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That is, dividing into squares in a manner that the pieces that can't be squares (the ones on the outline of the circle) are the same size as those who can?

I hope I explained myself :)

enter image description here

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By "size", you mean "area"? –  J. M. May 8 '12 at 15:36
    
Yes, sorry if that was misleading. –  kelmer May 8 '12 at 15:37
    
I think this might prove impossible. –  Pedro Tamaroff May 8 '12 at 16:38
    
Does bisecting the circle with a diameter count? –  Mark Bennet May 8 '12 at 16:43
    
No, I mean for any arbitrary number. I uploaded an image. @PeterTamaroff I hope it's not! –  kelmer May 8 '12 at 16:45
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As you see, you can't do it without giving up something because a whole square is always going to be bigger than a partial square. One thing you could give up is making the cuts straight. If you make the cuts bend in toward the center of the circle you can make each vertical strip have the same area, with the added width of the outer ones making up for the shorter height. Then make the same correction to the horizontal cuts. You will have a pattern that has four sided pieces in the middle, but the sides will not be the same length or straight. The pieces will have the same area, though.

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An easy case to visualise is with five pieces, one of which has four corners on the circumference. –  Mark Bennet May 8 '12 at 17:11
    
I think this settles it. I though that maybe not using squares but rectangles of some sort (i.e. not making vertical and horizontal lines equidistant) it would be possible, but I don't think that would solve it either. I guess I also failed to formulate the question right by using the word "square" :) –  kelmer May 8 '12 at 17:12
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