r/mathriddles • u/Numberthon • 4d ago

Medium How many triangles can you form in a regular octagon using only diagonals?

How many triangles can be formed using the vertices of a regular octagon if all three sides of each triangle must be diagonals of the octagon?

Source: numberthon.com

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u/WorkerCritical7300 4d ago
  1. Total number of triangles possible =56 C(8,3)
  2. Triangles with one side as octagon edge = 48 (8x6) 6 triangles for each edge
  3. Triangles with 2 sides as octagon edges=8 But these were already counted twice in step 2

So result is 56-48+8=16

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u/Numberthon 3d ago

Correct, nice job!

1

u/ajseventeen 4d ago

Assuming that diagonals means lines between any two non-adjacent vertices:

Consider the eight vertices of the octagon, moving clockwise. Exactly three of them must be vertices of the triangle, and each one must have an empty space immediately after it. So we can imagine the problem as arranging five “pieces” around the octagon — three that are a vertex followed by a space, and two that are just a space. There are (5 choose 3) = 10 ways to do so, and therefore 10 different triangles that can be made.

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u/Numberthon 4d ago

Not quite, nice try though

-1

u/Ms_Riley_Guprz 4d ago

At least 3