Fractal behind coin-reducing payment

Yamamoto, Ken; Yamazaki, Yoshihiro

doi:10.1016/j.chaos.2012.05.004

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Formulation Of the Problem And A Basic Solution1

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Cited by 3 publications

(2 citation statements)

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“…(n A , n B ) = (0, 12), (1,11), (2,10), (10,2), (11,1), (12,0). Note that we admit cases in which some players receive no coins.…”

Section: Formulation Of the Problem And A Basic Solutionmentioning

confidence: 99%

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Fractal patterns related to dividing coins

Yamamoto

2014

Chaos, Solitons & Fractals

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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.

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“…(n A , n B ) = (0, 12), (1,11), (2,10), (10,2), (11,1), (12,0). Note that we admit cases in which some players receive no coins.…”

Section: Formulation Of the Problem And A Basic Solutionmentioning

confidence: 99%

“…Even with a canonical coin system, which is easy to study, the change-making process possesses a rich mathematical structure. If we repeatedly pay money so that the number of coins in the purse after each payment is minimized, a fractal pattern is obtained from a sequence of change amounts [12,13].…”

Section: Introductionmentioning

confidence: 99%

Fractal patterns related to dividing coins

Yamamoto

2014

Chaos, Solitons & Fractals

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Emergence of the Sierpinski Gasket in Coin-Dividing Problems

Yamamoto

2013

J Stat Phys

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Multifractal Aspects of an Efficient Change-Making Process

Yamamoto

Yamazaki

2013

Fractals

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In this paper, we focus on the "minimal payment" which minimizes the number of coins left after the payment. Two kinds of multifractal properties of the minimal payment are studied. The first one is a frequency distribution of change amounts, and the second one is a visiting probability on the delay plot of successive change amounts. When the face values of coins are power of two (1, 2, 4, 8, …), we find that these two distributions are related to well-known multifractal models, and derive analytical expressions of multifractal spectra.

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Fractal behind coin-reducing payment

Cited by 3 publications

References 36 publications

Fractal patterns related to dividing coins

Fractal patterns related to dividing coins

Emergence of the Sierpinski Gasket in Coin-Dividing Problems

Multifractal Aspects of an Efficient Change-Making Process

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