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The following shape:

unwrapped cube

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 this be done?

plasticinsect
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1 Answers1

184

The shape

has an area of 30, if each square is taken to be 1 unit. So one face of the cube must have area 5: the easiest way to make an area-5 square on a lattice is by using a knight's move as your side.

Using this, we can make a guess for how the cube might be folded:

cube folding, part 1
Overlaying this net looks nice: the top of the cube is nearly done, the four sides around it look pretty good, and the back is mostly uncovered. The black shapes are good, but the holes in the middle are a problem. We can solve this problem by taking the gray sections along the border, and folding them along the lines between the four sides, as shown by the blue arrows. (Remember, the four sides really connect to each other! The blue arrows are legal folds, even though they might not seem like it at first.)

Once that fold is done, the shape looks more like this:

enter image description here
This is the same shape in 3D, and it's still connected as it was before, but I've moved some of the pieces across the seams of the net. Now it should be pretty clear how the rest folds up: each piece still hanging off the cube will be folded to cover 1/4 of the bottom face (as the lower right piece already shows), and then wrapped around to fill the small triangular hole in a different side of the cube.

A drawing of the finished product:

enter image description here
Here, one "arm" of the original shape has been colored dark gray to show how it folds up.

And an animation of the whole process:

enter image description here

Wossname
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Deusovi
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    +1, nice diagrams – micsthepick Jun 12 '19 at 07:54
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    Let's say I knew the answer right off the bat (I didn't). There's no way I could construct an answer like this within 1 hour of the question being asked. How much of that CPH4 did you actually take? – Strawberry Jun 12 '19 at 16:32
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    Impressive. This is a much better explanation than I could have come up with in twice the time. (and you presumably had to spend some of that time actually solving the problem!) – plasticinsect Jun 12 '19 at 16:44
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    Diagram 2 took me a couple of minutes to understand. Perhaps if the two topmost faces were drawn folded upright using orthagonal projection, the rest of the diagram would be more intuitive. – Vaelus Jun 13 '19 at 01:07
  • Wow. Very nice explanation, and diagrams. This wasn't the first time you'd seen the problem, surely? – ChrisA Jun 13 '19 at 22:35
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    @ChrisA Nope, it was the first time! Took about 10 minutes of fiddling to figure out that that maneuver was possible, then once I got that, everything fell into place. – Deusovi Jun 13 '19 at 22:36
  • I started thinking about area also, but see an area of 31. Doing 70hr weeks though, so perhaps I'm tired.. – Lamar Latrell Jun 14 '19 at 02:59
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    @LamarLatrell There are 4 + signs (area 5 each, so total area 20), a single square in the middle (area 1), and four triangles (each a quarter of a 3x3 square, so total area 9). – Deusovi Jun 14 '19 at 03:01
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    Ah, I've miscounted the corners as being area 2.5, when they're actually 2.25.. – Lamar Latrell Jun 14 '19 at 03:06
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    @Wossname Whoa, thanks for adding that animation! That's really cool (and I wish there was some way to give you a bounty for it, because you deserve credit for it). – Deusovi Jun 14 '19 at 06:22
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    @Wossname Wow, that animation is beautiful. May I ask what software you did it on? – plasticinsect Jun 14 '19 at 07:14
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    That animation is awesome! – Chris Jun 14 '19 at 16:42
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    @plasticinsect It was done in Blender. I mainly did it to convince myself the solution was real! – Wossname Jun 15 '19 at 03:13
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    Awesome animation, kudos to @Wossname. And more kudos to Deusovi for having a mind that can think like that. I wouldn't have known where to start with this one, and as for visualising it... wow. – ChrisA Jun 15 '19 at 08:04
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    @Wossname Nicely done. Thank you for posting it once you'd convinced yourself! – plasticinsect Jun 15 '19 at 19:53
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    @Wossname should post the animation as a separate answer to get some credit. – pacoverflow Oct 05 '19 at 05:14
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    @Wossname if you have the urge to make another animation, the OP has asked a similar question here. (Apologies to Deus about this ping.) – Brandon_J Jan 27 '20 at 19:55