You are given an equilateral triangle. What is the most number of such identical triangles you can place such that they do not overlap, but each one touches the original triangle?

  • $\begingroup$ Does a shared vertex count as touching? $\endgroup$ Commented Mar 28, 2021 at 11:57
  • $\begingroup$ @DanielMathias a shared vertex counts as touching, but not as overlapping. I hope that helps. $\endgroup$ Commented Mar 28, 2021 at 12:04
  • $\begingroup$ The solutions seems trivial? I'll have to think on it after answering the knights question. $\endgroup$ Commented Mar 28, 2021 at 12:09
  • $\begingroup$ The solution may well be trivial. This is not meant to be a difficult puzzle. $\endgroup$ Commented Mar 28, 2021 at 12:10
  • $\begingroup$ you should also mention that all the triangles should be in the same plane as that of the original traingle . $\endgroup$ Commented Mar 28, 2021 at 21:46

2 Answers 2


A trivial solution?

12 triangles:
enter image description here

  • $\begingroup$ That's it. You have found the kissing number for the triangle. $\endgroup$ Commented Mar 28, 2021 at 21:33

I think you can fit

An Infinite #
Place a triangle next to the original one so one edge is fully adjacent
Place a second triangle along that same edge, fully adjacent, but lift the far vertex vertically off the plane a bit so the new triangle rotates in 3-D
Place a third triangle along that same edge, but between the first two
Continue indefinitely. You will have an effect like the pages of a book spread open in a fan.
enter image description here


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