In a 3 x 3 grid where each cell has some no. of tokens.
In the figure below is one such example.
This is transformed as follows: in each step, every cell sends a token to all of its neighboring horizontal and veritcal cells. Incase there aren't enogh tokens to give to all neighboring cells , no tokens are sent at all. So, after one step this is the fiqure.
My Questions: Prove that every staring arrangement results in stable arrangement (one that will no longer change in this process), or will repeatedly cycle through n arrangements for some positve integr. n (i.e., those same n arrangements will appear again and again in the sequence over and over as the transformations are done).
I have some idea that this requires the use of pigeon-hole principle but am not too sure on how to solve this ?
There are only finitely many tokens, so there are only finitely many possible arrangements. After some number of transformations, an arrangement must be repeated. But if that happens, it happens infinitely often. That is, you simply cycle through the arrangements starting at the repeated pattern, until you get to the pattern again.