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The following slightly moth-eaten integral sign:

moth-eaten integral sign

... can be folded 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?

(You may notice that the placement of the two little "bite marks" is not quite symmetrical. This is not a mistake, it's a challenge.)

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I made folds like this:


enter image description here

The area is 150 units, each face area is 25, so side length is 5. The folds cannot be orthogonal with simple 5x5 squares, because the longest run is 34, which would be more than 4 face widths.

So I guessed that the grid will be based on a 3-4-5 right triangle.
The graphic shows the grid points stepping 3 units one way and 4 the other.

But how to align the grid?
After a couple of trials I noticed it can be arranged so the extreme tips are on grid points, and one corner of every notch too. Then when I cut and folded, I could see how tiny triangles would fold and meet at corners, such as the pairs at each end of the central bar. They are not exactly half-squares, but they join to each other when the cube corner is folded. There is also a minute triangle by one of the notches, which manages to fit somewhere, perhaps in the marked place shown in the next part-build photo. My crafting was wildly inaccurate with large gaps in places, but I am fairly sure I found the correct solution.

enter image description here

The red line marking on the photo was fairly tricky (and possibly incomplete), as I don't think there are two faces which are quite the same. As for the lack of symmetry noted in the question, either of those two notches could be moved along by one unit (so that the other internal corner is on the grid point) to make the shape symmetrical. A nice touch by @plasticinsect!

enter image description here

These puzzles are getting harder and harder. In this case it was impossible for me to eyeball where any face would be, and it only started to make sense once I started cutting and folding.

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  • $\begingroup$ Nice job, well done! On the subject of trickiness, I was thinking that since there are no "flaps" thin enough to cover the "moth-eaten" 1x1 holes, then there must be a fold through each of them, and from that the grid quickly becomes apparent. $\endgroup$
    – Bass
    Feb 12, 2020 at 13:15
  • $\begingroup$ Excellent work! As always, I especially appreciate the detailed description of your thought processes. I am indeed trying to make these progressively harder to solve, but I admit it is getting progressively harder for me to do that. :) As the puzzle-maker learns new tricks, so do the puzzle solvers! $\endgroup$
    – plasticinsect
    Feb 12, 2020 at 20:32
  • $\begingroup$ @Bass rot13(Lrf, vaqrrq. Gur "ovgrf" ng svefg nccrne gb or na naablvat pbzcyvpngvba gung zvtug znxr gur chmmyr uneqre, ohg gurl ner npghnyyl n inyhnoyr pyhr. Gur snpg gung gurl ner abg flzzrgevpnyyl cynprq npghnyyl cebivqrf n pyhr nf jryy, nf guvf zrnaf gur iregrk (bs gur phor) pna bayl or ybpngrq ba gur bar pbeare gung obgu ovgrf unir va pbzzba. Vs gurl jrer flzzrgevpny, gurer jbhyq or gjb cynprf gur iregrk zvtug tb.) $\endgroup$
    – plasticinsect
    Feb 12, 2020 at 20:42
  • $\begingroup$ I could not solve these puzzles without constructing a model, but that is not easy. Even printed at maximum size on A4 paper, the resulting cube dimension is less than 4cm making it hard to hold two parts aligned correctly together while applying stick tape to the inside. Then after a point, the tape has to go outside. If there is another puzzle, I will mark up the edges with a hiliter pen before folding. $\endgroup$
    – Weather Vane
    Feb 12, 2020 at 20:46
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    $\begingroup$ Have you considered making other shapes instead of cubes? Cuboids might be a new kind of challenge, or other platonic solids. $\endgroup$
    – Magma
    Feb 12, 2020 at 23:12

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