Nonplussed! by Julian Havil Read Free Book Online

occupy us for the rest of the chapter, is that level 5 is impossible to reach, no matter how many chequers are placed in whatever configuration below the barrier.
    Table 6.1 summarizes the situation. The result is indeed surprising, but then so is Conway’s ingenious method of proof, which, apart from anything else, brings in the Golden Ratio.
    The Solution
    To start with, fix any target square T on level 5 and, relative to it, associate with every square a nonnegative integer power of the variable x , that power being the ‘chequerboard distance’or ‘taxicab distance’ of the square from T. Such a distance is measured as the number of squares, measured horizontally and vertically from T, which gives rise to figure 6.4 .

    Figure 6.4. The labelling of the squares
    With this notation in place, every arrangement of chequer pieces, whether the initial configuration or the configuration at some later stage, can be represented by the polynomial formed by adding each of these powers of x together, for example, the starting positions to reach levels 1 to 4 might be represented by the polynomials x 5 +x 6 , x 5 +2 x 6 +x 7 , x 5 +3 x 6 +3 x 7 +x 8 and x 5 + 3 x 6 + 5 x 7 + 6 x 8 + 4 x 9 + x 10 , respectively.
    We now look at the effect of a move on the representing polynomial by realizing that, for this purpose, the choice of moves reduces to just three essentially different possibilities, which are characterized by the shaded cells in figure 6.4 , where counters in the light grey squares are replaced by the counter in the dark grey square in each case. The general forms of these are

    Any starting configuration will define a polynomial and, with every move that is made, that polynomial will change according to one of the three possibilities detailed above. The variable x is arbitrary and we are free to replace it with any value we wish and will look to do so by choosing a value (greater than 0) which will cause the numeric value of the polynomial to decrease in the second and third cases and remain unchanged in the first (this last is for later algebraic convenience) when this number is substituted into it. Since x > 0, evidently x n + x n- 1 > x n . If x n + x n+ 1 > x n+ 2 , we require that 1 + x > x 2 and this means thatwhich brings about the promised appearance of the Golden Ratio.

    Figure 6.5. The ultimate ‘polynomial’
    To cause the first move to leave the value of the polynomial unchanged we require that x n+ 1 +x n+ 2 = x n , which means x + x 2 = 1 andand the Golden Ratio appears once more.
    So, if we makewe are assured that the requirements are satisfied and further that, for this value of x , x + x 2 = 1.
    Whatever our starting configuration below the dividing line, there will be a finite number of squares occupied. This means that any starting position evaluated atwould be less than that of the ‘infinite’ polynomial generated by the occupation of every one of the infinite number of squares. We can find an expression for this by adding the terms in ‘vertical darts’, as illustrated in figure 6.5 .
    Adding terms in this way results in the expression

    The series in the brackets is a standard one (sometimes known as an arithmetic–geometric series) and is summed in the same way as a standard geometric series

    Multiplying by the x 5 term gives the final expression as

    Since our chosen value for x satisfies x + x 2 = x( 1 + x) = 1, it must be that 1 + x = 1 /x and also 1 - x = x 2 . Therefore,

    This means that the value of any starting position must be strictly less than 1 and since each move reduces or maintains the value of the position, the value of a position can never reach 1. It is impossible, therefore, to reach level 5.
    The proof can be seen to fail with the lower levels. For example, with level 4 we finish with the product

    leaving room for a reduction of the position to exactly 1.

Chapter 7

    THE TOSS OF A NEEDLE
    Philosophy is a game with objectives and no rules. Mathematics is a game with