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The following shape has an interesting property:

Cube-covering shape

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 this be done, and how many copies of the shape are needed?

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This can be done

with three copies. (There are 20 squares, and we need the total area to be divisible by 6 if we want a whole number of squares to fit on each face, so 3 is the minimum. If we don't have a whole number of squares on each face, there will be partial squares on faces, and the pieces won't fit together perfectly.)

To figure out how to do it,

the natural place to look is at the orange circle in this drawing:
annotated shape That lower-right hole looks like it'll be hard to fill. But if we can put a vertex of the cube at the orange circle, that means we can bend the shape along the orange arrow, so it occupies the dark gray line. Once that's in place, a similar maneuver can be done at the pink circle to bend the upper right side down, compacting the entire shape into a single blob. And now, this can be done with three copies of the shape, making the yellow circled points of each shape touch at a vertex.

Here are some images of the covering:

Front view (focused on the "orange circle" fold):
image centered on front vertex
Top and bottom view (focused on the "yellow circle" vertex, and the point directly opposite that)
image centered on top vertex image centered on bottom vertex

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    $\begingroup$ Jesus Christ, how did you see this?! $\endgroup$
    – greenturtle3141
    Jun 14, 2019 at 14:43
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    $\begingroup$ I am stunned by your near-superhuman puzzle solving ability. $\endgroup$
    – plasticinsect
    Jun 14, 2019 at 17:06
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    $\begingroup$ @greenturtle3141 - it's easy. ...when you live and see in another dimension and merely visit ours briefly to solve our puzzles. $\endgroup$
    – BruceWayne
    Jun 14, 2019 at 18:44
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    $\begingroup$ @greenturtle3141 My answer basically shows my thought process -- first I realized that I'd have to use a 3-forward-1-sideways line as the cube edge (based on counting squares, I needed a side length of √10). The orange spot came next, because it seemed like the only real way to fill the gap there, and then the pink spot came immediately after -- after that, all that was left was drawing it out to make sure that it would cover the cube as it seemed. $\endgroup$
    – Deusovi ♦
    Jun 14, 2019 at 19:21
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    $\begingroup$ @Deusovi - Your comment is not inconsistent with BruceWayne's hypothesis. :) $\endgroup$
    – plasticinsect
    Jun 14, 2019 at 19:35

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