The Exact Upper Bound of Diffy Problem

Given nonnegative integers \(a\), \(b\), \(c\), and \(d\), the transition function \(\nabla\) is defined by \(\nabla(a, b, c, d) = (|a-b|, |b-c|, |c-d|, |d-a|)\). The Diffy problem asks if it can reach \((0, 0, 0, 0)\) after some iterations of \(\nabla\) on the four numbers. If \((a, b, c, d)\) can transfer to \((0, 0, 0, 0)\) iterated by \(\nabla\) operations, the smallest \(N\) such that \(\nabla^N(a, b, c, d) = (0, 0, 0, 0)\) is called the stopping steps of the Diffy problem. In this paper, we will show that there exists \(N\) such that \(\nabla^N(a, b, c, d) = (0, 0, 0, 0)\) and the loose upper bound and exact upper bound of \(N\). In addition, we will also show that we can find a starting vector \((a, b, c, d)\) so that it reaches the zero vector \((0, 0, 0, 0)\) after exact \(k\) steps for any given positive integer \(k\).