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Imperfect Square (Posted on 2004-10-14) Difficulty: 3 of 5
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The above piece of chocolate looks like a square with 4 pieces stuck on to it. How can you divide it with 4 straight cuts that they may form four congruent squares when moved? (You may not move the pieces until all cuts have been made, and you may not rotate or flip over any of the pieces when moving them. Also, all the chocolate must be used to make the squares.)

No Solution Yet Submitted by Gamer    
Rating: 3.0000 (1 votes)

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Some Thoughts re(2): Method for solving this type of dissection problem | Comment 4 of 8 |
(In reply to re: Method for solving this type of dissection problem by Jer)

I see what you mean Jer - that makes a lot of sense. Your explanation made me think of something though… maybe your post addressed this and I just missed it. Anyways…

How come when I was cutting along the other diagonal (top left to bottom right, vs. bottom left to top right), I couldn’t get a solution?

I had some thoughts. First I thought because the way I first did it, I had one piece that was too big… I would need to chop off another corner of it to make it the correct size. I also noticed that in this same configuration, I had one "corner" of my tic-tac-toe thing empty. I thought maybe that had something to do with it.

So then I decided to still make the cuts in the same orientation, but not necessarily have all the cuts go through vertices. Now I had pieces of the chocolate in all 9 regions of my tic-tac-toe board, but I still couldn’t get a solution this way. Now I had the one solid center piece, and 4 large pieces that were all about 3 units^2, which is more than half the area allowed. That won’t work to get 4 squares total.

So then I thought it must be because the lines in my actual solution are parallel to the translated movement of the original piece when tessellating. The whole original piece moves up 4 units and to the left 2 units. That’s parallel to some of the lines I made in my solution.

But why is this so? Why couldn’t I find a solution the other way? Is there one? Is there a way to prove that one can’t be done the other way?

Is there any way to make the same tic-tac-toe cuts in the same orientation as my solution, but just translated around to a different area, and NOT get a solution? Hmmmm.


  Posted by nikki on 2004-10-14 17:41:34
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