Triangles on A Mountain

Probability Level 5

Consider the above 11 × 11 11 \times 11 triangular lattice of points. Find the number of distinct triangles of any size that can be formed in this lattice.

For extra credit, generalize this for an n × n n \times n triangular lattice.


The answer is 43861.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

I found the terms here , for the number of triangles on a triangular lattice. The generalized formula I found was 1 2 ( n 2 + n + 2 ) ( n + 2 4 ) − 3 2 ∑ k = 2 n ∑ m = 2 k ( n − k + 1 ) ( n − k + 2 ) gcd ⁡ ( k − 1 , m − 1 ) \frac {1}{2}(n^2+n+2)\binom{n+2}{4}-\frac {3}{2} \sum_{k=2}^{n} \sum_{m=2}^{k} (n-k+1)(n-k+2)\gcd(k-1, m-1) .

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I entered 43861 as a number of triangles on a lattice having 11 points on each side (66 points altogether). The number you are giving as a result is for a lattice having 12 points on each side. I think it would be better to have it clarified. Thanks.

Maria Kozlowska - 4 years, 7 months ago

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Thanks. I have updated the answer to 43861 and rephrased the problem accordingly.

Calvin Lin Staff - 4 years, 7 months ago

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Thank you.

Maria Kozlowska - 4 years, 7 months ago

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