How many Happy triangles?

Level pending

How many distinct equilateral triangles can we count in the diagram below such that

  1. every vertex of the triangle is a vertex on the diagram, and
  2. the triangle is oriented in the same direction as triangle A B C ? ABC?

Bonus and Details:

  • What's the general formula for the corresponding question with the diagram having n n rows in the triangular grid?
  • This is not an original question from me.


The answer is 35.

Roberto Nicolaides by Roberto Nicolaides , 26, United Kingdom

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1 solution

In general, the formula for ABC having side length n (assuming the length of an edge between adjacent vertices in the diagram is length 1) gives ( n + 2 3 ) n+2 \choose 3 . Here is a quick animation to try help see why this is true.

Here is a note going into more detail of how it works.

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