#### My namehih

##### New Member

- Joined
- Mar 9, 2020

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- Female

- HSC
- 2020

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.

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.

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