Pascal-like Triangles Mod k

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.