A Game That Does Not Produce Pascal-like Triangle
A natural question will be "Is there any game of cards that does not produce Pascal-like triangle? I present one of them here.
The ruel of the game is like this. Two players take turns, and each player takes out two cards in each turn. A player who takes out two red cards in his or her turn will lose the game. In other words, a player is the person who takes out two red cards in succession.
In the above figure, Player A plays first, and Player B plays second. This is an example of a case when Player A takes out reds cards in the first and sixth round, and Player B takes out red cards in the seventh and eigth round. Note that Player A does not lose the game, and Player B loses the game, since Player A does not takes out red cards in succession.
If you arrange the number of combination of cards in which the first player loses the game, then we have the triangle in the above. In most parts of this triangle, we have the Pascal-like propery, but in some parts we do not have the property. We marked these parts in red. The red parts seem to have a certain pattern, and we are now studying the pattern.