The following 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 this be done?
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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:
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:
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:
And an animation of the whole process: