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Crack-free Walls

Problem 215 Published on Friday, 31st October 2008, 01:00 pm; Solved by 2975;
Difficulty rating: 50%

Consider the problem of building a wall out of 2×1 and 3×1 bricks (horizontal×vertical dimensions) such that, for extra strength, the gaps between horizontally-adjacent bricks never line up in consecutive layers, i.e. never form a "running crack".

For example, the following 9×3 wall is not acceptable due to the running crack shown in red:

There are eight ways of forming a crack-free 9×3 wall, written W(9,3) = 8.

Calculate W(32,10).