The most basic game that produces Pascal-like triangle.
This is the most basic game that produces Pascal-like triangle. The rule is as the following.
There are p players, and each player takes turns and take out a card. If a player takes out a
red card, then he is the loser and the game ends.
The movies shows the game of four players taking out a card each time.
First, Player 1 takes out a white card, then Player 2, Player 3, Player 4 does the same. This is the first cycle.
Then, in the second cycle Player 1 takes a white card, Player 2 does the same. Then Player 3 takes out a red card, and
he is the loser.
Next, we present a Pascal-like triangle produced by this game.
Here we suppose that there are n cards, and m red cars among the cards. In other words, there are n-m white cards.
Let F(n,m) be Player 1's losing the game, and arrange F(n,m) as a triangle in the following.
This is a triangle made by F(n,m)
We are going to explain about the property of the triangle. In the following figure, F(6,2)=6/15, F(6,3)=10/20 and
F(7,3)=16/35. If you look at denominators, then 35=15+20. If you look at the numerators, 16=6+10. The same kind of
thing happen for any part of this triangle. Therefore this is a Pascal-like triangle of fractions! This fact was published at
Mathematical Gazette of England by our club.
