Squared Squares

Solution: $32\times 33$

The area of the rectangle is

$1+16+49+64+81+100+196+225+324=1056={ 2 }^{ 5 }\cdot 3\cdot 11$

To assemble the rectangle we must locate the square on side 18, the sides of the rectangle must measure at least 18.

The possible dimensions are $48\times 22$ , $32\times 33$ or $24\times 44$ . If you have a side of 24, when you put the square of 18 there are 6 left and you can not follow it. If you have a side of 22, after putting the square of 18 there are 4 left, putting the square of 4, below this is a gap that can not be filled. Then, the dimensions of the rectangle formed is $32\times 33$ , a configuration is given by the following figure.