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There is no backstory to this question; I just thought of it.

Take a piece of origami paper, a square with area 1. Then fold the paper in any way possible without cutting. There is no limit to the number of folds. Then find the smallest possible cuboid that would contain the entirety of the folded paper.

What is the largest possible volume of the cuboid? In what way(s) could the paper be folded into to create a cuboid of this volume? And how would other problems similarily relating to folded paper be solved?

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  • $\begingroup$ Are we considering just one fold? Regardless, I believe $\sqrt{2}$ is the largest "diameter" of the shape post-fold, which happens when the diagonal is preserved. I don't know whether the largest possible cube corresponds to a shape with diameter $\sqrt{2}$. $\endgroup$ – Kaj Hansen Jan 7 '18 at 1:45
  • $\begingroup$ @Kaj Hansen As many folds as needed. I will edit the question to clarify. $\endgroup$ – Tbw Jan 7 '18 at 1:49
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I would claim it is "obvious" you only want one fold. More folds make the package smaller. If you fold along the diagonal you have to have a segment of length $\sqrt 2$ fit in the cuboid. A right angle fold lets you use cuboids of $\sqrt 2 \times \frac {\sqrt 2}2 \times \frac {\sqrt 2}2$ with volume $\frac {\sqrt 2}2$ or $\sqrt 2 \times 1 \times \frac 12$ with the same volume. I would suggest that is the best, and invite others to find better.

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