Thoughts On Algorithms, Geometry etc...

Tuesday, February 07, 2012

Packing Rectangles

Here is a puzzle to which I dunno the answer yet.

Claim: Given N (N>1) rectangles all of the same area but all with different perimeters.
We cannot pack them without gaps to form a large rectangle.

IOW, if a rectangle is broken into N smaller rectangles all of same area, at least two of the pieces are identical.

Remark: it is not too hard to show rectangles which can be divided into N equal area rectangles with just 2 pieces identical, and all others having different perimeter.

Perhaps the claim may be easier to prove under the constraint: the sides of the rectangle and the pieces ought to be integers/rationals.

The following pages may be of interest.

Update (March 27th 2012) - (The portion above was not touched in this edit)
The claim above *does not* hold if the tiles can have incommensurate sides - if the ratio between the lengths can be irrational. has details. As of now, if all tiles ought to have rational side lengths, the claim remains open.
We posted the problem as we knew it a few days back at

And then, came a MAJOR SURPRISE: I just gathered from that the problem was known at least in its basic form long back (in the late 1930s!) to the 4 mathematicians - Brooks, Smith, Stone and Tutte - who did plenty of work on 'squaring the square'. This 'Blanche dissection' was not known to even some of the experts I approached with above question - a possible reason might be its odd name. I am not sure if the quartet attempted the rational edge lengths case or left it for us to ask.


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