Free Nonplussed! by Julian Havil Page A

The tree diagram pruned.
    The probability that A wins is, then,
    Finally, we need to deal with the middle four cases, and this is rather more subtle since the tree diagram does not fully resolve the situation in all cases.

    The three-stage tree diagram is shown in figure 9.12
    Once again, the branches can be trimmed, but by not so much this time. If the coin comes up HH, then A will assuredly win and similarly if it comes up THH. This results in figure 9.13 .
    To better analyse the remaining possibilities it is useful to add in an extra level, as in figure 9.14 , where p is the probability that A wins.

    Figure 9.14. The pruned tree diagram extended.
    If we work through the branches from left to right, we have the expression

    which makesand so
    The analysis is complete and we have established the nontransitivity.

Chapter 10

    A PURSUIT PROBLEM
    This book is written in mathematical language and its characters are triangles, circles and other geometrical figures, without whose help… one wanders in vain through a dark labyrinth.
    Galileo Galilei
    The defunct magazine Graham DIAL was circulated to 25 000 American engineers during the 1940s and featured a Private Corner for Mathematicians , edited by L. A. Graham himself and populated by problems posed by readers for other readers to solve. Akin to Martin Gardner’s articles in Scientific American , the articles spawned two books which discussed, commented on and sometimes extended the original contributions. The first book, Ingenious Mathematical Problems and Methods , was published in 1959 and contains the problem we will discuss here. It is framed as a chase on the high seas, and does not seem to have sufficient information provided to be able to solve it. For its solution we will need two special curves, the nature of which we will deal with first.
    A Linear Pursuit Curve

    Figure 10.1. A linear pursuit curve.
    Pursuit curves were first studied in 1732 by the French scientist Pierre Bouguer, who was also the first person to measure the Earth’s magnetic field. Their exact nature depends on the path of the pursued and the method of pursuit, but the common ground is that they are the paths a pursuer should take when attempting to intercept a quarry.
    Suppose that we assume the pursued to be moving in a straight line and that the pursuer always steers towards his quarry’s current position, continually altering course to achieve this. With this agreed, we arrive at the ‘linear pursuit curve’, solved by Arthur Bernhart, and which can be shown to have an equation of the form y = cx 2 − ln x ; it is shown in figure 10.1 .
    Pursuit Using the Circle of Apollonius
    Alternatively, the pursuer could catch the quarry more quickly by utilizing a special plane curve: the Circle of Apollonius, named after Apollonius of Perga (ca. 262 b.c.e. to ca. 190 b.c.e.), which can be defined in the following way.
    Take two distinct fixed points A and B and consider the set of all points P such that PA : PB = k for some positive constant k . If k = 1, the points form the perpendicular bisector of AB , otherwise they form a circle, the Circle of Apollonius, as shown in figure 10.2 .

    Figure 10.2. The circle of Apollonius.

    Figure 10.3. A certain capture.
    Now suppose that the speed of the pursuer is υ p and that of the quarry is υ q and that at some point the pursuer is at position A and the quarry at position B . Given that the pursuer knows υ q , he should mentally construct the Circle of Apollonius as the set of points P such that AP : PB = υ p : υ q , as shown in figure 10.3 . Given also that the pursuer knows the direction of flight, he will be able to calculate the point P ′ at which the pursued will cross the circle. He should head for P ′ and so ensure capture at that point.
    The Circle of Apollonius will not be suffcient in itself for the solution to our problem, but it will have its contribution to make.
    Pursuit Using the Logarithmic Spiral
    A pleasing generalization of