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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The dimensions are
69 x 61
and the tiling looks like this:
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.
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