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The following set of eight shapes (which sort of looks like XOXOXOXO if you squint with your brain a little):

Eight shapes, consisting of four X's and four 2x3 rectangles.

... can all be folded onto the surface of a single cube in a way that covers the entire cube with no gaps or overlaps. How can this be done?

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2 Answers 2

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The total area of the given shapes is 60 square units. Therefore, each face of the cube must be 10 square units. The edge length is the diagonal of a 1 x 3 rectangle. With this in mind, we find this solution:

The four hugs share a common vertex which is the center of one cube face. With the proper orientation, the four kisses can be wrapped around them to cover the cube as desired.
Unfolded cube view

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  • $\begingroup$ Ah, just noticed your answer, which is the same as mine. $\endgroup$ Commented 2 days ago
  • $\begingroup$ nice answer! A detail that will not change anything: shouldn't the OP have rather associated the X to a hug (like 4 arms) and the O with a kiss (like the mouth shape) ? $\endgroup$ Commented 16 hours ago
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    $\begingroup$ @mountrix - I agree that that would make more sense, but traditionally the X's have stood for kisses, and the O's have stood for hugs. Even less logically, 'XOXO...' is usually written with the X first, even though it is almost always spoken as "hugs and kisses" (not "kisses and hugs"). Tradition is not always logical. $\endgroup$ Commented 13 hours ago
  • $\begingroup$ @mountrix See Wikipedia Hugs and kisses. Little is known about the origin of using X and O to represent kisses and hugs. $\endgroup$ Commented 10 hours ago
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Here’s how I did it:

cube covering with Xs and Os

There’s a nice symmetry to the covering. The Xs are arranged on four adjacent faces that go around the cube (forming a cylinder) such that the bottom face is also covered. The Os can then be used to cover the top face.

We can calculate the side of the cube to be √10, which is the hypotenuse of a right triangle with legs 1 and 3. This suggests the orientation of the pieces.

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    $\begingroup$ This is one of those times where I really wish I could award more than one check mark. I like both your answers equally, but I'm going to have to give it to Daniel Mathias' answer because it was posted sooner. Nonetheless, nice work, +1. $\endgroup$ Commented 2 days ago
  • $\begingroup$ @plasticinsect. No worries! Thanks for the nice words! $\endgroup$ Commented 2 days ago

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