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It is well-known that a square can be cut into a finite number of squares all of mutually different sides (hence mutually non-congruent) - for example, see https://en.wikipedia.org/wiki/Squaring_the_square

Questions:

  1. Is there a non-rectangular, non-parallelogram polygon P that can be cut into a finite number of scaled down copies of P such that each copy is of a unique size (unique scale factor)?

  2. Is there a convex polygon P that can be cut into a finite number of (not necessarily convex) pieces all of which are mutually similar and of different sizes to one another but are not similar to P?

Remark: As has been pointed out by M Winter in comments below, one can cut a right triangle into 2 pieces similar to itself and non-congruent to each other with the altitude falling on the hypotenuse. This gives a simple answer to question 1 and naturally gives an equally simple answer to qn 2 - indeed, one can cut a rectangle into 3 right triangles, all mutually similar and non-congruent. Are there any other answers?

Bit added on 2nd August, 2021: Specific question: Is there any non-right triangle T that can be tiled by finitely many mutually non-congruent triangles all similar to T? Note: It was proved by Tutte (1948) that the equilateral triangle does not have this property.

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  • $\begingroup$ Parallelograms work too, but that probably isn't what you are looking for. $\endgroup$
    – zeb
    Commented Apr 4, 2021 at 13:06
  • $\begingroup$ Thanks... made an edit to the question. $\endgroup$
    – Nandakumar R
    Commented Apr 4, 2021 at 14:19
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    $\begingroup$ Trivial answer to 2.: a shape built from several squares, e.g. an L-shape. $\endgroup$
    – M. Winter
    Commented Apr 4, 2021 at 16:40
  • $\begingroup$ Thanks. Actually, question 2 needs to be restricted to cases where P is convex. Else, as you point out, on can simply patch together squares of all kinds of different sizes into a non convex P. $\endgroup$
    – Nandakumar R
    Commented Apr 4, 2021 at 18:33
  • $\begingroup$ An example for 1. that still might be "too simple" for you: the decomposition of a right triangle into two similar triangles that you obtain by drawing the altitude. $\endgroup$
    – M. Winter
    Commented Apr 4, 2021 at 21:12

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