Nonplussed!

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to imagine that all meaningful relationships between all pairs of objects are transitive: ‘older than’, ‘bigger than’, etc., but we do not need to look too far to produce examples for which transitivity fails: ‘son of’, ‘perpendicular to’, etc. This chapter is primarily concerned with a relationship which is seemingly transitive, but in fact need not be: that relationship is ‘better than’.
    For example, suppose that A is a better tennis player than B and that B is a better tennis player than C, then it’s pretty clearthat A is a better tennis player than C; ‘better than’ is evidently transitive in this case.

    Figure 9.1. A transitive relationship.

    Figure 9.2. Where transitivity is not defined.
    If we represent a relationship between pairs of elements by ‘ → ’ and call the elements A, B and C, it will be transitive if figure 9.1 holds.
    With the two nontransitive relationships above, it is the case that the diagram simply cannot be completed and so becomes figure 9.2 ; A and C simply do not share the relationship of A with B and B with C.
    Things become decidedly more confusing when the diagram completes to figure 9.3 , where the arrows chase each other’stails; this is altogether stranger. In particular, how can it be that A is better than B, B is better than C and yet C is better than A?

    Figure 9.3. Where transitivity is confounded.
    Do such relationships exist? The answer is decidedly yes . A familiar example from childhood is the game of scissors–paper–rock, where each of two players holds a hand behind his or her back. On the count of three, both players bring their hidden hand forward in one of three configurations. Two fingers in a ‘V’ to represent scissors, the whole hand flat and slightly curved to represent paper, and a clenched fist to represent rock. The winner is determined by the following sequence of rules: scissors cut paper, paper wraps rock and rock breaks scissors, where ‘better than’ has an appropriate definition in each of the three cases. There is no ‘best’ choice and, with A representing ‘scissors’, B representing ‘rock’ and C representing ‘paper’, that tail-chasing is evident.
    We will continue to more devious examples; in each case, ‘better than’ is given the specific interpretation ‘is more likely to win than’.
    The Lo Shu Magic Square
    The 4200-year-old Lo Shu magic square, shown in figure 9.4 , provides the basis of the first example and one can be confident that the mathematicians of the time of Emperor Yu would have had no idea of this hidden property of the design. It is, of course, a3 × 3 square with each of the nine squares filled with one of the integers from 1 to 9, the ‘magic’ stems from the fact that each row, column and diagonal add up to the magic number of 15.

    Figure 9.4. The Lo Shu magic square.

    Figure 9.5. The Lo Shu dice nets.
    Now take the three rows and number three six-sided die each with two repeats of the three numbers forming each row, as shown by the nets of figure 9.5 .
    We can use these curiously numbered dice to play a simple game of chance with an opponent: he chooses a die, then we choose a die and we roll them (say) 100 times and see who wins the most times. Table 9.1 lists the possible outcomes with each die matched against each, and shows that A → B → C → A, each with a probability ofWe have a situation which is modelled by figure 9.3 , which means that, if we allow our opponent first choice of die, we will always be in the better position.
    The choice of numbers is not unique. Toy collector and consultant Tim Rowett devised a set of three nontransitive dice where no face has a number higher than 6 (the highest number on a standard six-sided die); figure 9.6 gives the nets. Again, A → B → C → A; in this case, the reader may wish to check that the probabilities of winning in each case arerespectively.
    Table 9.1. The Lo Shu dice compared.

    Figure 9.6. The Rowett dice

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