# The \$1 question: Tiling a triangle with trapezoids (the hard way)

Take a triangular grid consisting of 64 equilateral triangular cells in the shape of a larger triangle, and remove a single triangle at one of the tips. Can you tile this shape with 21 trapezoidal tiles, each consisting of three little triangles?

The answer is yes, and it is not so hard to find a solution. However, what if you impose the additional rule that no three trapezoids in the tiling can form a triangle with side length 3? Both the upward pointing and downward pointing variations of the sub-triangle are prohibited.

If the answer is yes, exhibit a tiling. If the answer is no, prove it. I see the answer is found already, still I have a simpler one to present.

A 'special triangle' contains at least two trapezoids of 'diagonal' direction (green here). Here all diagonal triangles are far apart, so you don't have to look for errors. Bill Cipher on Christmas

Here's one possible solution I found:

I failed my first try, so I re-arraged a little bit and it works (I hope)

• With more diagonal trapezoids than not.
– user
Apr 6, 2018 at 3:01