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This is a four-legged walky-square:

four-legged walky-square

This shape has an interesting property: It is possible to map multiple copies of this shape onto the surface of a cube in a way that perfectly covers the entire cube with no gaps and no overlaps.

How can it be done, and how many copies of the shape are needed?

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The cube can be covered by

two copies

The shapes can be folded like this

enter image description here

Here is one half

enter image description here

As can be seen, I made several attempts before I found a solution.
Here is the "finished" cube, which proves the solution although not very expertly crafted.
In each half four edges need to be joined, where the white sticky paper shows through.

enter image description here

Method

I noticed two ways to mark out the folds, below is another one. In both cases, at first I could only make a parallelpiped shape. Then I managed to make a half-ish right cube from the image below, like a broken eggshell. However the triangle on each of the four sides projected too far, and would not make a cube with another equal part. So I reverted to the first arrangement, and bingo. Originally, I had not thought to extend the four edges of the marked square, which gives the needed folds.

enter image description here

As @Bass points out, there are two solutions: one where the shapes are folded forwards along the crease lines; the other when the folds are made backwards. So each solution is the inside-out version of the other.

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    $\begingroup$ I just love how you use actual folded paper as part of the solving process! I also appreciate the level of detail here, especially the way you included your whole thought process, including false leads. (But, as puzzle-maker, I reserve the right to use this information to make harder puzzles. Muahahaha!) $\endgroup$ – plasticinsect Jul 17 at 16:33
  • $\begingroup$ This must be one of the prettiest answers I've seen on this site! Maybe you could also mention the potential chirality issue: looks you can choose to fold the "legs" either up or down. One choice makes the "feet" point in the clockwise direction, and the other gives the mirror image, with the feet pointing the other way. Either fold works, but both pieces need to be folded in the same direction, or they won't fit together. $\endgroup$ – Bass Jul 18 at 11:47
  • $\begingroup$ @Bass thanks, that is true. During solving, I wondered if each part should be folded the same way (be identical) but soon saw that they must be. But I had wondered if there are two solutions, and you have answered that: there are, by folding the other way. The same is true for the OP's previous puzzle. $\endgroup$ – Weather Vane Jul 18 at 12:03
  • $\begingroup$ Interesting. I tend to think of it not so much as two solutions for the same puzzle, but more as one solution for the shape as given, and a mirror image of that solution for the shape's mirror image. But I think seeing it as one shape with two solutions is equally valid for the puzzles as I described them. It's just semantics. Maybe I interpret this differently because I tend to solve these things on the computer? It's easy to think of a piece of paper as having two sides, but for a digital image not so much. I'll try to be a little more specific with my descriptions next time. $\endgroup$ – plasticinsect Jul 18 at 16:43
  • $\begingroup$ If the final 3D solution is not mirror symmetric, then its mirror image will always also be a solution, just by folding the pieces in the opposite direction. You could see this as a consequence of the unfolded paper itself being symmetric in 3D - its plane of symmetry being the plane of the paper itself. Just imagine building it on a mirrored table - as you fold it up, your mirror image folds it down. $\endgroup$ – Jaap Scherphuis Jul 18 at 17:06

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