Extending the Sierpinski Property to all Cases in the Cups and Stones Counting Problem by Numbering the Stones
David Ettestad and Joaquin Carbonara
In 1992 Barry Cipra posed an interesting combinatorial counting problem. In essence, it asks for the number Sk,σ of configurations possible if a circular arrangement of k cups, each having σ stones, is modified by applying a particular transition rule that changes the distribution of stones. Ettestad and Carbonara (2010 and 2011) noted that this system is a finite Cellular Automaton, showed two interesting non-recursive formulas for Sk,1 and showed that the shape of the non-zero terms in the reduced matrix for the Cups and Stones Counting Problem (CSCP) with exactly 2n + 1 cups is equivalent to Sierpinski’s gasket. We call this the Sierpinski property. In this paper we extend the problem by numbering the stones, thereby revealing several new and interesting properties of the game. In particular we extend a slightly modified version of the Sierpinski property to the CSCP with any number of cups by defining a “home cup” and referencing all the other cups to the home cup.
Keywords: Cellular automata, fractals, Sierpinski gasket, Pascal’s triangle, combinatorics, enumeration, counting, discrete dynamical system, discrete mathematics, mancala