Über die Autorin bzw. den Autor

Julian Havil is a former Master at Winchester College, England, where he taught mathematics for more than thirty years. He is the author of Gamma: Exploring Euler's Constant and Impossible?: Surprising Solutions to Counterintuitive Conundrums (both Princeton).

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"Nonplussed!, as the title suggests, is a marvelous study of some two dozen choice mathematical problems that boggle the mind. Unlike so many books on recreational math, Havil doesn't hesitate to give crystal-clear proofs and their necessary equations. John Conway's great checker-jumping puzzle is here, along with amazing nontransitive betting paradoxes and other confounding results almost impossible to believe. No one interested in recreational mathematics on an intermediate advanced level should pass up this stimulating, delightful volume."--Martin Gardner

"In Nonplussed!, his new book of fascinating discussions of mathematical questions, Julian Havil's literary signature is evident even without seeing his name on the cover. The presentation always displays his strong ability to weave together the historical with what is often a surprising mathematical twist, even for those problems that have been around long enough to be called classic. Nonplussed! will be a classic, too."--Paul J. Nahin, author of Dr. Euler's Fabulous Formula

"Nonplussed!! is a very interesting and eclectic mix of paradoxes that has the potential to be a very useful and lasting contribution to popular mathematics."--Christopher J. Sangwin, author of Mathematics Galore

"I greatly enjoyed this book and I imagine that anyone who liked high school math will too. Nonplussed! certainly succeeds in surprising-and in giving insightful proofs of the mathematical results discussed. It is a magnificent demonstration of how far even rather straightforward mathematics can take you. And by describing many of the historical characters involved and the problems that motivated them, it makes mathematics seem like an adventure."--Nick Huggett, author of Space from Zeno to Einstein

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Nonplussed!

PRINCETON UNIVERSITY PRESS

Contents

Preface..........................................................xiAcknowledgements.................................................xiiiIntroduction.....................................................1Chapter 1 Three Tennis Paradoxes.................................4Chapter 2 The Uphill Roller......................................16Chapter 3 The Birthday Paradox...................................25Chapter 4 The Spin of a Table....................................37Chapter 5 Derangements...........................................46Chapter 6 Conway's Chequerboard Army.............................62Chapter 7 The Toss of a Needle...................................68Chapter 8 Torricelli's Trumpet...................................82Chapter 9 Nontransitive Effects..................................92Chapter 10 A Pursuit Problem.....................................105Chapter 11 Parrondo's Games......................................115Chapter 12 Hyperdimensions.......................................127Chapter 13 Friday the 13th.......................................151Chapter 14 Fractran..............................................162The Motifs.......................................................180Appendix A The Inclusion-Exclusion Principle.....................187Appendix B The Binomial Inversion Formula........................189Appendix C Surface Area and Arc Length...........................193Index............................................................195

Chapter One

So that as tennis is a game of no use in itself, but of great use in respect it maketh a quick eye and a body ready to put itself into all postures; so in the mathematics, that use which is collateral and intervenient is no less worthy than that which is principal and intended. Roger Bacon

In this first chapter we will look at three examples of sport-related counterintuitive phenomena: the first two couched in terms of tennis, the third intrinsically connected with it.

Winning a Tournament

The late Leo Moser posed this first problem during his long association with the University of Alberta. Suppose that there are three members of a club who decide to embark on a private tournament: a new member M, his friend F (who is a better player) and the club's top player T.

M is encouraged by F and by the offer of a prize if M wins at least two games in a row, played alternately against himself and T.

It would seem sensible for M to choose to play more against his friend F than the top player T, but if we look at the probabilities associated with the two alternative sequences of play, FTF and TFT, matters take on a very different look. Suppose that we write f as the probability of M beating F and t as the probability of M beating T (and assume independence).

If M does choose to play F twice, we have table 1.1, which lists the chances of winning the prize.

This gives a total probability of winning the prize of

[P.sub.F] = ftf + ft(1 - f) + (1 - f)tf = ft(2 - f).

Now suppose that M chooses the seemingly worse alternative of playing T twice, then table 1.2 gives the corresponding probabilities, and the total probability of winning the prize becomes

[P.sub.T] = tf t + tf (1 - t) + (1 - t)ft = ft(2 - t).

Since the top player is a better player than the friend, t < f and so 2 - t > 2 - f, which makes ft(2 - t) > ft(2 - f) and PT > PF. Therefore, playing the top player twice is, in fact, the better option.

Logical calm is restored if we look at the expected number of wins. With FTF it is

[E.sub.F] = 0 x (1 - f)(1 - t)(1 - f) + 1 x {f(1-t)(1-f) + (1-f)t(1-f) + (1-f)(1-t)f} + 2 x {ft(1 - f) + f(1 - t)f + (1 - f)tf} + 3 x ftf = 2f + t

and a similar calculation for TFT yields ET = 2t + f .

Since f > t, 2f - f > 2t - t and so 2f + t > 2t + f, which means that EF > ET - and that we would expect!

Forming a Team

Now let us address a hidden pitfall in team selection.

A selection of 10 tennis players is made, ranked 1 (the worst player, W) to 10 (the best player, B). Suppose now that W challenges B to a competition of all-plays-all in which he can chose the two best remaining players and B, to make it fair, must choose the two worst remaining players.

The challenge accepted, W's team is TW = {1, 8, 9} and B's team is [T.sub.B] = {10, 2, 3}. Table 1.3 shows the (presumed) inevitable outcome of the tournament; at this stage we are interested only in the upper left corner. We can see that W's disadvantage has not been overcome since [T.sub.B] beats [T.sub.W] 5 games to 4.

The remaining players are {4, 5, 6, 7} and W reissues the challenge, telling B that he can add to his team one of the remaining players and then he would do the same from the remainder; of course, both B and W choose the best remaining players, who are ranked 7 and 6 respectively. The teams are now [T.sub.W] = {1, 8, 9, 6} and [T.sub.B] = {10, 2, 3, 7} and the extended table 1.3 now shows that, in spite of B adding the better player to his team, the result is worse for him, with an 8-8 tie.

Finally, the challenge is reissued under the same conditions and the teams finally become [T.sub.W] = {1, 8, 9, 6, 4} and [T.sub.B] = {10, 2, 3, 7, 5} and this time the full table 1.3 shows that [T.sub.W] now beats [T.sub.B] 13-12.

A losing team has become a winning team by adding in worse players than the opposition.

Table 1.4 shows, in each of the three cases, the average ranking of the two teams. We can see that in each case the [T.sub.B] team has an average ranking less than that of the [T.sub.W] team and that the average ranking is increasing for [T.sub.B] and decreasing (or staying steady) for [T.sub.W] as new members join. This has resonances with the simple (but significant) paradox known as the Will Rogers Phenomenon.

Interstate migration brought about by the American Great Depression of the 1930s caused Will Rogers, the wisecracking, lariat-throwing people's philosopher, to remark that

When the Okies left Oklahoma and moved to California, they raised the intellectual level in both states.

Rogers, an 'Okie' (native of Oklahoma), was making a quip, of course, but if we take the theoretical case that the migration was from the ranks of the least intelligent of Oklahoma, all of whom were more intelligent than the native Californians(!), then what he quipped would obviously be true. The result is more subtle, though. For example, if we consider the two sets A = {1, 2, 3, 4} and B = {5, 6, 7, 8, 9}, supposedly ranked by intelligence level (1 low, 9 high), the average ranking of A is 2.5 and that of B is 7. However, if we move the 5 ranking from B to A we have that A = {1, 2, 3, 4, 5} and B = {6, 7, 8, 9} and the average ranking of A is now 3 and that of B is 7.5: both average...