Sums and Products

Explanation

Assume at a certain stage of the process we have a sequence S of integers, S = {A₁, A₂, ..., A_n}. Let P stand for the product of all integers present each increased by 1 less 1: P = (A₁+1)(A₂+1)...(A_n+1) - 1. When two numbers A and B are replaced with (A·B + B + C) the product P does not change. Indeed, it loses two factors (A+1)(B+1) but gains another one (A·B + B + C + 1). The two are of course equal.

It then follows that P is an invariant quantity that does not change in the course of the game. It could be computed at the very beginning. For example, if we are given the sequence 1, 2, 3, 4, 5, 6 then P = 7! - 1 which you can easily verify.

References