2001 Pan African MO Problems/Problem 2
Let be a positive integer. A child builds a wall along a line with identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
From smaller values of , there is 1 wall with 1 cube, 2 walls with 2 cubes, 4 walls with 3 cubes, and 8 walls with 4 cubes. Thus, we can suspect that there are walls with cubes.
To prove our claim, we can calculate the number of walls with blocks and columns. We can use ball-and-urn counting to determine the number of walls. Since there are columns, there would be dividers. There are a total of blocks, but each column must have at least one block, so there are blocks left to sort. Thus, there are walls that have blocks and columns.
Summing all possible values of means that there are a total of walls with cubes.
There are cubes. After placing the first cube down, we have cubes left. Now for each of these remaining remaining cubes, we have two options; stack the cube or put it in front. This then gives that since there are options for each of the cubes, the answer as .
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