## Vanishing Triangles

Consider a triangular grid made of matchsticks. Three matches are needed to make a single triangle. To break it, you can remove any single match. Nine matches are needed to make a length 2 grid of triangles (6 for the perimeter and 3 internal.) There are 5 triangles in such a grid, four of size 1 and one of size 2. To eliminate all triangles, three matches must be removed (an external match must be removed to break the size 2 triangle, but this leaves three size 1 triangles, and at least two more matches must be removed to leave no full triangles.)
 ``` / \ --- / \ / \ --- --- / \ / \ / \ --- --- --- ``` What is the fewest number of matches which must be removed to break all triangles - of any size - in: a size 3 grid (originally 18 matches, as shown)? a size 4 grid (originally 30 matches)? a size 5 grid (originally 45 matches)?
Source: Original extension of previous Vanishing Squares puzzle.
Solutions were received from Philippe Fondanaiche and Joseph DeVincentis, who provided a very nice proof that his answers are the minimum:

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.

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