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# Generalization of the $$m \times n$$ grid

How many triangles are there in a $$m \times n$$ grid?

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Note by Jos Dan
2 years ago

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Let $$a = \max(m, n)$$ and $$b = \min(m, n)$$. Let $$T_n$$ denote the $$n^{\text{th}}$$ triangular number.

If $$b \geq \frac{a}{2}$$, then number of triangles is equal to $$2 T_a + 2 T_b + 4 b$$.
If $$b < \frac{a}{2}$$, then number of triangles is equal to $$2 ( T_a - T_{a - 2b}) + 2 T_b + 4b$$. · 2 years ago