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?
Answer
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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