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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"?2012-05-08
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    Yes, sorry if that was misleading.2012-05-08
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    I think this might prove impossible.2012-05-08
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    Does bisecting the circle with a diameter count?2012-05-08
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    No, I mean for any arbitrary number. I uploaded an image. @PeterTamaroff I hope it's not!2012-05-08
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    To get a diameter, choose a square of side longer than the diameter of the circle. To get two perpendicular diameters (quarters) choose squares with side greater than the radius, and put the corner of a square at the centre of the circle. I suspect you want to specify that the circle contains at least one whole square.2012-05-08
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    @MarkBennet Not necessarily. What I want to do is divide the circle in equal pieces, using a checkerboard-like pattern.2012-05-08
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    Yes, and I've given you two ways of doing that. If there is a square internal to the circle think about why the pieces on the edge can't be squares.2012-05-08
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    Yes, I understand your answer, and for the cases of two pieces and four pieces it works because there is no whole square and the pieces are equal even in shape (only they are rotated). But for a greater number of pieces like the one in the image, the whole squares that fill the center are obviously bigger in size than the ones in the laterals, let alone the four on the corners.2012-05-08
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    And of course, if it is simply not possible, I want to know :)2012-05-08

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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.2012-05-08
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    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" :)2012-05-08