7. Unsolved Problems
There are a lot of unsolved problems in the research of Pascal-like triangles. Let’s introduce some of them.
If you are interested in these problems, please contact us!
It is easy to see that the above triangle is also Pascal-like triangle. We have not proved that the list {U12(n,m), n=2,... and m ≤n} forms Pascal-like triangle mathematically.
The first number in each row is {1,1,3,4,6,8,11,13,17,20,24,28,33,37, ...}
We found out that this sequence satisfies the following formula.
Definition 4. We play the game of Definition 1 with 3 players. When one of the players becomes a loser, remaining two players play the game of Definition 1 again. When one of them becomes the loser, the game ends.
Let U12(n,m) be the number of the cases that the first and the second player’s losing in the game.
Then for fixed number p and v with v≤p the list {U12(n,m), n=2,... and 2≤ m ≤n} forms the following triangle.
Here the condition 2≤ m is essential, since there should be two loser.
We have not proved the above formula mathematically. This formula was discovered by Mr. Soh Tatsumi and Mr. Masakazu Naito who are new members of our research group.
They are not registered member of this project.
