# A dissection puzzle where you're allowed to use dilation

You may be familiar with Dudeney's famous dissection of the equilateral triangle into a square. (A nice physical version is demonstrated here.) His dissection uses four pieces. I believe this to be the minimum (though I have never seen a proof).

Now, I shall give you superpowers: you may dilate (aka resize, aka grow and shrink) any piece or pieces.

With this new ability, how few pieces are required to dissect the equilateral triangle into the square? (Said another way, you must partition the equilateral triangle and square each into pieces such that each piece of one is similar (but not necessarily congruent) to a corresponding piece in the other.)

This is a golf-style question: fewest pieces wins.

• You can model this in the real world using shadows, I suppose. Dec 15, 2022 at 1:22
• Are you aware of an optimal solution? Dec 15, 2022 at 1:23
• @bobble It's doable in 3, but I have no idea if it is optimal (I suspect yes). Dec 15, 2022 at 1:25
• @AkivaWeinberger - would shadows be a valid real-world equivalent? Shadows can be skewed (i.e. not resized proportionally in both dimensions), is that still a "resize" per the terms of your puzzle. Dec 15, 2022 at 1:50
• @Phylyp No, it wouldn't, you're right. This question does not allow skews. Shadows are too high-powered. (EDIT: In fact, all triangles are 'shadow-equivalent' to each other, which would make this way too easy.) Dec 15, 2022 at 1:52