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Given a $9\times 16$ sq. unit rectangle. You have one change, you can cut the $9\times 16$ sq. unit rectangle only once and join the two parts to get a square of dimension $12\times 12$ sq. unit.

Note : $9\times 16=12\times12$

I don't have any proof but I think that answer is not possible because (case i) you cut the $9\times 16$ rectangle into two rectangles, with this you cannot make the $12\times 12$ rectangle, (case ii) you cut the $9\times 16$ rectangle into two hexagons.

I'm defending my case (i) with the reason that $9$ and $16$ are co-primes. I don't have any mathematical reason to defend my case (ii).

My questions

1. Is it possible to transform $9\times 16$ sq. unit rectangle to a $12\times 12$ sq. unit square with only one chance to cut and rejoin the original rectangle(note that rotation of parts are allowed).

2. If not possible then what is the smallest number of chance(cutting and rejoining) required to accomplish the job.

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Per @Michael's suggestion, a zigzag cut of the 16x9 rectangle solves the problem. Once you have decided that the right-hand piece will shift one "stair up and left" there are not too many degrees of freedom (stairs need to be kept orthogonal etc.). Each stair in the diagram is 4 units wide and 3 units high.

enter image description here

As long as difference between the longer side (16) and the new side length (12) evenly divides the longer side length, this sort of dissection is always possible (there will be an integer number of steps). So if, for example, you started off with an 18x8 rectangle you could dissect this using stairs 6 wide and 4 high.

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