Can you cut the following black shape into exactly three pieces, and then rearrange those pieces into a square?

exotic bulbfish

  • $\begingroup$ Definitely 3, not 4? $\endgroup$
    – Stiv
    Nov 7, 2019 at 7:06
  • 1
    $\begingroup$ @Stiv Yes, definitely three. Four is possible too (and much easier), but it can be done with three. $\endgroup$ Nov 7, 2019 at 7:09
  • $\begingroup$ Yes, 4 would be rather simple! 3 with those curves becomes interesting... Nice head-scratcher! +1 $\endgroup$
    – Stiv
    Nov 7, 2019 at 7:11
  • $\begingroup$ 4 would be simple - two lines from the left two points crossing and touching the circle, then rearrange to diamond. $\endgroup$
    – Gloweye
    Nov 7, 2019 at 15:46

1 Answer 1


Cut along the red lines and move the pieces as indicated by the yellow arrows.

enter image description here

As is usual with this kind of dissection, it helps if you look at the area to work out the length of the side of the final square. Given the grid lines, you also know the orientation of the square, so you can try to place a square over the original picture at such a location that the pieces reveal themselves.

It is similar to this cross dissection, as this fish shape tiles the plane too, and the final square is a fundamental region of that tiling. This is also why the pieces do not have to be rotated.

  • 2
    $\begingroup$ Wow, you posted this 8 minutes after I posted the question. I'm impressed. $\endgroup$ Nov 7, 2019 at 17:32
  • 1
    $\begingroup$ When you post a long-published problem, yes, the solution will appear quickly. $\endgroup$
    – Prune
    Nov 7, 2019 at 23:20
  • 6
    $\begingroup$ @Prune I honestly had no idea this was a long-published problem (or published at all, for that matter). I don't suppose you can point me at an example? (I don't doubt you. I'm just curious.) $\endgroup$ Nov 7, 2019 at 23:36
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    $\begingroup$ No, I can't -- I'd have to dig through my "mental floss" library (a distributed collection in at least three rooms), old Mathematical Games columns, and my high-school's mathematics section. I've seen this published in at least three collections. I'd try starting with Martin Gardner's "Aha!" and his collections of puzzles and recreations. $\endgroup$
    – Prune
    Nov 7, 2019 at 23:42

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