrangements, decimal
    expansions, rational triangles, finite arithmetics, D. H. Lehmer's number
    sieve for prime factors, and concludes with a discussion of p'erfect
    numbers, Mersenne's numbers, and Fermais theorem.
        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
 3. GEOMETRICAL RECREATIONS
    mainly geometrical fallacies and paradoxes, problems in dissection,
    cyclotomy and area-covering. The deltoid solution to Kakeya's minimal
    problem, erroneously attributed to Kakeya (p. 100) was really suggested
    by Professors Osgood and Kabota according to Question 39, American
    Mathematical Monthly (1921), p. 125.
        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 99 | Addendum on a solution
 4. GEOMETRICAL RECREATIONS (continued)
    statical and dynamical games of position.  Among topics discussed are
    some extensions of the game of three in a row, tessellations of the
    plane, problems with moving counters, and the effect of cutting a Möbius
    strip in various ways.
        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
 5. POLYHEDRA
    comprehensive elementary discussion of the relations between the faces,
    edges, and vertices and the associated angles of the regular solids and
    the Archimedean solids, which is well illustrated by good
    figures. Stellated polyhedra, solid tessellations, and the kaleidoscope
    each receive some attention. The use of the term Platonic for the regular
    solids might be questioned since they were known before Plato.
        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
 6. CHESS-BOARDRECREATIONS
    recreations associated with the chessboard and with magic
    squares. Similar problems with dominoes and with magic cubes are also
    discussed.
        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 | Routes on a chess-board
        | Guarini's problem | Latin squares | Eulerian squares | Euler's
        officers problem | Eulerian cubes
 7. MAGIC SQUARES
        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 Loubere'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
 8. MAP-COLOURING PROBLEMS
    general theory of the four-colour problem more elaborately than the
    earlier editions of this book, mentions briefly such matters as
    orientable surfaces and dual maps, and more fully the seven-colour
    mapping problem on the torus, and finally considers various colouring
    problems on the regular polyhedra.
        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
 9. UNICURSAL PROBLEMS
    mazes and other similar problems whose solutions depend on the unicursal
    tracing of a route through prescribed points (nodes) over various given
    paths.
        Euler's problem | Number of ways of describing a unicursal figure |
        Mazes | Trees | The Hamiltonian game | Dragon designs
10. COMBINATORIAL DESIGNS
    certain combinatorial problems known under the title of Kirkman's
    school-girl problems, and ends with a similar problem about arranging
    members of a bridge club at tables so that different members shall play
    together in successive rubbers.
        A projective plane | Incidence matrices | An Hadamard matrix | An
        error - correcting 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
11. MISCELLANEOUS PROBLEMS
    the Fifteen Puzzle, the Tower of Hanoï, Chinese Rings, and various
    mathematical card
	The fifteen puzzle | The Tower of Hanoi | 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
12. THREE CLASSICAL GEOMETRICAL PROBLEMS
    famous classical problems concerning the duplication of the cube,
    trisection of an angle, and quadrature of the circle.
        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 symbol it | Geometrical
        methods of approximation to the numerical value of Pi | 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 |
        Approximations by the theory of probability
13. CALCULATING PRODIGIES
    calculating prodigies which introduces over a dozen famous mental
    calculators beginning with Jedediah Buxton and Thomas Fuller in the
    eighteenth century, and including two American calculators Zerah Colburn
    and Trueman Henry Safford, and gives something of their histories and the
    type of problems they could solve.
              John Wallis, 1616-1703 ; Buxton, circ. 1707-1772 ; Fuller,
              1710-1790; Amp,6re ; Gauss, Whately ; Colburn,1804-1840 ;
              Bidder, 1806-1878 ; Mondeux, Mangiamele ; Dase, 1824-1861 ;
              Safford, 1836-1901 ; Zamebone, Diamandi, Ruckle ; Inaudi, 1867- ]
        | Types of memory of numbers
        | Bidder's analysis of methods used
              [ Multiplication ; Digital method for division and factors ;
              Square roots. Higher roots ; Compound interest ; Logarithms ]
        | Alexander Craig Aitken
14. CRYPTOGRAPHY AND CRYPTANALYSIS
    cryptography and cryptanalysis written by Dr.  Abraham Sinkov. It
    presents in easily understandable form the chief elements in a
    cryptographic system, and gives various possible ways for attempting to
    solve such a cipher.
        Cryptographic systems | Transposition systems | Columnar
        transposition | Digraphs and trigraphs | Comparison of several
        messages | The grille | Substitution systems | Tables of frequency |
        Polyalphabetic systems | The Vigenere square | The Playfair cipher |
        Code | Determination of cryptographic system | A few final remarks