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What is the fewest number of matches which must be removed to break all
triangles  of any size  in:

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*(n1)/2 triangles of another color  all the upwardpointing triangles will be one color, and all the downwardpointing 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.