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What is the fewest number of matches which must be removed to break all
triangles - of any size - in:
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For a size n grid of triangles, color the small triangles alternately in two different colors so that no two adjacent triangles are the same color. You will end up with n*(n+1)/2 triangles of one color, and n*(n-1)/2 triangles of another color -- all the upward-pointing triangles will be one color, and all the downward-pointing triangles the other. No one match is part of two small triangles of the same color, so at least n*(n+1)/2 matches must be removed to remove all the small triangles. However, if we do this by removing a match from each of the n*(n+1)/2 triangles of the same color, such that all matches removed point in the same direction, we have removed *all* the matches in that direction, so no triangle can be formed.
For a size 3 grid, 6 matches must be removed; for a size 4 grid, 10; for a size 5 grid, 15.