A rectangle with side length a and b are in ratio $a : b = (n+1)^2 : n^2$, where n is a positive integer.
Is it possible to cut each such rectangle into two pieces, which can be put together to build a square?
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2 Answers
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The answer to your question:
Yes, this is always possible
Proof:
This solution, shown with $n = 6$, can be applied for all $n$. The rectangle with dimensions $n^2$ by $(n+1)^2$ is divided into $n\times(n+1)$ congruent rectangles with dimensions $n$ by $n+1$ such that there are $n$ rows of height $n$ and $n+1$ columns of width $n+1$. The rectangle is cut into two congruent pieces along a stair-step diagonal. These pieces can be arranged to form a square with edge length $n\times(n+1)$, divided into $n+1$ rows of height $n$ and $n$ columns of width $n+1$.
answered Dec 2, 2021 at 23:48
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4$\begingroup$ The cut is very hard to make out in your picture. Could you color it stronger? And maybe add a text description of what you are doing for visually impaired users ? $\endgroup$– FalcoCommented Dec 3, 2021 at 10:32
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1$\begingroup$ Your rectangle seems to have side in ratio n:(n=1) - 6 and 7 specifically. The case for n=6 would have sides of lengths 49 and 36, which is not the same shape, and does not enable a simplistic 1x1 "staircase" cut. $\endgroup$– AdamVCommented Dec 3, 2021 at 12:44
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3$\begingroup$ @AdamV if you read the (less than 20 words of) explanation given, you'll notice that the non-square looking small rectangles (that go into the square 6 times horizontally and 7 times vertically) do not actually represent unit squares. $\endgroup$– BassCommented Dec 3, 2021 at 14:44
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$\begingroup$ @Bass it's not less than 20 words ;-P $\endgroup$ Commented Dec 3, 2021 at 19:57
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3$\begingroup$ @Falco Your suggestions have been applied. $\endgroup$ Commented Dec 3, 2021 at 22:48
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I managed to do it like this:
Start with a rectangle of size 16 x 9 and cut into two parts as shown below:
Then place the pieces together like this to form a 12 x 12 square:
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2$\begingroup$ The question is asking if this is possible for all n. $\endgroup$ Commented Dec 2, 2021 at 23:51
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$\begingroup$ @DanielMathias OK I see the "each" in the question now. Looks like you supplied the correct answer for that. Now I wonder if other rectangles can be cut/pasted into squares or only these kinds? $\endgroup$– JS1Commented Dec 2, 2021 at 23:57
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$\begingroup$ The steps of this staircase are each n x (n+1). That will always work because there will always be n divisions of n into n^2, and n+1 divisions of n+1 into (n+1)^2, enabling the "shift and lift" to equalise them so you end up with sides of (n+1) x n and n x (n=1) which are of course the same. $\endgroup$– AdamVCommented Dec 3, 2021 at 12:48