Squares

[cinelli]

Take three square pieces of card, sizes 12 inches, 15 inches and 16 inches. Since 12^2 + 15^2 + 16^2 = 144 + 225 + 256 = 625 = 25^2, it is tempting to wonder if, with suitable cuts, these pieces could be re-arranged to form a square of size 25. At one extreme you could cut 625 unit squares and re-arrange them. But in this puzzle you are asked to make appropriate cuts and re-arrange just six pieces to form a big square.

Cinelli

[GoSeigen]

Take three square pieces of card, sizes 12 inches, 15 inches and 16 inches. Since 12^2 + 15^2 + 16^2 = 144 + 225 + 256 = 625 = 25^2, it is tempting to wonder if, with suitable cuts, these pieces could be re-arranged to form a square of size 25. At one extreme you could cut 625 unit squares and re-arrange them. But in this puzzle you are asked to make appropriate cuts and re-arrange just six pieces to form a big square.

Cinelli

Quick question: are diagonal cuts permitted/anticipated in this puzzle? Is the idea to maintain integer unit sizes?

GS

[jfgw]

Partial spoiler:

The 16 x 16 square stays as it is;
The 12 x 12 square is cut into two;
The 15 x 15 square is cut into three, and two of these pieces are the same.


Julian F. G. W.

[cinelli]

Quick question: are diagonal cuts permitted/anticipated in this puzzle? Is the idea to maintain integer unit sizes?

GS

I hadn't anticipated diagonal cuts but if you can come up with a solution like this, please go ahead.

Cinelli

[jfgw]

Quick question: are diagonal cuts permitted/anticipated in this puzzle? Is the idea to maintain integer unit sizes?

My answer does not involve angled cuts, nor unit fractions.


Julian F. G. W.

[modellingman]

Quick question: are diagonal cuts permitted/anticipated in this puzzle? Is the idea to maintain integer unit sizes?

My answer does not involve angled cuts, nor unit fractions.


Julian F. G. W.

Indeed it does not.

Spoiler...


Cut the 12x12 square into pieces A and B and re-assemble these to create a 9x16 rectangle which is added to the right of the 16x16 square.
Cut a 15x6 strip from the 15x15 square and place the remaining 15x9 strip on top of the 16x16 square aligning left edges. This leaves a 10x9 rectangle missing in the top-right and this can filled by cutting the 15x6 strip into pieces C and D and re-assembling to fill the gap.

Picture here


modellingman

[jfgw]

There are a few different but very similar solutions.

I had the 16 square bottom right, the 16 x 9 rectangle (made from the 12 square) above that, the 9 x 15 rectangle top left, and the 9 x 10 rectangle (made from the 6 x 15 offcut) bottom left, with the steps going the same way as modellingman's.


Julian F. G. W.

[cinelli]

Excellent. I did wonder if there were alternative solutions but modellingman's matches mine exactly.


Cinelli

[jfgw]

If I am not mistaken, there are 64 solutions including rotations and reflections.


Julian F. G. W.