27
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ I doubt that there is a four piece solution, but that's hard to prove. $\endgroup$ – greenturtle3141 Mar 18 '17 at 1:24
15
$\begingroup$

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

$\endgroup$
21
+50
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Not what I had in mind, but clever. (+1) $\endgroup$ – Franklin Pezzuti Dyer Dec 17 '18 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 '18 at 14:13
  • 3
    $\begingroup$ Very nice! It's even a translation-only dissection, i.e. the pieces are not rotated. $\endgroup$ – Jaap Scherphuis Dec 17 '18 at 14:29
  • 1
    $\begingroup$ I'd say that this is a valid answer, as all the pieces touch each others at corners. $\endgroup$ – a guy Dec 18 '18 at 8:06
  • 1
    $\begingroup$ Gets my vote for "best" answer. Very "puzzling" and a bit out of the box. $\endgroup$ – BmyGuest Dec 18 '18 at 22:47
7
$\begingroup$

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!

$\endgroup$
7
$\begingroup$

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

holey octomino dissection

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 '18 at 1:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.