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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.

enter image description here

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3 Answers 3

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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.

Merry Christmas from Bill Cipher!!!Bill Cipher on Christmas

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Here's one possible solution I found:

enter image description here

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

enter image description here

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

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