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Here are two identical cubes:

two identical cubes

Your challenge:

  1. Start with two cubes of exactly the same size.
  2. Cut the surface of each of these two cubes along its edges and unfold the surface into a 2D shape. (So you now have two 2D shapes.)
  3. Now show how to re-fold these two 2D shapes to form the surface of a single larger cube, with twice the surface area of one of the originals.

Rules and clarifications:

  • The original cubes must be the same size.
  • The original cubes may only be cut along their edges.
  • Each of the two original cubes must be unfolded into a single continuous shape.
  • The two shapes must cover the entire surface of the final cube, with no gaps and no overlaps.
  • The edges of the 2D shapes do not need to line up with the edges of the final cube.
  • I am aware of exactly one solution (not counting its mirror image), but I can't absolutely guarantee it's the only one. Any solution that meets the above requirements is valid. (And if you can find one that's different from mine, I will be very impressed!)
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  • $\begingroup$ I understand the puzzle but don't know why the image is needed/helpful. Maybe an indication that one can use 3D painting software to solve it? $\endgroup$
    – WhatsUp
    Jun 28 at 9:14
  • $\begingroup$ @WhatsUp - I agree it's not really necessary. I added it because all of my other cube-wrapping puzzles have big images at the top, and this one just seemed naked without one. :) $\endgroup$ Jun 28 at 16:28
  • $\begingroup$ I can cut them into pieces and reassemble them into a sphere, but I need to use the Axiom of Choice. :P $\endgroup$ Jun 29 at 21:22

1 Answer 1

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I have found several solutions.

There are three cube nets that cover the same-shaped half of the larger cube surface, so you can choose any two copies of those (the same or different) and combine them to make the larger cube. In the image below I have only shown the three cases where you take two copies of the same net. In these solutions, the two halves of the large cube have the same handedness - a 180 degree rotation around a face centre maps one half to the other.

There is another solution in which the two halves are mirror images of one another. A point reflection through the centre of the large cube maps one half to the other.

Cube nets

Note that the large goal cube has twice the surface area of the two smaller ones, so its edge length is $\sqrt{2}$, the length of a diagonal of the face of one of the smaller cubes. Therefore, the 12 edges of the goal cube are formed by a diagonal from each face of the two cube nets.

To find the solution I looked at all 11 cube nets, and drew one diagonal in each face, such that the ends of the diagonals meet. Those diagonals can be drawn in two ways, unless there is symmetry. In each case I looked at what shape this would cover on the larger cube, where its edges were formed by the drawn diagonals of these nets. Note that no net can cover a complete face (this would require a 2x2 region in the net) so if a net leaves a complete face uncovered, it cannot be part of a solution. That left only a few cases to try.

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  • $\begingroup$ Very nice! Your solution in the middle of the top row is the one I had in mind, but, of the ones you show, my favorite is the top right. $\endgroup$ Jun 28 at 16:22
  • $\begingroup$ @plasticinsect I think the bottom one is the nicest once you see it folded as the large cube, with the two parts in different colours, but in the unfolded state the top right one is my favourite too. $\endgroup$ Jun 28 at 20:34

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