Pascal-like Triangles Made from a Game

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The triangle of fractions has properties that makes it very similar to Pascal's triangle: suppose two adjacent fractions in the same row are and . Then the fraction below them is , which is how the fractions and are added in the Farey sequence.


Let , , be fixed natural numbers such that . There are players seated in a circle. The game begins with the first player. Proceeding in order, a box is passed from hand to hand. The box contains red cards and white cards. When a player gets the box, he draws a card from it. Once a card is drawn, it is not returned to the box. If a player draws a red card, he loses and the game ends. Let be the probability that the player loses the game. Then for fixed numbers and with , the numbers form a Pascal-like triangle.


Contributed by: Hiroshi Matsui, Toshiyuki Yamauchi, Daisuke Minematsu, and Ryohei Miyadera (September 2007)
Open content licensed under CC BY-NC-SA



Originally this game was studied as a game of Russian roulette.

For the mathematical background see T. Hashiba, Y. Nakagawa, T. Yamauchi, H. Matsui, S. Hashiba, D. Minematsu, M. Sakaguchi, and R. Miyadera, "Pascal-like Triangles and Sierpinski-like Gaskets," Visual Mathematics: Art and Science Electronic Journal of ISIS-Symmetry [online], 9(1), 2007.

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