Extending the Sierpinski Property in the Cups and Stones Problem

Big open problems in science and mathematics have a way of surprisingly showing up in simple puzzles. One big problem involves the famous fractal known as the Sierpinski gasket and how much space it takes in the plane as an object of fractal dimension Log_2(3). Ettestad and Carbonara showed that a simple puzzle posed by Barry Cipra in 1992 is an example of the Sierpinski gasket, and it can be used to study its properties, hopefully shedding light on some of the big mysteries associated with fractals and in particular to the Sierpinski gasket. In 1992 Barry Cipra posed an interesting combinatorial counting problem (call it the CSCP problem). In essence, it asks for the number of configurations possible if a circular arrangement of k cups, each having s 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 it and showed that the shape of the non-zero terms in the reduced matrix for the CSACP Problem with exactly 2^n+1 cups is equivalent to the Sierpinski gasket. (We call this the Sierpinski property.) In this presentation we extend the problem by numbering the stones, thereby revealing several new and interesting properties of the game. In particular we extend 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.