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This is a pure dissection problem, with no added twists. Cut the holey octomino (i.e., a square with the middle third removed) into several pieces, and reassemble those pieces into a square with no hole. Aim for as few pieces as possible.

An illustration of the problem.

My solution uses 5 pieces. Is it possible to do better?

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    $\begingroup$ I doubt that there is a four piece solution, but that's hard to prove. $\endgroup$
    – greenturtle3141
    Mar 18, 2017 at 1:24

5 Answers 5

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I seriously doubt that this can be done in 4 pieces or less. It would be a miracle if it was possible, but it obviously isn't a walk in the park to prove. Regardless, to get people started, I have found two solutions that use 5 pieces:

First:

solution 1

Second:

solution 2

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Well, from my viewpoint this is a four-piece dissection, since parts of each piece don't move relatively to each other. They are even connected, to some extent. However, I would completely agree that there are about 24 pieces in this dissection, from a pragmatic viewpoint.

Technically a four-piece

At least evaluate an hour-long fiddling with MS paint here.

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  • $\begingroup$ Not what I had in mind, but clever. (+1) $\endgroup$
    – Franklin Pezzuti Dyer
    Dec 17, 2018 at 14:04
  • $\begingroup$ @Frpzzd - I'm guessing a 'proper' answer is going to involve some tedious case-analysis proof that no 4-piece solution exists, and you'd have better luck on math.SE... but that's just a guess. $\endgroup$
    – deep thought
    Dec 17, 2018 at 14:13
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    $\begingroup$ Very nice! It's even a translation-only dissection, i.e. the pieces are not rotated. $\endgroup$
    – Jaap Scherphuis
    Dec 17, 2018 at 14:29
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    $\begingroup$ I'd say that this is a valid answer, as all the pieces touch each others at corners. $\endgroup$
    – TakingNotes
    Dec 18, 2018 at 8:06
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    $\begingroup$ Gets my vote for "best" answer. Very "puzzling" and a bit out of the box. $\endgroup$
    – BmyGuest
    Dec 18, 2018 at 22:47
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greenturtle3141 found two solutions with 5 pieces. Since this ties my own solution, I'm accepting it now. Now that there's an answer, I'll post my own solution:

solution

Even though I've already accepted an answer, if you find a 4-piece solution, please post it!

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Here's a dissection into $5$ pieces that only uses $3$ cuts.

holey octomino dissection

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Looking for a square with no hole, hmmm...

You can make two cuts on opposite sides of the square, two thirds of the way across one side and one third across the other - giving you two "L" shaped pieces. Cut a third from the longer side of one "L" (three pieces), and move the resulting square into the angle of it's bend - giving you a four-square, with no hole in the center. As a bonus, a single extra cut a third of the way on the long side of the second "L" can give a second four-square.

Perhaps you were looking for a way to assemble all the pieces into a single solid square, in which case I don't have an answer. But, these cuts will allow "a square with no hole" to be made from three pieces, or two squares made from four.

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  • $\begingroup$ Well, given that OP checks greenturtle3141's answer, it seems OP is looking for a way to dissect into only 4 pieces, and rearrange into an eight-square. $\endgroup$
    – Hakdo
    Dec 20, 2018 at 1:31

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