The following blocky shape:
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?
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asked Jul 15, 2019 at 6:22
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$\begingroup$ The dimensions of the various parts are not clear. BY cover you mean the 6 faces? $\endgroup$– MotiCommented Jul 15, 2019 at 7:09
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$\begingroup$ By 'cover', I mean one copy of this shape can be folded to cover all six faces of a cube, with no gaps or overlaps. The dimensions may appear to be unclear because this shape is designed so no edge lines up with any other edge. This is by design, and should be considered part of the challenge. The image is precise down to the width of a pixel. $\endgroup$– plasticinsectCommented Jul 15, 2019 at 7:22
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8$\begingroup$ The fact that both answers printed it out made cube delights me! $\endgroup$– aslumCommented Jul 15, 2019 at 18:41
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2 Answers
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This seems to work:
Below, I printed out the shape, and cut off the excess. The white parts are for glueing; if everything works out as planned, all of them will be covered by the coloured bits around the black squares.
Joy, it all worked! Here's the final cube, with some white "intentionally" showing through between the pieces, highlighting the borders:
The key to solving this was to
notice that the big squares are way too big to be the side of the cube. Also, because the big squares need to be folded, and the cube sides are going to be smaller than them, all the 4 corners of one cube side really want to be touching the edges of the big square: if you were to place a 3D corner anywhere inside a solid area, the folds would unavoidably cause a wrinkle.
After that, it was a matter of figuring out the proper size and orientation of the cube's edge that's inside the big square. I first tried "rotated 45 degrees", but that didn't seem to pack well. The natural next guess was to put a fold through the 270 degree "inside" angle, which fixed the cube side's position inside the big square, and that happened to work all the way through.
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4$\begingroup$ Props to you for the bounty. Well and honorably played. :) $\endgroup$– Rubio ♦Commented Jul 18, 2019 at 4:37
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1$\begingroup$ Since the posteriority will only be able to see Rubio's comment above, I'll put a short history here: User @WeatherVane answered this question 51 seconds before me. When both answers (nearly identical) were at 40-50 upvotes, OP decided that my answer deserved the tick (I agree, look at all the pretty colours!), so mine started to be listed first. This made little difference, until disaster struck: An unnamed mod accidentally deleted WeatherVane's answer. Eventually everything was straightened out, but I felt I had got undue advantage, so splitting the extra rep seemed the right thing to do. $\endgroup$– BassCommented Jul 20, 2019 at 18:19
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The shape can be folded like this
Then it should become obvious.
Method of solving:
I started out by cutting out the shape, noticed some equal lengths, and initially folded it in the wrong places (as can be detected by creases in my model). I then wondered if the four smaller rectangles would wrap around four sides of the cube. I marked the parallel diagonals of the four small rectangles, and noticed that a line touching the tips of their saw-tooth edges is at 90° to them. I could then mark out the two other faces in the largest squares, and from there it all fell into place, leaving the larger squares as the top and bottom caps, folded to fill in the gaps.
As I was working with a print of the black image in the question, it proved rather difficult getting a photogenic shot, and I ended up retouching it to show where the original edges are.
answered Jul 15, 2019 at 8:43
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1$\begingroup$ Nice job with the new photo, would upvote a second time if I could. $\endgroup$– BassCommented Jul 15, 2019 at 11:29
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$\begingroup$ @Bass thanks, I had already made a loose cube at the time of first posting and needed several attempts at the photo, which still needed graphics on it. $\endgroup$ Commented Jul 15, 2019 at 11:32
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2$\begingroup$ I'm sorry I can't accept both your posts. You and Bass posted good answers pretty much simultaneously. $\endgroup$ Commented Jul 15, 2019 at 16:40
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7$\begingroup$ I'm not short of 15 points. I enjoyed solving this puzzle. $\endgroup$ Commented Jul 15, 2019 at 17:18
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2$\begingroup$ @Bass awarding a bounty is most generous and chivalric of you. Although it comes off your own reputation, it augments you. Many thanks. $\endgroup$ Commented Jul 19, 2019 at 8:30