Fractal patterns related to dividing coins

Abstract

The present paper formulates and solves a problem of dividing coins. The basic form of the problem seeks the set of the possible ways of dividing coins of face values 1,2,4,8,… between three people. We show that this set possesses a nested structure like the Sierpinski-gasket fractal. For a set of coins with face values power of r, the number of layers of the gasket becomes r. A higher-dimensional Sierpinski gasket is obtained if the number of people is more than three. In addition to Sierpinski-type fractals, the Cantor set is also obtained in dividing an incomplete coin set between two people.