Richard K. Guy (1916-2020)

Mathematician Richard K. Guy died on March 9, 2020, at the age of 103. Active to the very end, he made contributions to a wide range of mathematical fields, particularly number theory, combinatorics, graph theory, and geometry. Much of his work was inspired by his passion for recreational mathematics and games and seasoned with his wry sense of humor.

I first met Richard in 1986 when he hosted, at the University of Calgary, the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. He was kind enough to review the manuscript of my first book, The Mathematical Tourist: Snapshots of Modern Mathematics (W.H. Freeman, 1988), for accuracy and had many helpful suggestions.

I wrote about some of Richard's work in my second book, Islands of Truth: A Mathematical Mystery Cruise (W.H. Freeman, 1990).

A Shortage of Small Numbers

Mathematician Richard K. Guy is a collector. He patiently and painstakingly searches far and wide for the unexpected and quirky among the family of whole numbers. He looks for unusual patterns.

Identifying patterns and asking the right questions are two of the most important ingredients of mathematical research. Lamentably, there's no foolproof recipe for generating good questions and no formula for recognizing whether an observed pattern will lead to a significant new theorem or is merely a lucky coincidence. Until a mathematical proof is constructed to settle the question, a mathematician must rely on fallible empirical evidence.

Consider the remarkable sequence of integers 31, 331, 3331, 33331, 333331, 3333331. Each of these integers is a prime number, that is, divisible only by itself and the number one. Is the sequence's next number, 33333331, a prime?

The answer is yes. Unfortunately, the pattern falls apart with the succeeding number, 333333331, which turns out to be the product of 17 and 19,607,843. A promising pattern is slain by a cruel counterexample.

Guy's specimens are all instances of sequences that depend on the whole-number value, n, of some parameter. In the first example, n represents the number of threes in each integer. The pattern works for n = 1, 2, 3, 4, 5, 6, and 7, but fails when n = 8.

For any sequence that depends on the value of n, experience shows that sometimes a pattern persists, but frustratingly often the pattern is simply a figment of the smallness of the values of n for which the example has been worked out.

For many years, Guy has been trying to encapsulate his findings in the form of a universal law. So far, the best he can manage is the statement, "There aren't enough small numbers to meet the many demands made of them." He calls it the Strong Law of Small Numbers.

"It is the enemy of mathematical discovery," Guy argues."When you notice a mathematical pattern, how do you know it's for real? We are easily led astray by spurious patterns, which do not continue as the numbers get larger. On the other hand, genuine patterns are often hidden by a few exceptions near the beginning."

As an instance of the misleading behavior of small numbers, Guy cites the fact that 10 percent of the first 100 numbers are perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, and 100). On the basis of this pattern, one could conjecture that 10 percent of the first 1,000 numbers would be perfect squares, but some quick calculations show the conjecture to be ill-founded. Only about 3 percent are perfect squares.

On the other hand, the statement that all prime numbers are odd is almost true. The only exception occurs right at the beginning. In a sense, as Guy points out, two is the "oddest" prime.

Guy's tussles with such aberrant numerical behavior have led him to formulate an important, elegantly simple theorem: "You can't tell by looking." The theorem, he insists,"has wide application, outside mathematics as well as within," and it can be "proved by intimidation."

Many of Guy's examples, gathered from numerous sources, concern prime numbers. One of the most famous involves numbers of the form P = 2²ⁿ + 1. When n = 0, P = 2²⁰ = 2¹ + 1 = 2 + 1 = 3, a prime number. For n = 1, P = 2²¹ = 5, another prime. For n = 2, P = 17; for n =3, P = 257; for n = 4, P = 65,537. The numbers 3, 5, 17, 257, and 65,537 are all primes.

Does the pattern continue? Mathematician Pierre de Fermat (1607-1665) thought so when, more than three centuries ago, he proposed that all numbers of the form 2²ⁿ + 1 are prime. Alas, when n = 5, the number is not a prime but the product of 641 and 6,700,417. The Strong Law strikes again.

A tastier problem concerns slicing a round cake into pieces—not in the conventional way but in a fashion that probably only a mathematician would find appetizing. The idea is to define a certain number of points, n, along the cake's rim, then to slice the cake so that the cuts join all possible pairs of points. The question is how many separate pieces of cake are created by the cuts.

The answer for n = 1 (one point on the rim) is, of course, one. With only one point, no cuts can be made. When n = 2, a cut joins two points, dividing the cake into two pieces. For n = 3, the number of pieces, p, is four; for n = 4, p = 8; for n = 5, p = 16. The sequence 1, 2, 4, 8, 16 looks familiar. Does the pattern hold for larger numbers of points?

Joining in every possible way a number of points marked on a circle slices the circle into a certain number of regions. How does the number of regions depend on the number of points?

…
The answer is no. The number of pieces for n = 6 is 31. The sequence continues 57, 99, 163, 256, 386, 562, 794, 1,093... . However, it is possible to work out a formula that gives you every term in this sequence (n⁴ – 6n³ + 23n² – 18n + 24)/24.

Pennies show up in a low-budget hexagon construction project. Seven pennies can be laid out to form a hexagon in which each side is two pennies long. A hexagon with each side made up of three pennies consists of a total of 19 pennies.

As the length of the hexagon's side goes from 1 penny to 5 pennies, the total number of pennies involved in each case is 1, 7, 19, 37, and 61. The numbers of this sequence are called "hex" numbers.

Adding together the so-called hex numbers produces the partial sums 1, 8, 27, 64, 125, which are perfect cubes.

Interestingly, 1 + 7 = 8, 8 + 19 = 27, 27 + 37 = 64, 64 + 61 = 125. Each of these partial sums is a perfect cube. For example, 8 = 2³ = 2 x 2 x 2, 27 = 3 x 3 x 3, and so on. Does this pattern continue when larger hexagons built from pennies are included?

The pattern is genuine. It's handy to regard the nth hex number as comprising the three faces at one corner of a cubic stack of unit cubes. Expressed algebraically, the (n + 1)th hex number, 1 + 6 + 12 + … + 6n (or equivalently, 3n² + 3n + 1), when added to n³, gives (n + 1)³.

To explain the pattern shown by partial sums of hex numbers, it's useful to regard the nth hex number as comprising the three faces at one corner of a cubic stack of n³ unit cubes.

Another problem involves writing down a string of positive integers, say, from 1 to 11 (although you can go as high as you like). Cross out every second number; that is, all the even numbers. Then add up the remaining numbers, writing down the answers, or partial sums, along the way.

Write down the positive integers, delete every second number, and form the partial sums of those remaining. The result is a sequence of perfect squares.

The resulting sequence (1, 4, 9, 16, 25, 36) consists of consecutive squares of positive integers. Does the pattern continue if the string consists of more than the first 11 positive integers? Yes. This is a mathematically proper way of generating a sequence of squares.

What happens if you delete every third number, compute the partial sums, then delete every second partial sum and calculate new partial sums? This time, the resulting sequence consists of consecutive cubes.

Delete every third number, calculate the partial sums, then delete every second partial sum. The result is a sequence of perfect cubes.

Guy's collection of problems illustrates the major role that disinformation in the form of misleading patterns plays in the pursuit of mathematical truth. Guy and his fellow collectors could fill many volumes with examples of how the Strong Law of Small Numbers has led to significant mathematical theorems, or has misled investigators into looking for theorems that don't exist.

It's all part of the trial-and-error effort that characterizes much of mathematical research.

See also "The Ladies' Diary" and "Powerful Sequences."

Reference:
"The Strong Law of Small Numbers," Richard K. Guy in The American Mathematical Monthly, Vol. 95, October 1988, pages 697-712.