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A rectangle is tiled by nine squares with side lengths $2,5,7,9,16,25,28,33$ and $36$ (without overlapping and without gaps).

What are the side lengths of the rectangle? What does the tiling look like?

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An obvious question would be: Which rectangles can be tiled by squares of all different sizes? (Note: There is a proof that if a rectangle can be tiled by squares, then the ratio of its sides is a rational number and the ratio of its sides to the sides of each square is rational. So we can scale this up and ask about rectangles with integer sides and integer squares without loss of generality). – gnasher729 Feb 25 at 13:25
up vote 27 down vote accepted

The dimensions are

69 x 61

and the tiling looks like this:

solution

Working out the dimensions of the rectangle is quite easy. We know its total area is $4209$ (i.e., $2^2 + 5^2 + 7^2 + 9^2 + 16^2 + 25^2 + 28^2 + 33^2 + 36^2$). This factorizes as $3 \times 23 \times 61$, and in order to fit in a square with a side length of 36, the rectangle must be $3 \times 23$ units long on one side, and $61$ units on the other. Fitting the squares into this rectangle only takes a few minutes.

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1  
Great work! Maybe you could add a short argument in the beginning detailing that the solution rectangle has to have integer side-lengths to make it complete :-) – Falco Feb 26 at 11:50

1. Size of rectangle = 61 x 69
2. Tiling: enter image description here

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tricky is where to place the 2x2 square. It can't go on the corner or the sides because it will create a gap. Place it in the center. Experimentally add squares to it's sides. 9-2=7. 7-2=5. Continue fitting all squares until a rectangle is formed. Check widths and heights are the same. Answer:69*61

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