More on the Four-Numbers Game
Daniel H. Ullman · Mathematics Magazine · 1992
Place arbitrary integers on the four corners of a square. Then place on the midpoint of each side of the square the absolute value of the difference of the numbers associated with the adjacent corners. Connect the midpoints of the sides of the square to form a new square with integers on its corners. Now repeat the process. FIGURE 1 shows an example starting with the numbers 1, 2, 4, and 7. Much has been written about this process ([1], [3], [4], [5], [6], [7])-the so-called four-numbers game-and its generalizations [8]. The earliest published reference seems to be in [2], where it is attributed to E. Ducci of Italy. It is a simple exercise to show that upon iteration the procedure eventually produces a square of zeroes. What is surprising is how fast this convergence actually happens in practice. Ask someone to pick four numbers play the game, and you will probably find it takes eight or fewer iterations to converge to the zero square. This is despite the fact that the convergence time is unbounded-a slightly harder exercise. It is the purpose of this note to calculate the distribution of convergence times with respect to the natural probability measure on labeled squares and thereby to explain the surprising speed of convergence. Let us generalize the process slightly by permitting real numbers on the corners of the squares. For convenience, we formulate the problem as follows. Let T: R4 R4 be defined by T(a, b, c, d) = (la-bl, lb-cl, Ic-dl, Id-al). For vtE R4, let the convergence time of the four-numbers game starting v be n(vG) = min{m: m > 0 and Tm(vt) = (0, 0, 0, 0)). We wish to calculate the probability that n(v ) = k for small natural numbers k. Since we have no way of making sense of choosing an integer or a real number at random, let us assume that the numbers are chosen according to a uniform distribution on [0, N] for some very large N. (In fact, the distribution of the function n is easily seen to be independent of the choice of N.)