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We have an equilateral triangle and we want to cut it into $n$ equal pieces.

For which $n$ is it possible?

My Attempt: I found these possible numbers $2,3,4,6$ and also I proved every $n$ of the form $4^n$

note: I don't mean equal areas I mean equal triangles

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  • $\begingroup$ By equal do you mean congruent? You can always cut it into pieces that are equal in area. Given any $n$ for which you are successful, you can do all numbers of the form $n\cdot 4^k$ by cutting the triangle into $4^k$ equilateral triangles, then cutting each into $n$ $\endgroup$ – Ross Millikan Jun 15 '16 at 16:03
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    $\begingroup$ All $n$ of the form $k^2$ would do. So would $2k^2$, and probably many more. Your findings leave 5 as the smallest unknown number; turns out that it kinda can be done, too! (The text is in Russian, but you don't really need it; the picture pretty much says it all. Yes, the pieces are not connected, but still congruent.) $\endgroup$ – Ivan Neretin Jun 15 '16 at 16:13
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    $\begingroup$ Previously: Splitting equilateral triangle into 5 equal parts $\endgroup$ – Rahul Jun 15 '16 at 16:36
  • $\begingroup$ Do you mean congruent pieces or do you also require that the pieces be triangles? $\endgroup$ – almagest Jun 15 '16 at 17:57
  • $\begingroup$ no it should be triangle $\endgroup$ – Taha Akbari Jun 15 '16 at 17:58

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