Pascal-like Triangles Mod k


	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
	Show Initialization Code

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.