A Proof Without Words: Triangles in the Triangular Grid

Abstract

This proof without words demonstrates that there are $\left(\genfrac{}{}{0ex}{}{n+2}{4}\right)$ equilateral triangles in the regular $n$-vertices-per-side triangular grid by describing a map from four-element subsets of $\{1,2,\dots ,n+2\}$ into the set of equilateral triangles in this grid. Specifically, we illustrate the triangle that corresponds to the subset $\{4,5,8,11\}$ under this bijection when $n=10$.

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