Coxeter, Harold Scott Macdonald; Walter William Rouse Ball;
Mathematical Recreations and Essays
Courier Dover Publications, 1987, 448 pages
ISBN 0486253570, 9780486253572
topics: | math | puzzle | history | geometry |
p.6 [source: Bachet, problem VI, p. 84]
Ask anyone to select a number n less than 60. Request him to perform the
following operations.
(i) Divide by 3 and mention the remainder; say a.
(ii) Divide by 4, remainder b;
(iii) divide by 5, remainder c.
Then the number selected is the remainder obtained by dividing 40a + 45b
+ 36c by 60.
[Why this works:
Let: A = 40 = multiple of 4,5, that is 1 more than some multiple of 3;
B = 45 = multiple of 3,5, that is 1 more than some multiple of 4;
C = 36 = multiple of 3,4, +1 of some multiple of 5;
[can have more than three multiples; their product p must be
> chosen number n]
p = 60 = product of 3,4,5...
N = Aa + Bb + Cc + ...
now, 40%3 =1, so 40a%3 = a.
45b, 36c etc are divisible by 3.
Hence N = Aa + Bb + Cc divided by p has remainder = a
Since n % 3 is also = a,
hence, N-n is divisible by 3,4,5... hence by p.
since p>n, N % p = n
[summary of the general argument]
Let a', b', c', ... be a series of numbers prime to one another, such that
their product p is > the chosen number n.
Let a, b, c, ... be the remainders when n is divided by a', b', c',
Find a number A which is a multiple of the product b'c'd'...
and which is +1 of a multiple of a'. Find a number B which is a
multiple of a'c'd' ... and which exceeds by +1 a multiple of b', and find
analogous numbers C, D, ...
in general, if the numbers a', b', c' are small, the corresponding numbers A, B, C can be found by inspection. I proceed to show that n is equal to the remainder when Aa + Bb + Cc + ... is divided by p. Let N = Aa + Bb + Cc + ... Now A % a' = 1, therefore Aa % a' = a. The other terms, Bb, Cc etc. are divisible by a'. Hence, if N is divided by a', the remainder is a. Also if n is divided by a', the remainder is a. Therefore N - n is divisible by a', b', ... - hence by p. Hence N = kp + n. Since n is less than p, hence if N is divided by p, the remainder 18 n. In the above example from Bachet, a',b', c' = 3,4,5. So p = 60, By inspection, A = 40 [multiple of 4*5, s.t. A-1 is divisible by 3] B = 45, C = 36. EXAMPLE 1: 3,4,7 (<84) : N = (28a + 21b + 36c) % p guess 33: 0,1,5 : N = 0+ 21 + 180 = 201 % 84 = 33 EXAMPLE 2 (for all 2-digit nums): 3,5,7 (p=105)-> 70.a + 21.b + 15.c guess 33: 0,3,5 -> N=0 + 63 + 75 = 138 % 105 = 33 guess 47 : 2,2,5 -> N= 140 + 42 + 75 %105 = 35 + 12 = 47
son of a small farmer in Cabut, Vermont, USA. While still less than six years old he showed extraordinary powers of mental calculation, which were displayed in a tour in America. Two years later he was brought to England, where he was repeatedly examined by competent observers. He could instantly give the product of two numbers each of four digits, but hesitated if both numbers exceeded 10,000. Among questions asked him at this time were to raise 8 to the 16th power; in a few seconds he gave the answer 281,474,976,710,656, which is correct. He was next asked to raise the numbers 2, 3, ... 9 to the lOth power: and he gave the answers so rapidly that the gentleman who was taking them down was obliged to ask him to repeat them more slowly; but he worked less quickly when asked to raise numbers of two digits like 37 or 59 to high powers. He gave instantaneously the square roots and cube roots (when they were integers) of high numbers, e.g. the square root of 106,929 and the cube root of 268,336,125; such integral roots can, however, be obtained easily by various methods. More remarkable are his answers to questions on the factors of numbers. Asked for the factor of 247.4SS. he replied 941 and 263; asked for the factors of 171,395, he gave 5, 7, 59, and 83; asked for the factors of 36,083, he said there were none. He, however, found it difficult to answer questions about the factors of numbers higher than 1,000,000. His power of factorizing high numbers was exceptional, and depended largely on the method of two-digit terminals... His English and American friends, raised money for his education, and he was sent in succession to the Lycee Napoleon in Paris and Westminster School in London. With education his calculating powers fell off, and he lost the frankness which when a boy had charmed observers.
p.357 Bidder was born in 1806 at Moreton Hampstead, Devonshire, where his father was a stone-mason. At the age of six he was taught to count up to 100, but though sent to the village school, learnt little there, and at the beginning of his career was ignorant of the meaning of arithmetical terms and of numerical symbols. Equipped solely with this knowledge of counting, he taught himself the results of addition, subtraction, and multiplication of numbers (less than 100) by arranging and rearranging marbles, buttons, and shot in patterns. In after-life he attached great importance to such concrete representations, and believed that his arithmetical powers were strengthened by the fact that at that time he knew nothing about the symbols for numbers. p.358 Seven years old: he heard a dispute between two of his neighbours about the price of something which was being sold by the pound, and to their astonishment remarked that they were both wrong, mentioning the correct price. After this exhibition the villagers delighted in trying to pose him with arithmetical problems. Nine years old: His father started to take him around to fairs, exhibiting his powers. Some questions : What is the interest on £11,111 for 11,111 days at 5 per cent. a year 1 answer, £16,911 11s., given in one minute. How many hogsheads of cider can be made from a million of apples, if 30 apples make one quart 1 answer, 132 hogsheads 11 gallons] quart and 10 apples over, given in 35 seconds... At age 11, some Cambridge mathematicians raised a fund and started his education. But a few months later, his father took him out again as a showman. Age 14 (1819): In addition to answering questions on products of numbers and the number of specified units in given quantities, he was, after 1819, ready in finding square roots, cube roots, etc., of high numbers, it being assumed that the root is an integer. [many others also, e.g. Colburn] Qs: Find two numbers whose difference is 12 and whose product, multiplied by their sum, is equal to 14,560? answer, 14 and 26. Find a number whose cube less 19 multiplied by its cube shall be equal to the cube of 6: answer, 3, given instantly. Eventually at Edinburgh university, some members persuaded his father to let him study, and he graduated as a Civil Engineer. He rose to high distinction in the profession. 358 In 1818, in the course of a tour, young Bidder was pitted against Colburn (ages 13 and 14), and on the whole proved the abler calculator. 358 In factorizing numbers he was less successful than Colburn, and was generally unable to deal at sight with numbers higher than 10,000. 359
Bidder retained his power of rapid mental calculation to the end of his life, and as a constant parliamentary witness in matters connected with engineering it proved a valuable accomplishment. Just before his death an illustration of his powers was given to a friend who, talking of then recent discoveries, remarked that if 36,918 waves of red light which only occupy one inch are required to give the impression of red, and if light travels at 190,000 miles a second, how immense must be the number of waves which must strike the eye in one second to give the impression of red. "You need not work it out," said Bidder; "the number will be 444,433,651,200,000." 362 members of the Bidder family have also shown exceptional powers of a similar kind as well as extraordinary memories. Of Bidder's elder brothers, one became an actuary, and on his books being burnt in a fire, he rewrote them in six months from memory, but, it is said, died of consequent brain fever Bidder's eldest son, a lawyer of eminence, was able to multiply together two numbers each of fifteen digits. Neither in accuracy nor rapidity was he equal to his father, but, then, he never steadily and continuously devoted himself to developing his abilities in this direction. He remarked that in his mental arithmetic, he worked with pictures of the figures, and said, "If I perform a sum mentally it always proceeds in a visible form in my mind; indeed I can conceive no other way possible of doing mental arithmetic ": this, it will be noticed, is opposed to his father's method. Two of his children, one son and one daughter, representing a third generation, inherited analogous powers. p.362
Jacques Inaudi was born in 1867 at Onorato in Italy. Be was employed in early years as a shepherd, and spent the long idle hours in which he had no active duties in pondering on numbers, but used for them no concrete representations such as pebbles. 366 His calculating powers first attracted notice about 1873. Shortly afterwards his elder brother sought his fortune as an organ-grinder in Provence, and young Inaudi, accompanying him, came into a wider world, and earned a few coppers for himself by street exhibitions of his powers. In 1880 he visited Paris, where he gave exhibitions... He was then ignorant of reading and writing: these arts he subsequently acquired. A performance before the general public rarely lasted more than 12 minutes... A normal programme included the subtraction of one number of twenty-one digits from another number of twenty-one digits: the addition of five numbers each of six digits: the multiplying of a number of four digits by another number of four digits : the extraction of the cube root of a number of nine digits, and of the fifth root of a number of twelv..e digits: the determination of the number of seconds in a period of time, and the day of the week on which a given date falls. Of course, the questions were put by members of the audience. To a professional calculator these problems are not particularly difficult. As each number was announced, Inaudi repeated it slowly to his assistant, who wrote it on a blackboard, and then slowly read it aloud to make sure that it was right. Inaudi then repeated the number once more. By this time he had generally solved the problem, but if he wanted longer time he made a few remarks of a general character, which he was able to do without interfering with his mental calculations. Throughout the exhibition he faced the audience: the fact that he never even glanced at the blackboard added to the effect.
Bidder did not visualize a number like 984 in symbols, but thought of it in a concrete way as so many units which could be arranged in 24 groups of 41 each. It should also be observed that he, like Inaudi, relied largely on the auditory sense to enable him to recollect numbers. "For my own part," he wrote, in later life, "though much accustomed to see sums and quantities expressed by the usual symbols, yet if I endeavour to get any number of figures that are represented on paper fixed in my memory, it takes me a much longer time and a very great deal more exertion than when they are expressed or enumerated verbally." For instance, suppose a question put to find the product of two numbers each of nine digits, if they were "read to me, I should not require this to be done more than once; but if they were represented in the usual way, and put into my hands, it would probably take me four times to peruse them before it would be in my power to repeat them, and after all they would not be impressed so vividly on my imagination. " ... 361-2 It is probable that the majority of oalculating prodigies rely on the speech-muscles as well as on the eye and the ear to help them to recollect the figures with which they are dealing. It was formerly believed that they all visualized the numbers proposed to them, and certainly some have done so. Inaudi, however, trusted mainly to the ear and to articulation. Bidder also relied partly on the ear, and when he visualized a number, it was not as a coIlection of digits, but as a concrete coIlection of units divisible, if the number was composite, into definite groups. Ruckle relied mainly on visualizing the numbers. 367-8 So it would seem that there are different types of the memories of calculators. Inaudi could reproduce mentaIly the sound of the repetition of the digits of the number in his own voice, and was confused, rather than helped, if the numbers were shown to him in writing. The articulation of the digits of the number also seemed necessary to enable him fully to exhibit his powers, and he was accustomed to repeat the numbers aloud before beginning to work on them - the sequence of sounds being important.
Numbers were ever before Inaudi : he thought of little else, he dreamt of them, and sometimes even solved problems in his sleep. His memory was exceIlent for numbers, but normal or subnormal for other things. At the end of a seance he could repeat the questions which had been put to him and his answers, involving hundreds of digits in all. Nor was his memory in such matters limited to a few hours. Once, eight days after he had been given a question on a number of twenty-two digits, he was unexpectedly asked about it, and at once repeated the number. He was repeatedly examined, and we know more of his work than of any of his predecessors, with the possible exception of Bidder. 368 Bidder gave an account of the processes he had discovered and used in a lecture given by him in 1856 to the Institution of Civil Engineers. [Ball remarks that these may be his perfected analysis rather than may not be how he actually did them, esp as a child; modern cognitive scientists like Ericsson would completely agree.] 369
[see Barlow, F. Mental prodigies. (1952). , and Smith, S.B. The great mental calculators (1983)] [by Arthur Jensen, School of Education, UC Berkeley: "Speed of Information Processing in a Calculating Prodigy"; INTELLIGENCE, v.14, 259-274 (1990) Material added for relevance] Shakuntala Devi was born in Bangalore, India, in 1940, to a 15-year-old mother and 61-year-old father, who was a circus acrobat and magician. Devi travelled with him since she was 3, performing card tricks, from which she cultivated her facility with numbers. Her talent in this sphere was manifested early; at age 5 she could already extract cube roots quickly in her head, and she soon began supporting herself and the rest of her family as a stage performer, travelling throughout India billed as a calculating prodigy. Even before she was in her teens, she began travelling around the world, performing numerical feats, usually before audiences in colleges and universities. In addition to mathematics texts such as Figuring: The joy of numbers (1978), Shakuntala Devi has also written a book on The world of homosexuals (1977).
It seems hard to believe, but the following is reported in the Guinness Book of Records (1982), which has a reputation for the authenticity of its claims: Mrs. Shakuntala Devi of India demonstrated the multiplication of two 13-digit numbers of 7,686,369,774,870 x 2,465,099,745,779 picked at random by the Computer Department of Imperial College, London on 18 June 1980, in 28 s. Her correct answer was 18,947,668,177,995,426,462,773,730. An article in the New York Times (November 10, 1976, cited in Smith, 1983, p. 306) reported that Shakuntala Devi added the following four numbers and multiplied the result by 9,878 to get the (correct) answer 5,559,369,456,432: 25,842,278 111,201,721 370,247,830 55,511,315 She was reported to have done this calculation in "20 seconds or less." At Southern Methodist University, in 1977, Devi extracted the 23rd root of a 201-digit number in 50 s. Her answer--546,372,891--was confirmed by calculations done at the U.S. Bureau of Standards by the Univac 1101 computer, for which a special program had to be written to perform such a large calculation (Smith, 1983).
in 1988, Devi visited the San Francisco Bay Area, when I had the opportunity to observe a demonstration she gave at Stanford University before an audience filled with mathematicians, engineers, and computer experts, who had come with their electronic calculators or printouts of large problems that had been submitted to the University's main-frame computer. I was curious, fast of all, to see if Devi had the kind of autistic personality so commonly associated with such unusual mental feats. Also, I wanted to measure her performance times myself, to see if they substantiated the astounding claims I had read of her calculating prowess. But mainly, if the claims proved authentic, I hoped I could persuade her to come to Berkeley to be tested in my chronometric laboratory, so we could measure her basic speed of information processing on a battery of elementary cognitive tasks (ECTs) for which the results could be compared with the reaction time (RT) data we had obtained on the same ECTs in large samples of students and older adults. Indeed, Devi kindly consented to come to my laboratory and spent about 3 h taking various tests. In addition, she spent some 2 h with me, discussing her life and work. At her Stanford appearance, Shakuntala Devi, in a colorful silk sari, sat at a table in front of the blackboard in a lecture hall. The demonstration lasted almost 90 rain. (Engaging in such intense mental activity beyond that length of time, Devi said, she begins to feel tired.) Problems involving large numbers were written on the blackboard by volunteers from the audience, many of whom knew of Devi's reputation and had brought along computer printouts with the problems and answers. Devi would turn around to look at a problem on the blackboard, and always in less than 1 rain (but usually in just a few seconds) she would state the answer, or in the case of solutions involving quite large numbers she would write the answer on the blackboard. [AJ and his wife are on row 1, measuring her response time with a stopwatch. Answers are always correct, and it almost never takes more than a minute. ] When I handed Devi two problems, each on a separate card, thinking she would solve first one, then the other, my wife was taken by surprise, as there was hardly time to start the stopwatch, so quick was Devi's response. Holding the two cards side-by-side, Devi looked at them briefly and said, "The answer to the first is 395 and to the second is 15. Right?" Right, of course! (Her answers were never wrong.) Handing the cards back to me, she requested that I read the problems aloud to the audience. They were: (a) the cube root of 61,629,875 (= 395), and (b) the 7th root of 170,859,375 (= 15). I was rather disappointed that these problems seemed obviously too easy for Devi, as I had hoped they would elicit some sign of mental strain on her part. After all, it had taken me much longer to work them with a calculator. cuberoot of a 10-digit number (13343) : 10s 8th root of 14-digit number (468) : 10s 7th root of 33-digit number (462957) : 40s Devi also does noninteger exact fractional roots almost as fast as integer roots-- averaging about 3 to 4 s longer... [not irrationals]. Apparently she does not apply a standard algorithm uniformly to every problem of a certain type, such as square roots, or cube roots, or powers. Each number uniquely dictates its own solution, so to speak. The presence of commas only interferes with the "natural" (and virtually automatic) dissolution of the number in Devi's mind. I have since learned from an Indian professor that commas are not used in India's number system, and it seems likely that their interfering effect for Devi could stem in part from her intensive childhood experience in working with large numbers lacking commas or any other form of triplet grouping. Calendar calculations: To determine if anything besides sheer calculation enters Devi's thought process while she is doing calendrical calculations, I called out "January 30, 1948," to which she instantly answered, "That was a Friday-- and the day that our great leader Mahatma Gandhi was assassinated." Obviously her calendrical calculating does not entirely usurp her other memory or thought processes. Personal Characteristics. The first thing most observers would notice about Devi is that her general appearance and demeanor are quite the opposite of the typical image of the withdrawn, obsessive, autistic savant, so well portrayed by Dustin Hoffman in the recent motion picture, Rain Man. Devi comes across as alert, extraverted, affable, and articulate. Her English is excellent, and she also speaks several other languages. She has the stage presence of a seasoned performer, and maintains close rapport with her audience. At an informal reception after her Stanford performance, I noticed that among strangers she was entirely at ease, outgoing, socially adept, self-assured, and an engaging conversationalist. To all appearances, the prodigious numerical talent resides in a perfectly normal and charming lady. She is divorced and has a daughter attending college in England, who, Devi remarks with mock dismay, uses a computer in her science and math courses. In fact, Devi claims none of her relatives has ever shown any mathematical talent.
method used by Bidder (and Colburn): when it was exactly divisible. for example, in dividing (say) 25,696 by 176, he first argued that the answer must be a number of three digits, and obviously the left-hand digit must be 1. Next he had noticed that there are only 4 numbers of two digits (namely, 21, 46, 71, 96) which when multiplied by 76 give a number which ends in 96. Hence the answer must be 121, or 146, or 171, or 196; and experience enabled him to say without calculation that 121 was too small and 171 too large. Hence the answer must be 146. If he felt any hesitation, he mentally multiplied 146 by 176 (which he said he could do "instantaneously"), and thus checked the result. It is noticeable that when Bidder, Colburn, and some other calculating prodigies knew the last two digits of a product of two numbers, they also knew, perhaps subconsciously, that the last two digits of the separate numbers were necessarily of certain forms.
Multiply 79,532,853 by 93,758,479: asked by Schumacher, answered in 54 seconds. In answer to a similar request to find the product of two numbers each of twenty digits, he took 6 minutes; to find the product of two numbers each of forty digits, he took 364 MATHEMATICAL RECREATIONS AND ESSAYS [cn. 40 minutes; to find the product of two numbers each of a hundred digits, he took 8 hours 45 minutes. Gauss thought that perhaps on paper the last of these problems could be solved in half this time by a skilled computator. Like all these calculating prodigies, he had a wonderful memory, and an hour or two after a performance could repeat all the numbers mentioned in it. Be had also the peculiar gift of being able after a single glance to state the number (up to about 30) of sheep in a Hock, of books in a case, and so on; and of yisualizing and recolle£ting a large number of objects. For instance, after a second's look at some dominoes he gave the sum (1 17) of their points; asked how many letters were in a certain line of print chosen at random in a quarto page, he instantly gave the correct number (63); shown twelve digits, he had in half a second memorized them and their positions so as to be able to name instantly the particular digit occupying any assigned place. Dase - cooperated with mathematicians - compiled table of natural logarithms - computed the first 1,005,000 numbers to 7 places of decimals in his spare time from 1844 to 1847, when occupied by the Prussian survey.
Review of chapters from:
J. S. Frame, Bull. Amer. Math. Soc. Volume 46, Number 3 (1940), 211-213.
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:
Bachet, Ozanam, Montucla |
To find a number selected by someone |
Prediction of the result of certain operations |
Problems involving two numbers | Digit questions |
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-BOARD RECREATIONS
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