Cutting an equilateral triangle into $n$ equal pieces

Question

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

For which $n$ is it possible?

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

Note: I don't mean equal areas, I mean equal triangles.

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 $n$ — Ross Millikan, Jun 15 '16 at 16:03
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.) — Ivan Neretin, Jun 15 '16 at 16:13
Previously: Splitting equilateral triangle into 5 equal parts — , Jun 15 '16 at 16:36
Do you mean congruent pieces or do you also require that the pieces be triangles? — almagest, Jun 15 '16 at 17:57
Then the allowed values of $n$ seem to be limited to $k^2,;2k^2,;3k^2,\text{ and }6k^2$. — Ivan Neretin, Jun 15 '16 at 20:02

score 4 · Accepted Answer · answered Dec 08 '20 at 21:57

Michael Beeson has a series of papers on tiling triangles by smaller congruent triangles listed here, of which a recent one dealing with the equilateral case is here.

In it, it is shown that an equilateral triangle can be tiled by $N$ congruent triangles when $N$ is of the form $m^2, 2m^2, 3m^2,$ or $6m^2$. Shockingly, these are not the only possibilities! Among others, the case $N=109345=3^7\cdot 5$ is possible, and shown in the paper; for convenience, I'll reproduce the diagram below.

However, it is proven that no prime $N$ beyond $N=2,3$ are possible.

The paper also shows that the question of whether $N$ tiles are possible is computable, though the given algorithm is not efficient enough to make a search over large $N$ tractable.

Cutting an equilateral triangle into $n$ equal pieces

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