Take four arbitrary natural numbers, a, b, c and d.
Prove that if use them to find the four numbers a’, b’, c’ and d’, which are equal, respectively, to the differences between a and b, b and c, c and d, d and a (taking the positive difference each time), and then we repeat this process with a’, b’, c’ and d’ to obtain four other numbers a”, b”, c” and d”, and so on, we eventually must obtain four zeroes.
For example, if we begin with the numbers 32, 1, 110, 7, we obtain the following pattern:
32 1 110 7
31 109 103 25
78 6 78 6
72 72 72 72
0 0 0 0
It is enough to show that the number is decreasing in each row (except in case if there are zeros, which is an easy case to study.)