Build an All-red Cube

This is one of my favorite problems given at the 2017 Moscow Olympiad to grades 6 and 7. It was suggested by one of my favorite problem writers: Alexander Shapovalov.

Problem. We are given eight unit cubes. The third of the total number of their faces are blue, and the rest are red. We build a large cube out of these cubes so that exactly the third of the unit cube’s visible faces are red. Prove that you can use these cubes to build a large cube whose faces are entirely red.

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2 Comments

  1. Andrew:

    There are 8 cubes, with 48 faces — 32 red faces and 16 blue faces.

    When assembled into a large cube, there are 24 visible faces and 24 hidden faces — each unit cube shows three and hides three.

    The demonstrated cube has 8 visible red faces and 16 visible blue faces, so it must have 24 hidden red faces and 0 hidden blue faces.

    Every hidden face is opposite a visible face, so by disassembling the large cube and reassembling it with every cube in the “opposite” position (e.g. the top-front-right cube is placed in the bottom-back-left), every hidden face is made visible and every visible face is hidden.

    The resulting large cube has 24 visible red faces and 0 visible blue faces.

  2. Andrew:

    Put more simply: to make the figure in the problem, we had to put every available blue face on the outside. If we can do that, it’s also geometrically possible to put them all on the inside 🙂

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