# Remembering Richard Kenneth Guy: Games and Taking on Mountains

This essay is dedicated to the memory of Richard K. Guy (1916-2020) whose life story and accomplishments display a variety of aspects of the complicated landscape of finding fulfilling work on the part of individuals, the concern of society with nourishing those who can produce and use mathematics, and the community of people who have devoted their lives to encourage the flourishing of mathematics.. …

Joseph Malkevitch

York College (CUNY

### Introduction

What background is required to do research in mathematics? Who produces new mathematical results and why do such individuals do mathematical research? While mathematics has become a profession and there are individuals who, when asked what they do for a living, respond that they are mathematicians, in the grand scheme of human history of mathematics, its being a profession is of quite recent origin. This essay is dedicated to the memory of Richard K. Guy (1916-2020), whose life story and accomplishments display a variety of aspects of the complicated landscape of finding fulfilling work on the part of individuals, the concern of society with nourishing those who can produce and use mathematics, and the community of people who have devoted their lives to encourage the flourishing of mathematics.

Certainly there are many people who today would say they are “mathematicians” rather than that they teach mathematics, are professors of mathematics or work in industry or government as mathematicians. This group is also notable for having typically earned a doctorate degree in mathematics. Note, however, that many of the individuals who are thought of as great mathematicians of the 19th century and early 20th century who were British or Irish (e.g. Arthur Cayley, James Joseph Sylvester, Augustus de Morgan, William Rowan Hamilton) did not earn doctorate degrees. This was because of the uneven history of where doctorate degrees were awarded. Whereas doctorate degrees were awarded in France, Italy, and Germany as early as the 17th century and even in the United States in late 19th century, well into the 20th century it was common for British mathematicians not to get doctorate degrees. Galileo (1564-1642), though known as a physicist by the public, taught mathematics. And there were early universities in Paris and Bologna where mathematics was part of the curriculum and scholars of mathematics were trained.

However, one can also pursue a “career” involving mathematics if one studies mathematics education, physics, chemistry, economics, biology, computer science, statistics, and a multitude of other “academic” preparations for being employed and one builds on one’s mathematical skills or abilities. I will return to this issue later after taking a look at Richard Guy’s career, noteworthy for the variety of experiences he had in pursuit of his interest and love of mathematics but whose only doctorate was an honorary doctorate. Another thread I will touch on is the view that when mathematics leaps forward it is through the work of relatively young practitioners, and Richard Guy lived and actively worked into old age. He died at 103! Guy was an inspiration for those who love mathematics and are nervous that contributing to mathematics is done only by the relatively young.

### Richard Guy–a brief biography

Richard Kenneth Guy was born in England in the town of Nuneaton, which is in the part of the country known as Warwickshire.

Figure 1 (Photo of Richard Guy by Thane Plambeck, from Palo Alto, CA, courtesy of Wikipedia.)

Eventually he made his way to Cambridge University, whose structure is that one studies at one of various “competing” colleges. In the area of mathematics perhaps the best known of the Cambridge Colleges is Trinity College, partly because this was the college associated with Isaac Newton (1642-1727) but also because such other famous mathematicians as Isaac Barrow (1630-1677), Arthur Cayley (1821-1895), G.H. Hardy (1837-1947), Srinivasa Ramanujan (1887-1920), Bertrand Russell (1892-1970), William Tutte (1917-2002), and William Timothy Gowers. However, Guy went to Caius College, though the “official” name of the college is Gonville and Caius College. For a young country like America it is hard to realize that Gonville and Caius was founded in 1348 (well before Christopher Columbus voyaged to the New World). Some other mathematicians or mathematical physicists who were students at Caius and whose name you might recognize were:

George Green (1793-1841), John Venn (1834-1903), Ronald Fisher (1890-1962), Stephen Hawking (1942-2018)

Figure 2 (Photo of Gonville-Caius College at Cambridge, courtesy of Wikipedia.)

John Conway (1937-2020), who died recently was another person well known to the mathematics community who attended Caius College. Conway’s and Richard Guy’s careers crossed many times! Perhaps Guy’s most influential writing was his joint book with Conway and Elwyn Berlekamp, *Winning Ways*.

Figure 3 (Photo of John Horton Conway.)

Guy served during World War II in the Royal Air Force where he did work related to forecasting the weather–a critical part of the British war effort.

Subsequently, at various times Guy worked as a teacher or took courses at various places. Guy married Nancy Louise Thirian in 1940. She died in 2010. Richard and Louise had three children. His son Michael Guy also did important work in mathematics. Guy relocated from Britain to Singapore in 1950 and taught mathematics at the University of Malaya, where he worked for 10 years. After leaving Singapore he spent time at the new Indian Institute of Technology in Delhi. In 1965, he “settled down” by moving to Canada where he taught in the mathematics department of the University of Calgary. Although he eventually “retired” from the University of Calgary he continued to show up at his office regularly until not long before he died. In Calgary he supervised several doctoral students. The following information is drawn from the Mathematics Genealogy Project

Roger Eggleton, University of Calgary, 1973 Nowakowski, Richard, University of Calgary, 1978 Dan Calistrate, University of Calgary, 1998 Jia Shen, University of Calgary, 2008

In turn Eggleton had at least 8 doctoral students and Nowakowski had at least 10.

While at Calgary, Guy helped organize various mathematics conferences, some of which took place at a location in the mountains not far from Calgary.

Figure 4 (Photo of the Banff Center for the Arts and Creativity, in Banff, Alberta [Canada], site for some conferences organized by Richard Guy. Photo courtesy of Wikipedia.)

Richard Guy and Louise both enjoyed the exercise and challenge of hiking. Well after many people give up heading out on the trail, he and Louise continued to hike. Guy also continued to hike after his wife died. A point of pride for Guy was that a Canadian Alpine Hut was named for him and Louise. The hut is not open year-round, but in a policy that would have pleased Guy and his wife, “to avoid pressuring the bear habitat, the Guy Hut will be closed between May 1 and November 30 annually.”

Richard Guy eventually got a doctorate degree but it was an honorary degree awarded by the University of Calgary in 1991, relatively late in his long and very productive career as an active mathematician. Much of Guy’s career was in academia but the qualifications for a career in academia vary considerably from one country to another and from one era to another. While most people who are famous for their contributions to mathematics in recent years have earned doctorate degrees, there are some famous examples, including Richard Guy, who found different routes to becoming distinguished contributors to mathematics. Guy, having been born in 1916, presents a relatively common case because of the history of granting the doctorate degree in Great Britain. Let me say something about this situation.

For those attempting to understand the history of mathematics, the development of particular branches of mathematics–geometry, combinatorics, non-associative algebras, partial differential equations, Markov chains, entire functions (a topic in the theory of functions of a complex variable), Banach spaces, etc. a wonderfully useful tool is the previously noted Mathematics Genealogy Project. On this website, searches for British mathematicians yield mathematical ancestors but often the names of the people shown as an ancestor were not a doctoral thesis supervisor but a person who “tutored” or was a primary influence for someone who went on to obtain a reputation in mathematical research but who did not have a doctorate degree. Another historical source of information about individual mathematicians (though not Guy, yet) and various aspects of mathematics is the MacTutor History of Mathematics Archive. Also see this fascinating history of the doctorate in Britain.

### Richard Guy’s collaborators

MathsciNet lists a large number of people whom Guy collaborated with as an author. He edited quite a few books as well. As mentioned earlier, his best-known book, *Winning Ways*, was a joint project with John Horton Conway and Elwyn Berlekamp.

Figure 5 (Photo of Elwyn Berlekamp, courtesy of Wikipedia.)

Figure 6 (Photo of John Conway)

But during his long life, Guy had many collaborators. Some of these were “traditional” academics who taught at colleges and universities but two of his collaborators stand out for having benefited mathematics somewhat differently: Paul Erdős (1913-1996) and Martin Gardner (1914-2010).

Paul Erdős was a mathematical prodigy, born in Hungary. He not only showed talent for mathematics as a youngster, but also skill at developing new mathematical ideas and proving new results while quite young. (Musical prodigies show a split, too: those who at an early age can wow one with how well they play the violin or piano but also prodigies like Mozart and Mendelssohn who at amazingly young ages compose new and beautiful music.) Some prodigies continue to wow with their accomplishments as they age but some die young–Mozart died at 35, Schubert at 31, and Mendelssohn at 38. One can only be sad about the wonderful music they would have created had they lived longer, but we can be thankful for all of the wonderful music they did create.

Paul Erdős did not die young but his career did not follow a usual path. Rather than have a career at one academic institution, he became noted for becoming a traveling ambassador for innovative mathematical questions and ideas, particularly in the areas of number theory, combinatorics, and discrete geometry (including graph theory questions). Erdős’s creativity and brilliance created interest in how closely connected mathematicians were to Erdős via chains of joint publications (books and jointly edited projects do not count). This leads to the concept of *Erdős number*. Erdős has Erdős number 0. Those who wrote a paper with Erdős have Erdős number 1 and those who wrote a paper with someone who wrote a paper with Erdős have Erdős number 2. Richard Guy’s Erdős number was one, and some might argue that since he wrote 4 papers with Erdős that his Erdős number is 1/4. If you have written a joint paper with a mathematician and are curious to know what your Erdős number is you can find out by using a free “service” of the American Mathematical Society: Find your Erdős number!

Figure 7 (Photo of Paul Erdős, Ron Graham (recently deceased) Erdős number 1, and Fan Chung, Ron Graham’s wife–who also has Erdős number 1, courtesy of Wikipedia)

One can compute an Erdős number for people who lived in the distant past. Thus, Carl Gauss has an Erdős number of 4. (Enter names at the link above in the form Euler, Leonard. For this particular name, though, no path can be found!)

Martin Gardner (1914-2010) stands out as perhaps the most unusual promoter of mathematics for the general public, the science community, and research mathematicians in history. Not only did Gardner never earn a doctorate degree in mathematics, he did not earn any degree in mathematics. He did earn a degree in philosophy from the University of Chicago in 1936. Eventually he became involved with writing a mathematical recreations column for *Scientific American *magazine, and over a period of time these columns were published, anthologized, and augmented to create a constant stream of books that were well-received by the general public. Given the title of the column that Gardner wrote for Scientific American, Mathematical Games, it is not surprising that he drew on the work of Berlekamp, Conway, and Guy. These individuals made suggestions to Gardner for columns to write, and he in turn, when he had ideas for a column/article, often drew on their expertise to “flesh out” his ideas. Over the years, an army of *Scientific American *readers (including me) were stimulated by Gardner’s columns and books to make contributions to problems that Gardner wrote about or to pursue careers related to mathematics. Readers of Gardner’s books and columns also branched out and learned about mathematical topics that they had not seen in mathematics they had learned in school.

Figure 8 (Photo of Martin Gardner, courtesy of Wikipedia.)

### Grundy’s game

It may be helpful to give an example of a combinatorial game that is easy to describe and which is perhaps new to some readers. One of my favorite combinatorial games is known as Grundy’s game, named for Patrick Michael Grundy (1917-1959). Like many combinatorial games it is played using a pile of identical coins, beans, or in Figure 9 line segments. The game is an example of an *impartial* game, which contrasts with partisan games where each player has his/her own pieces, as in chess or checkers.

Figure 9 (An initial configuration for Grundy’s game.)

Figure 10 (One can get to this position in Grundy’s game from the position in Figure 9 by a legal move.)

Grundy’s game works as follows. The game starts with a single pile of stones (here shown as short line segments in a column). The players alternate in making moves. A move consists of taking an existing pile of segments and dividing it into two piles of unequal size. Thus, for the pile in Figure 9, the player making a move from this position could move to a position where there would be one pile of 6 segments and the other of 2 segments. This would be a legal move. A player having to make a move from the position in Figure 10 can NOT make a move based on the pile with two segments, because the only way to divide this pile would create two equal sized piles, which is not allowed. However, there are several moves from this position. Divide the the first pile into piles of size 5 and 1 or of size 4 and 2. The game terminates when a player can’t make a move. John Conway promoted this rule of “normal” play in combinatorial games, namely, “if you can’t move, you lose.” If the termination rule is “if you can’t move, you win,” usually called *misère play*, then an analysis of optimal play is much harder, for reasons that are clear.

Rather remarkably, whatever the rules of a two-person impartial game, it turns out that the game is equivalent to playing the game of Nim (see below) with a single pile. Charles Bouton in 1902 found a way for any game of Nim, with one or many piles, to determine from the size of these piles whether the player making the next move would win or lose. In the language of combinatorial game theory, any position with legal moves that remain can be classified as being in an N position or a P position. An N position is one in which the next player to play can win with proper perfect play, while a P position is one where the previous player. having had this position, could win with proper perfect play.

Note, however, for a given game, knowing if a position is an N or a P position does not tell a player how to play optimally–that is, what moves to make to guarantee a win no matter what responses the opponent may make, or, if a position is hopeless, what moves to make to force the opponent to make as many moves as possible to win, and perhaps force a mistake to turn a losing position into a win. For a particular “board” of a combinatorial game that has moves that can be made, it can be thought of as either an N position or a P position.

*Nim* is played starting from a position with one or more piles of identical objects. A move consists of selecting a single pile and removing any number of objects from the pile, including removing all of the objects from the pile. Thus, treating the segments of Figure 9 as a Nim position, one could remove, 1, 2, 3, 4, 5, 6, 7, or 8 segments from the single pile. This position in Figure 9 is an N position. The player who moves from this position can win by removing all 8 segments. For Figure 10 can you see, with more thought, why this is also an N position. Bouton’s solution to checking a Nim position to see its status as an N position or P position involves representing the number of items in each pile as a binary number–binary number notation represents any positive integer using only the symbols 0 and 1. Summarizing what Bouton showed:

a. From a P position, every “move” leads to an N position

b. From an N position, there is some move that leads to a P position

c. P positions have a “value” of 0; N positions have a value that is positive.

The theory that showed all combinatorial two-person impartial games can be shown to have a single Nim value is due independently to the German-born mathematician Roland Percival Sprague (1894-1967) and English mathematician Patrick Michael Grundy (1917-1959). Remarkably, Richard Guy, also independently showed the same result.

Let us come back to Grundy’s game. In light of the discussion above, every initial position of *n* (positive integer) segments (coins, beans) has a Nim value, that is, a single number that represents the “value” of that position. Suppose one looks at the sequence of Nim values for Grundy’s game starting from a pile of *n* objects where each computation can take a fair amount of work. Starting with *n* = 0 this sequence looks like:

0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 3, 2, 1, 3, 2, 4, 3, …..

Remember that the meaning of a zero is that if it is currently your move, you have lost the game, because you have no move with optimal play by your opponent that will allow you a victory. If the value in the sequence is not zero this means that you have a position that allows you to make a move which will result in a win on the next move or allows you to move to a position from which your opponent cannot win!

What can one say about this sequence? The pattern seems rather irregular and even if you saw a much larger swath of this sequence it would not seem there was a pattern. However, Richard Guy conjectured that this sequence will eventually become “periodic.” This means that after neglecting some long initial piece of the sequence at some point a block of numbers, perhaps large in size, will repeat over and over!

Perhaps the work Guy is best known for is the joint work with Berlekamp and Conway on combinatorial games, which resulted in a book through which all three of these men are known to the public–*Winning Ways*. Taking off from the many new and old games described in this book there have been hundreds of research papers loosely belonging to combinatorial game theory but reaching into every branch of mathematics. While much of *Winning Ways* is devoted to discrete mathematics (which includes both finite set ideas as well as those involving discrete infinities), the book also treats topics that lead into complex areas related to “infinitely small” and “infinitely large” numbers.

Eventually John Conway wrote about the remarkable ways that studying combinatorial games of various kinds leads to ideas of numbers that go beyond the integers, rational numbers and real numbers that are the most widely studied discrete infinite and “dense” number systems, for which given any two numbers there is some other number of the system between them. This work of Conway is described in the book *Surreal Numbers* by Donald Knuth which is aimed at a general audience, and in Conway’s more technical book *On Numbers and Games*. Guy’s work helped popularize many of Conway’s more technical contributions.

There are two major ways mathematics has contributed to understanding “games.” One strand of game theory concerns conflict situations that arise in various attempts to understand issues related to economics–different companies or countries take actions to get good outcomes for themselves in situations that involve other “players.” The other is combinatorial games like Nim, Grundy’s game, and checkers and chess.

### Richard Guy’s mathematics

Richard Guy’s influence as a mathematician comes from a blend of the influence he had through broadcasting the ideas and accomplishments of his collaborators and his role in calling attention to problems which could be understood and attacked with more limited tools than the most abstract parts of mathematics. He did not invent dramatic new tools to attack old problems, or chart out new concepts that would create new fields in mathematics or new avenues for mathematical investigation. He did not get awarded any of the growing number of prizes for dramatic accomplishments in an old field of mathematics (e.g. the work that earned James Maynard the Cole Prize in Number Theory in 2020) or for “lifetime” achievement such as the Abel Prize awarded to Karen Uhlenbeck in 2019–the first woman to win the Abel Prize.

To get an idea of Guy’s work, it helps to look at a sample of the titles of the articles he authored and co-authored. As you can see, they reflect the cheerful and playful aspects of his personality that he showed in his personal interactions with people.

- All straight lines are parallel
- The nesting and roosting habits of the laddered parenthesis
- What drives an aliquot sequence?
- John Isbell’s game of beanstalk and John Conway’s game of beans-don’t-talk
- Primes at a glance
- The second strong law of small numbers
- Graphs and the strong law of small numbers
- Nu-configurations in tiling the square
- The primary pretenders
- Catwalks, sandsteps and Pascal pyramids
- The number-pad game
- Don’t try to solve these problems!
- Some monoapparitic fourth order linear divisibility sequences
- Richard Rick’s tricky six puzzle: S5 sits specially in S6
- A dozen difficult Diophantine dilemmas
- New pathways in serial isogons

His most important books were:

- Berlekamp, E. and J. Conway, R. Guy,
*Winning Ways for Your Mathematical Plays*, Two volumes, Academic Press (1982) - Berlekamp, E. and J. Conway, R. Guy,
*Winning Ways for Your Mathematical Plays*, Four volumes, AK Peters (2004) - Guy, Richard.
*Unsolved problems in number theory*, Springer Science & Business Media, 2004 (with 2546 citations on Google Scholar as of July, 2020) - Croft, Hallard T. and Kenneth Falconer, Richard K. Guy,
*Unsolved problems in geometry*, Springer Science & Business Media, 2012

These problem books have been reissued at times to update progress on old problems and give new problems. They have resulted in gigantic progress on a wide array of number theory and geometry problems.

Different countries over the years have had varying “success stories” in reaching a general public about the delights of mathematics. In America one of those success stories was Martin Gardner’s contributions mentioned above. In the Soviet Union a different intriguing phenomenon emerged. In America, the typical way one reached students in schools was to provide the students with materials that were not written by especially distinguished research mathematicians but rather by materials that “translated” what researchers had done and or thought about down to the level of the students. But in the Soviet Union MIR Publishers developed a series of books by very distinguished mathematicians and aimed directly for students. These books, originally published in Russian, were eventually made available to an English-speaking audience by a variety of publishers by translating books of these Soviet authors into English. There was the Little Mathematics Library which was marketed through MIR, there was Popular Lectures in Mathematics which were distributed by the University of Chicago, (under an NSF project lead by Izaak Wirszup), there was Topics in Mathematics published by Heath Publishing and another version published via Blaisdell Publishing (which was a division of Random House). Some of the Soviet books appeared in several of these series. While not up-to-date about some of the topics that were treated, to this day these books are remarkable examples of exposition. They are books I keep coming back to and which always reward me by another look. Perhaps my personal favorite is *Equivalent and Equideomposable Figures*, by Y. G. Boltyanskii. This book is an exposition of the amazing theorem now often called the Bolyai-Gerwien-Wallace Theorem. In brief this theorem says that if one has two plane simple polygons of the same area, one can cut up the first polygon with straight line cuts into a finite number of pieces which can be reassembled to form the second polygon, in the style of a jigsaw puzzle. Figure 11 shows a way to cut a square into four pieces and reassemble the parts into an equilateral triangle of the area of the original square!! Lots of current work is to find the minimal number of pieces needed to carry out dissections between two interesting polygon shapes of the same area.

Figure 11 (A square cut into 4 pieces which have been reassembled to form an equilateral triangle of the same area. Image courtesy of Wikipedia.)

In the mid-1980s, the Consortium for Mathematics and Its Applications (COMAP) approached the National Science Foundation about creating a similar series of books as those pioneered in the Soviet Union. Figure 12 shows the cover of one such book developed and written by Richard Guy, aimed at pre-college students and devoted to combinatorial games. Guy’s book *Fair Game* was published in 1989. His manuscript was submitted as a handwritten text which had to be retyped prior to publishing!

Figure 12 (Cover of the book *Fair Game*, by Richard Guy. Image courtesy of the Consortium for Mathematics and its Applications.)

Richard Guy sometimes referred to himself as an “amateur” mathematician.

While he wrote many papers and books, his legacy is not the theorems that he personally proved but his role as someone who loved mathematics and who was a mathematics enthusiast–writing about, teaching about, and doing mathematics. His wonderful promotion of mathematics will live on long after those who knew him personally are gone.

### References

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society’s MathSciNet can be used to get additional bibliographic information and reviews of some of these materials. Some of the items above can be found via the ACM Digital Library, which also provides bibliographic services.

Some books which Richard Guy authored, co-authored or edited:

Guy, R., The Book of Numbers (joint with John Horton Conway), Springer-Verlag, 1996.

Guy, R., Fair Game, Consortium for Mathematics and Its Applications, 1989.

Guy, R.. and R. Woodrow, The Lighter Side of Mathematics, Proc. E. Strens Memorial Conf. on Recr. Math. and its History, Calgary. 1986.

Guy, Richard. Unsolved problems in number theory. Vol. 1. Springer Science & Business Media, 2004.

other references:

Albers, D. and G. Alexanderson (eds.), Fascinating Mathematical People, Princeton U. Press, 2011. (Includes Richard Guy.)

Albert, M. and R. Nowakowski, (eds), Games of No Chance 3, Cambridge U. Press, NY, 2009.

Conway, J., On Numbers and Games, 2nd. edition, AK Peters, Natick, 2001.

Croft, H. and Kenneth Falconer, Richard K. Guy. Unsolved problems in geometry: unsolved problems in intuitive mathematics. Vol. 2. Springer Science & Business Media, 2012.

Erdős, Paul, and Richard K. Guy, Crossing number problems, The American Mathematical Monthly 80 (1973): 52-58.

Guy, R.K. and Smith, C.A., The G-values of various games. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 52, No. 3, pp. 514-526). Cambridge University Press, 1956.

Guy, R. K., & Nowakowski, R. J. (1995). Coin-weighing problems. The American mathematical monthly, 102(2), 164-167.

Nowakowski, R. (ed.), Games of No Chance, Cambridge U. Press, 1996.

Nowakowski, R. (ed)., More Games of No Chance, Cambridge U. Press, 2002.