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Problem 215


Problem 215


Crack-free Walls

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).


没有裂纹的墙

考虑这样一个问题:用大小为2×1和3×1的砖块(均为宽度×高度)来筑墙,而且为了保证强度,两个砖块之间的空隙的上下方均不能是砖块的空隙,也就是说不能有“延伸裂纹”。

例如,如下的9×3的墙就是不可行的,因为其中有用红色标出的延伸裂纹:

筑一面没有裂纹的9×3的墙一共有八种方式,记为W(9,3) = 8。

试计算W(32,10)。