The Gergonne p-pile problem and the dynamics of the function

The Gergonne p-pile problem is concerned with repeatedly dealing (based upon some “fixed scheme” for collecting and redistributing) a deck of np cards into an n by p matrix and attempting to identify the row position of any elected card in the matrix after some fixed number of deals. Many special cases of this problem give rise to “magic tricks”. In Section 1 of this paper, this problem will be precisely defined and the motivation for studying the “dynamical” properties of the function x |→ ⌊(x + r)p⌋ will be provided. In Section 2, a general formula for the /th iterate of the aforementioned function will be developed and used to determine the position of any card after the /th deal. Also, the cases in which the aforementioned function has a unique fixed point will be identified and a value (based on n and p) will be given for how many times the deck must be dealt to ensure that the selected card has reached the fixed position. Finally, in Section 3, the results of Section 2 will be extended to more “complicated” schemes in which the collecting and redistributing of the deck is based upon the definition of a periodic omega word in which all of the values are between 0 and p − 1 inclusive.