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If there are N coins total, can you show that there are at most, say, 3^N configurations containing those coins?

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...