#### Nonplussed!

** MATHEMATICAL PROOF OF IMPLAUSIBLE IDEAS **

** By Julian Havil **
** PRINCETON UNIVERSITY PRESS **

** Copyright © 2007 ** ** Julian Havil **

All right reserved.
** ISBN: 978-0-691-14822-9 **

#### Chapter One

**THREE TENNIS PARADOXES**

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* = 2*f + t*

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

Since *f < t*, 2*f - f* < 2*t - t* and so 2*f + t* < 2*t + 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 intelligence levels have risen.

If we move from theoretical intelligence levels to real-world matters of the state of health of individuals, we approach the medical concept of *stage migration* and a realistic example of the Will Rogers phenomenon. In medical stage migration, improved detection of illness leads to the fast reclassification of people from those who are healthy to those who are unhealthy. When they are reclassified as not healthy, the average lifespan of those who remain classified as healthy increases, as does that of those who are classified as unhealthy some of whose health has been poor for longer. In short, the phenomenon could cause an imaginary improvement in survival rates between two different groups. Recent examples of this have been recorded (for example) in the detection of prostate cancer (I. M. Thompson, E. Canby-Hagino and M. Scott Lucia (2005), 'Stage migration and grade inflation in prostate cancer: Will Rogers meets Garrison Keillor', *Journal of the National Cancer Institute* 97:1236-37) and breast cancer (W. A. Woodward et al. (2003), 'Changes in the 2003 American Joint Committee on cancer staging for breast cancer dramatically affect stage-specific survival', *Journal of Clinical Oncology* 21:3244-48).

**Winning on the Serve**

Finally, we revert to lighter matters of tennis scoring and look at a situation in which an anomaly in the scoring system can, in theory, be exposed.

The scoring system in lawn tennis is arcane and based on the positions of the hands of a clock. For any particular game it is as follows.

If a player wins his first point, the score is called 15 for that player; on winning his second point, the score is called 30 for that player; on winning his third point, the score is called 40 for that player, and the fourth point won by a player causes the player to win, unless both players have won three points, in which case the score is called deuce; and the next point won by a player is scored 'advantage' for that player. If the same player wins the next point, he wins the game; if the other player wins the next point the score is again called deuce. This continues until a player wins the two points immediately following the score at deuce, when that player wins.

The great tennis players of the past and present might be surprised to learn that, with this scoring system, a *high quality tennis player serving at 40-30 or 30-15 to an equal opponent has less chance of winning the game than at its start*.

We will quantify the players being evenly matched by assigning a fixed probability *p* of either of them winning a point as the server (and *q* = 1 - *p* of losing it); for a high quality player, p will be close to 1. The notation *P(a, b)* will be used to mean the probability of the server winning the game when he has a points and the receiver *b* points; we need to calculate *P*(40, 30) and *P*(30, 15) and compare each of these with *P*(0, 0), which we will see will take some doing!

First, notice that the position at 'advantage' is the same as that at (40, 30), which means that the situation at deuce, when divided into winning or losing the next point, is given by

*P*(40, 40) = *pP*(40, 30) + *qP*(30, 40),

also, using the same logic, we have

*P*(30, 40) = *pP*(40, 40) and *P*(40, 30) = *p + qP*(40, 40).

If we put these equations together, we get

*P*(40, 40) = *p(p + qP*(40, 40)) + *q(pP*(40, 40))

and so

*P*(40, 40) = [*p*.sup.2]/1 - 2*pq*.

Using the identity 1 - 2**pq** = [(*p + q*).sup.2] - 2*pq* = [*p*.sup.2] + [*q*.sup.2] we have the more symmetric form for the situation at deuce,

*P*(40, 40) = [*p*.sup.2]/[*p*.sup.2] + [*q*.sup.2].

and this makes

*P*(30, 40) = *pP*(40, 40) = [p.sup.3]/[p.sup.2] + [q.sup.2]

and the first of the expressions in which we have interest is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we will find the expression for *P*(30, 15), which takes a bit more work, made easier by the use of a tree diagram which divides up the possible routes to success and ends with known probabilities, as shown in figure 1.1.

Every descending route is counted to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and so we have found the second of our expressions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We only need the starting probability *P*(0, 0), which is by far the hardest goal, and to reach it without getting lost we will make use of the more complex tree diagram in figure 1.2, which again shows the ways in which the situations divide until a known probability is reached. We then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the final expression needed is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Plots of the three probabilities, shown in figures 1.3-1.5, for all values of *p* (remembering that *q* = 1 - *p*) show that they have very similar behaviour to one another, but there are intersections and if we plot the pairs {*P*(0, 0), *P*(30, 15)} and {*P*(0, 0), *P*(40, 30)} on the same axes for large *p* we can see them. This is accomplished in figures 1.6 and 1.7.

Of course, to find those intersections we need to do some algebra.

*The Intersection of P*(30, 15) *and P*(0, 0)

To find the point of intersection we need to solve the formidable equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

again remembering that *q* = 1 - *p*.

Patience (or good mathematical software) leads to the equation in *p*,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which has repeated trivial roots of *p* = 0, 1 as well as the roots of the quadratic equation 8[*p*.sup.2] - 4*p* - 3 = 0.

The only positive root is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... and for any *p* < 0.911 437 ... we will have *P*(0, 0) < *P*(30, 15) and the result for this case is established.

*The Intersection of P*(40, 30) *and P*(0, 0)

This time the equation to be solved is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and, after a similarly extravagant dose of algebra, this reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which again has trivial roots of *p* = 0, 1.

The remaining cubic equation 8[*p*.sup.3] - 4[*p*.sup.2] - 2*p* - 1 = 0 has the single real root,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which evaluates to *p* = 0.919 643 ....

Again, for any *p* < 0.919 6 ..., we will have *P*(0, 0) < *P*(40, 30), with the paradox once again established.

In conclusion, two equal players who are good enough to win the point on their serve just over 90% of the time are better off at the game's start than they are when the score is either 30-15 or 40-30 in their favour.

#### Chapter Two

**THE UPHILL ROLLER**

Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics. Leonardo da Vinci

**An Advertisement for a Book**

*The Proceedings of the Old Bailey* dated 18 April 1694 chronicles a busy day devoted to handing down justice, in which 29 death sentences were passed as well as numerous orders for brandings; there would have been 30 death sentences had not one lady successfully 'pleaded her belly' (that is, proved that she was pregnant). The business part of the document ends with a list of the 29 unfortunates and continues to another list; this time of advertisements (rather strange to the modern mind), which begins with the following paragraph:

THE Ladies Dictionary: Being a pleasant Entertainment for the Fair Sex; Work never attempted before in English. The Design of this Work is universal, and concerns no less than the whole Sex of Men in some regard, but of Women so perfectly and neatly, that 'twill be serviceable to them in all their Concerns of Business, Life, Houses, Conversations.

Tempting though it is to delve into the details of what suggests itself as a bestselling book, we move to the second advertisement.

Pleasure with Profit: Consisting of Recreations of divers kinds, viz. Numerical, Geometrical, Mathematical, Astronomical, Arithmetical, Cryptographical, Magnetical, Authentical, Chymical, and Historical. Published to Recreate Ingenious Spirit, and to induce them to make further scrutiny how these (and the like) Sublime Sciences. And to divert them from following such Vices, to which Youth (in this Age) are so much inclin'd. By William Leybourn, Philomathes.

Presumably, those who were tried at the assizes had been given insufficient access to the work and we will touch on only a small part of it ourselves, to be precise, pages 12 and 13.

William Leybourn (1626-1719) (alias Oliver Wallingby) was in his time a distinguished land and quantity surveyor (although he began his working life as a printer). Such was his prestige, he was frequently employed to survey the estates of gentlemen, and he helped to survey the remnants of London after the great fire of 1666. Also, he was a prolific and eclectic author. In 1649 he published (in collaboration with one Vincent Wing) *Urania Practica*, the first book in English devoted to astronomy. After this came *The Compleat Surveyor*, which first appeared in 1653 and ran to five editions, and is regarded as a classic of its kind. His 1667 work, *The Art of Numbering by Speaking Rods: Vulgarly Termed Napier's Bones*, was significant in bringing them further into the public eye.

In 1694 he had published the recreational volume *Pleasure with Profit*, the opening page of which is shown in figure 2.1.

We can readily agree with the following sentiment expressed in the book:

But leaving those of the Body, I shall proceed to such Recreations as adorn the Mind; of which those of the Mathematicks are inferior to none.

And having done so we can then concentrate on a delightful mechanical puzzle described in the book and attributed to one 'J.P.', which has become known as the Uphill Roller.

Figures 2.2 and 2.3 show pages 12 and 13 of the book, which detail the construction of a double cone and two inclined rails along which the cone can roll - *uphill*. His final paragraph explains the paradox, pointing out that the important issue is that, even though the cone does ascend the slope, its centre of mass will descend if the measurements are just right, which ensures that, although one's senses might be confounded, the law of gravity is not.

**An Explanation**

Before we examine Leybourn's explanation, we will look at the matter through modern eyes, using elementary trigonometry to study it. Figures 2.4, 2.5 and 2.6 establish the notation that we need and parametrize the configuration in terms of three angles: a, the angle of inclination of the sloping rails; s, the semi-angle between the rails, measured horizontally at floor level; [??], the semi-angle at an apex of the double cone. Write *a* and *b* as the heights of the lower and upper ends of the rails and *r* as the radius of the double cone. An *x/y* coordinate system is then set up as shown in figure 2.4.

*(Continues...)*

Excerpted from **Nonplussed!** by **Julian Havil** Copyright © 2007 by Julian Havil. Excerpted by permission.

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