Consider a with 3 × 3 grid where each cell contains a number of coins; for example, the following represents a possible configuration of coins on the grid (the integer in each cell is the number of coins in that cell):

12	3	1
1	8	4
2	13	0

This configuration is transformed in stages as follows: in each step, every cell sends a coin to all of its neighbors (horizontally or vertically, not diagonally), but if there aren’t enough coins in a cell to send one to each of its neighbors, it sends no coins at all. For example, the above would result in the following after one step:

11	2	3
4	7	2
1	12	2

  

a) Show that every staring configuration results in stable configuration (one that no longer changes in this process), or repeatedly cycles through k configurations for some positive integer k (i.e., those same k configurations appear repeatedly in the sequence over and over as the transformation is applied). b) In the case that the initial configuration eventually cycles through k configurations, what are the possible values of k? c) Either prove that for some positive integer B, every configuration will reach a stable configuration or a repetition of a k-cycle in B or fewer steps, or prove there is no such B.