TL;DR
This paper explores the fractal structures, specifically Sierpinski-gasket and Cantor sets, that emerge from dividing coins with specific face values among people, revealing complex nested patterns.
Contribution
It introduces a mathematical framework linking coin division problems to fractal structures like Sierpinski-gasket and Cantor sets, extending to higher dimensions.
Findings
Division of coins with face values as powers of r forms a Sierpinski gasket with r layers.
Higher-dimensional Sierpinski gaskets arise with more than three people.
Incomplete coin sets can produce Cantor set patterns in division.
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 layer 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.
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