The rectangular

Geometry Level 2

A rectangle with integer side lengths is cut into $12$ squares with side lengths $2, 2, 3, 3, 5, 5, 7, 7, 8, 8, 9, 9.$

What is the perimeter of this rectangle?

The answer is 90.

2 solutions

Stephen Mellor
Sep 26, 2017

The area of the rectangle is the sum of the squares of the 12 numbers which equals 464. The prime decomposition of $464 = 2^4 \times 29$ . Therefore, the dimensions of the rectangle are either:

$1 \times 464$

$2 \times 232$

$4 \times 116$

$8 \times 58$

$16 \times 29$

The first 4 of these possibilities can't fit a square of side length 9 inside them. Therefore, the dimensions are $16 \times 29$ , meaning that the perimeter $= 2(16 + 29) = \boxed{90}$

Here is what the rectangle looks like before it is cut into the 12 squares:

Andy Hayes - 3 years, 8 months ago

(I did the solution in the same way as this.) But I was still wondering what the original rectangle looked like. How did you find this out? When you say 'this is what the rectangle looks like, do you mean that this is the only way that one can arrange these squares, or do you not know that. if you do how do you know? Yours most sincerely, David

David Fairer - 3 years, 8 months ago

(I think that it is possible to 'edit' a post, but maybe only for a little while?) Did you start with the 9x9 square and then add the 7x7 square in order to fill the side or the rectangle with length 16. And then wonder how you were going to fill the 7x2 rectangle, which you could only do with the 2x2 and the 5x5 squares. And with this the other way around to the way you have drawn it you would have an area that couldn't be filled. And at that point your configuration of the squares is taking shape nicely. And you've done well to find this. But I still wonder if this solution is unique?! Regards, David

David Fairer - 3 years, 8 months ago

@David Fairer – As per Stephen's solution, the rectangle must be 29x16. I believe that this is the only way to arrange the squares into the rectangle. The 9x9 square and 7x7 square must be adjacent to each other. Then the 8x8 squares must be adjacent to each other.

Andy Hayes - 3 years, 8 months ago

Stephen Brown
Sep 26, 2017

The squares, and therefore the rectangle, have total area of 464 = 2^4*29. The rectangle must have integer sides, and since there was a 9x9 square cut out, each side must have length at least 9. The only possible choice of side lengths is 16 and 29, giving perimeter 90.

The rectangular

The answer is 90.

2 solutions

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