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Problem of the Week 855

Dizzying Triangles

Consider a grid of 16 points arranged in a uniform 4×4 array. If all orthogonal connecting lines were drawn there would be 9 small squares. For each square, draw its two diagonals. How many triangles are there that have each of its vertices on the grid, and each of its edges on the network of lines?

For enthusiasts: Find the general formula for f(n), which counts the number of grid triangles on the n×n grid (which has n2 grid points). For example: f(1) = 0, f(2) = 4.

Source: Harris Kwong, SUNY College at Fredonia ( or He has a general formula and a proof, but would be interested in seeing other proofs; in particular, he wonders if there is an inudction proof.

© Copyright 1998 Stan Wagon. Reproduced with permission.

The Math Forum

2 October 1998