A Game That Use More Than Two Cards.

In previous games we used only two kinds of cards, namely, red cards and white cards. We change this rule, and make a game using three kinds of cards. We use red cards, white cards and blue cards. This game produces a 3D version of Pascal-like triangle.

"We have three kinds of card in a box, where there are p red cards, q blue cards and r white cards. If you pick up a red card, a blue card and a white card, then you get one point, two points and zero point respectively.

Here, we give a example of 3D Pascal-like triangle made by the following game.

"Players A and B take turns and pick up two card in each turn. Player A loses the game if he or she picks up a red card. Player B loses the game, if he or she collects two red cards."

Let U(p, q, r) be the number of combinations of Player A's losing the game when the number of the red, blue and white cards are p, q and r.

If you arrange U(p, q, r) as the following, you find a 3D Pascal-like triangle.

We separate two 2D triangles.