Ducci's Four Number Game with a Mutation

Anish Suresh, Dominic Klyve

Abstract: Ducci’s Four-Number Game begins with a square labelled with four positive integers, one on each corner. The game proceeds by labelling the midpoints of each side with the positive difference of the side’s two corners. We formalize this notion using points in N^4 (including 0) to represent each set of numbers, and a map D such that D([a,b,c,d]) = [|a − b|,|b − c|,|c − d|,|d − a|] to represent each turn of the game. This game serves as a fun activity for young kids. Naturally, these children make arithmetic mistakes, which begs the question: how do small mistakes impact the game? Inspired by this question, we introduce a variation to this game by allowing errors or “mutations” in the subtraction step. That is, every fixed number of turns, we use a different map Dn such that D_n([a,b,c,d]) = [|a − b|,|b − c|,|c − d|,|d − a + n|] where the positive integer n is the size of the error and the mutation randomly occurs in one of the four spots. In this paper, we show that if a mutation occurs every two, three, or four iterations, we have two cases. If n is even, any set of initial numbers will reach all 0s, like they would in the original game. If n is odd, no set of initial points reach all 0s. However, if there is a mutation five or more turns, every set of initial points reach all 0s, regardless of the parity of n. On the other hand, if there is a mutation every turn, no set of initial points reach all 0s, irrespective of n. 

Type: Article.

Status: Accepted at Rocky Mountain Journal of Mathematics.