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enter image description here

A 3 x 6 rectangle has 2 holes in it as shown. Can you cut it into 3 polyominoes with different areas so that they can form a square? The pieces can’t be flipped when they form the square and two solutions are the same if they are identical after rotation and/or reflection.

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  • $\begingroup$ If diagonal cuts are allowed, are they really polyominoes? $\endgroup$ – Rand al'Thor Jun 27 '20 at 21:00
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    $\begingroup$ Oh sorry, you can’t make diagonal cuts. @Randal'Thor $\endgroup$ – Display maths Jun 27 '20 at 21:08
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    $\begingroup$ Yes you’re right @hexomino $\endgroup$ – Display maths Jun 27 '20 at 21:08
  • $\begingroup$ You say "a rectangle has 2 holes". Is that the statement of the problem, and the picture is just one possible example? Or is the picture the only one you're asking about? $\endgroup$ – msh210 Jun 27 '20 at 21:23
  • $\begingroup$ If we can't flip pieces, how do we get reflections? $\endgroup$ – msh210 Jun 27 '20 at 21:25
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For this one, you don't even need to rotate the pieces:

enter image description here

Text version:
rrbggg
r br g
rrrrgg
becomes
gggb
rrgb
rggr
rrrr

And the final ones, which are closely related:

enter image description here    enter image description here
Text version:
ggggbb ggggbb
g gg b g gg b
grgbbb rrgbbb
becomes
gggg gggg
gbgg gbgg
gbgb rbgb
rbbb rbbb

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  • $\begingroup$ Nice! There is one left. $\endgroup$ – Display maths Jun 28 '20 at 12:53
  • $\begingroup$ @Displaymaths found it! Nice puzzle! $\endgroup$ – Glorfindel Jun 28 '20 at 13:05
  • $\begingroup$ Oh sorry, there is a last one. $\endgroup$ – Display maths Jun 28 '20 at 13:11
  • $\begingroup$ Oh, of course, it's a small variation. I'll add it in a few minutes. $\endgroup$ – Glorfindel Jun 28 '20 at 13:13
  • $\begingroup$ There are only two more if you allow flipping, if you want to get those as well since you are on a roll... $\endgroup$ – theonetruepath Jun 28 '20 at 16:05
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Here's one solution:

enter image description here

Other solutions may be possible.

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1
  • $\begingroup$ True, there are more. $\endgroup$ – Display maths Jun 27 '20 at 22:09

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