Pascal-like Triangles Mod k

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This Demonstration shows Pascal-like triangles reduced modulo , where can vary between 2 and 7. These arrays are made from the probabilities of winning a -player game described in Details.

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


Snapshots


Details

Besides the colored pixels, you can also see numbers for the Pascal-like triangles reduced modulo . Please see the third snapshot.

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

The denominators of the list form Pascal-like triangles, and these triangles, reduced modulo , are displayed. These triangles depend on the number of players. To make the situation simple, .

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

Because of the Pascal-like property, the triangles can be made by a kind of cellular automaton that has a kind of boundary condition.



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