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It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals with identical polylines to get 3 congruent, non-convex pieces) and this partition is easily generalized to 12, 27, ... number of pieces.

Question: Are there other triangles that can be cut into N polygons that are non-convex and mutually congruent? If so, for what values of N?

Note 1: I don't know if there is any partition of any triangle into 7 mutually congruent polygons, convex or otherwise - although it has been proved (https://arxiv.org/pdf/1811.09723.pdf) that no triangle can be cut into 7 or 11 congruent triangles.

Note 2: It is known that there are convex polygons that can be cut into a finite number of non-convex and mutually congruent pieces but NOT into convex congruent pieces (eg: A claim on partitioning a convex planar region into congruent pieces).

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Here is a solution with $24$ pieces:

enter image description here

In general proving impossibility results will be extremely hard, as it is with almost all tiling problems of this form.

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    $\begingroup$ Thanks. Nice construction with the sphinx that you had put up here as well: math.stackexchange.com/questions/3953136/… . The triangle being tiled is again equilateral. Perhaps no other triangle admits a tiling by any number of non-convex tiles. It might be easier to prove that no triangle with all sides different has this property. $\endgroup$
    – Nandakumar R
    Jun 18, 2022 at 4:43
  • $\begingroup$ Indeed, one would be surprised if equilaterals are the only triangles that can be tiled by non-convex tiles. $\endgroup$
    – Nandakumar R
    Jun 18, 2022 at 5:13

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