The Gergonne \(p\)-pile problem and the dynamics of the function \(x\mapsto\lfloor (x + r)/p\rfloor\) (Q1383371)

Let \(r,p\) be fixed positive integers with \(p\geq 2\), and let \(f\) be the function with \(f(x)= [(x+r)/p]\), where \([(x+r)/p]\) is the integral part of \((x+r)/p\). For any positive integer \(l\), let \(f^{(l)} (x)\) be the \(l\)th iterate of \(f(x)\). The authors find a nice formula for \(f^{(l)} (x)\) and use this formula for solving the Gergonne \(p\)-pile problem.