Foreword: Recreational Mathematics

FREEMAN DYSON

Hobbies are the spice of life. Recreational mathematics is a splendid hobby which young and old can equally enjoy. The popularity of Sudoku shows that an aptitude for recreational mathematics is widespread in the population. From Sudoku it is easy to ascend to mathematical pursuits that offer more scope for imagination and originality. To enjoy recreational mathematics you do not need to be an expert. You do not need to know the modern abstract mathematical jargon. You do not need to know the difference between homology and homotopy. You need only the good old nineteenth-century mathematics that is taught in high schools, arithmetic and algebra and a little geometry. Luckily for me, the same nineteenth-century mathematics was all that I needed to do useful calculations in theoretical physics. So, when I decided to become a professional physicist, I remained a recreational mathematician. This foreword gives me a chance to share a few of my adventures in recreational mathematics.

The articles in this collection, The Best Writing on Mathematics 2011, do not say much about recreational mathematics. Many of them describe the interactions of mathematics with the serious worlds of education and finance and politics and history and philosophy. They are mostly looking at mathematics from the outside rather than from the inside. Three of the articles, Doris Schattschneider's piece about Maurits Escher, the Fergusons' piece on mathematical sculpture, and Dana Mackenzie's piece about packing a circle with circles, come closest to being recreational. I particularly enjoyed those pieces, but I recommend the others too, whether they are recreational or not. I hope they will get you interested and excited about mathematics. I hope they will tempt a few of you to take up recreational mathematics as a hobby.

I began my addiction to recreational mathematics in high school with the fifty-nine icosahedra. The Fifty-Nine Icosahedra is a little book published in 1938 by the University of Toronto Press with four authors, H.S.M. Coxeter, P. DuVal, H. T. Flather, and J. F. Petrie. I saw the title in a catalog and ordered the book from a local bookstore. Coxeter was the world expert on polyhedra, and Flather was the amateur who made models of them. The book contains a complete description of the fifty- nine stellations of the icosahedron. The icosahedron is the Platonic solid with twenty equilateral triangular faces. A stellation is a symmetrical solid figure obtained by continuing the planes of the twenty faces outside the original triangles. I joined my school friends Christopher and Michael Longuet-Higgins in a campaign to build as many as we could of the fifty-nine icosahedra out of cardboard and glue, with brightly colored coats of enamel paint to enhance their beauty. Christopher and Michael both went on later to become distinguished scientists. Christopher, now deceased, was a theoretical chemist. Michael is an oceanographer. In 1952, Michael took a holiday from oceanography and wrote a paper with Coxeter giving a complete enumeration of higher-dimensional poly- topes. Today, if you visit the senior mathematics classroom at our old school in England, you will see the fruits of our teenage labors grandly displayed in a glass case, looking as bright and new as they did seventy years ago.

My favorites among the stellations are the twin figures consisting of five regular tetrahedra with the twenty vertices of a regular dodecahedron. The twins are mirror images of each other, one right-handed and the other left-handed. These models give to anyone who looks at them a vivid introduction to symmetry groups. They show in a dramatic fashion how the symmetry group of the icosahedron is the same as the group of 120 permutations of the five tetrahedra, and the subgroup of rotations without reflections is the same as the subgroup of 60 even permutations of the tetrahedra. Each of the twins has the symmetry of the even permutation subgroup, and any odd permutation changes one twin into the other.

Another book which I acquired in high-school was An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright, a wonderful cornucopia of recreational mathematics published in 1938. Chapter 2 contains the history of the Fermat numbers, F_n = 2² + 1, which Fer- mat conjectured to be all prime. Fermat was famously wrong. The first four Fermat numbers are prime, but Euler discovered in 1732 that F₅ = 2³² + 1 is divisible by 641, and Landry discovered in 1880 that F₆ = 2⁶⁴ + 1 is divisible by 274,177. Since fast computers became available, many larger Fermat numbers have been tested for primality, and not one has been found to be prime.

Hardy and Wright provide a simple argument to explain why F₅ is divisible by 641. Since 641 = 1 + 5 a = 2⁴ + 5⁴ with a = 2⁷, we have

F₅ = 2⁴a⁴ + 1 = (1 + 5a — 5⁴)a⁴ + 1 = (1 + 5a)a⁴ + (1 — (5a)⁴),

which is obviously divisible by 1 + 5a. I was always intrigued by the question of whether a similarly elementary argument could be found to explain the factorization of F₆. Sixty years later, I found the answer. This was another joyful piece of recreational mathematics. The answer turned out to depend on a theorem concerning palindromic continued fractions. If a and q are positive integers with a < q, the fraction a/q can be expressed in two ways as a continued fraction:

where the partial quotients p. are positive integers. The fraction is palindromic if p. = p_n+1— for each j. The theorem says that the fraction is palindromic if and only if a² + ( — 1)ⁿ is divisible by q.

Landry's factor of F₆ has the structure

where f is a factor of 2²⁴ — 1, so that

and

The partial quotients of the fraction a/q are (17, 1, 3, 1, 5, 5, 1, 3, 1, 17). A beautiful palindrome, and the palindrome theorem tells us that a² + 1 = qu with u integer. Therefore F₆ = 1 + (gq — a)² = q(g²q — 2ga + u) is divisible by q. I was particularly proud of having discovered the palindrome theorem, until I learned that I had been scooped by the French mathematician Joseph Alfred Serret, who published it in 1848. To be scooped by one of the lesser-known luminaries of the nineteenth century is part of the game of recreational mathematics. On another occasion I was scooped by Riemann, but that is a long story and I do not have space to tell it here.

When I was an undergraduate at Cambridge University, I was intrigued by a famous discovery of the Indian prodigy Ramanujan concerning the arithmetical properties of the partition function p(n). This discovery was lovingly described in Hardy and Wright's An Introduction to the Theory of Numbers. For any positive integer n, p(n) is the number of ways of expressing n as a sum of positive integer parts. Ramanujan discovered that p(5k 1 4) is always divisible by 5, p(7k + 5) is divisible by 7, and p(11k + 6) is divisible by 11. I wanted to find a way of actually dividing the partitions of 5k + 4 into 5 equal classes, and similarly for 7 and 11. I found a simple way to do the equal division. The “rank” of any partition is defined as the biggest part minus the number of parts. The partitions of any n can be divided into 5 rank classes, putting into class m the partitions that have rank of the form 5j 1 m for m = 1, 2, 3, 4, 5. I found to my delight that the 5 rank classes of partitions of 5 k 1 4 are exactly equal. The same trick works for 7 but not for 11. It was easy to check numerically that the rank classes were equal for the partitions of 5k 1 4 and 7k 1 5 all the way up to 100, but I failed to find a proof. I also conjectured the existence of another property of a partition that would do the same job for the partitions of 11k + 6, and I called that hypothetical property the “crank.”

Ten years later, Oliver Atkin and Peter Swinnerton-Dyer succeeded in proving the equality of the rank-classes for 5k 1 4 and 7k 1 5, and 45 years later, Frank Garvan and George Andrews identified the crank. Garvan and Andrews not only found the crank, but also proved that it provides an equal division of the crank classes for all three cases, 5, 7, and 11. More recently, in 2008, Andrews made another discovery as beautiful as Ramanujan's original discovery. Andrews was looking at another function S(n) enumerating the smallest parts of partitions of n. S(n) is defined as the sum, over all partitions of n, of the number of smallest parts in that partition. Andrews discovered and also proved that S(5k 1 4) is divisible by 5, S(7k 1 5) is divisible by 7, and S(13k + 6) is divisible by 13. The appearance of 13 instead of 11 in this statement is not a typographical error. It is a big surprise and adds a new mystery to the mysteries discovered by Ramanujan.

The question now arises whether there exists another property of partitions with their smallest parts, like the rank and the crank, allowing us to divide the partitions with their smallest parts into 5 or 7 or 13 equal classes. I conjecture that such a property exists, and I offer a challenge to readers of this volume to find it. To find it requires no expert knowledge. All that you have to do is to study the partitions and smallest parts for a few small values of n, and make an inspired guess at the property that divides them equally. A second challenge is to prove that the guess actually works. To succeed with the second challenge probably requires some expert knowledge, since I am asking you to beat George Andrews at his own game.

My most recent adventure in recreational mathematics is concerned with the hypothesis of Decadactylic Divinity. Decadactylic is Greek for ten-fingered. In days gone by, serious mathematicians were seriously concerned with theology. Famous examples were Pythagoras and Descartes. Each of them applied his analytical abilities to the elucidation of the attributes of God. I recently found myself unexpectedly following in their footsteps, applying elementary number theory to answer a theological question. The question is whether God has ten fingers. The evidence in favor of a ten-fingered God was brought to my attention by Norman Frankel and Lawrence Glasser. I hasten to add that Frankel and Glasser were only concerned with the mathematics, and I am solely responsible for the theological interpretation. Frankel and Glasser were studying a sequence of rational approximations to ? discovered by Derek Lehmer. For each integer k, there is a rational approximation [Rj(k)/R₂(k)] to P, with numerator and denominator defined by the identity

The right-hand side of this identity has interesting analytic properties which Frankel and Glasser explored. The approximations to ? that it generates are remarkably accurate, beginning with 3, 22/7, 22/7, 335/1 13, for k = 1, 2, 3, 4. Frankel and Glasser calculated the first hundred approximations to high accuracy, and found to their astonishment that the kth approximation agrees with the exact value of ? to roughly k places of decimals. I deduced from this discovery that God must calculate as we do, using arithmetic to base ten, and it was then easy to conclude that He has ten fingers. It seemed obvious that no other theological hypothesis could account for the appearance of powers of ten in the approximations to such a transcendental quantity as P.

Unfortunately, I soon found out that my theological breakthrough was illusory. I calculated precisely the magnitude of the error of the kth Lehmer approximation for large k, and it turned out that the error does not go like 10² but like Qj^k, where Q = 9.1197…is a little smaller than ten. For large k, the approximation is in fact only accurate to 0.96k places of decimals, where 0.96 is the logarithm of Q to base ten. Q is defined as the absolute value of the complex number q = 1 1 (2pi/ ln(2)). When we are dealing with complex numbers, the logarithm is a many-valued function. The logarithm of 2 to the base 2 has many values, beginning with the trivial value 1. The first nontrivial value of log₂(2) is q. This is the reason why q determines the accuracy of the Lehmer approximations. This calculation demonstrates that God does not use arithmetic to base ten. He uses only fundamental units such as ? and ln(2) in the design of His mathematical sensorium. The number of His fingers remains an open question.

Two of these recreational adventures were from my boyhood and two from my old age. In between, I was doing mathematics in a more professional style, finding problems in the understanding of nature where elegant nineteenth-century mathematics could be usefully applied. Mathematics can be highly enjoyable even when it is not recreational. I hope that the articles in this volume will spark readers' interest in digging deeper into some aspect of mathematics, whether it is puzzles and games, history of mathematics, mathematics education, or perhaps studying for a professional degree in mathematics. The joys of mathematics are to be found at all levels of the game.