Imagine you have a cube with sides of 1.4cm each. And you attach to its faces six flat squares, each on a little stick so that they sit 1.4cm away from the central cube. Now around these squares-on-sticks you pack cubes equal in size to the central cube until you’ve filled in all the spaces and you have a new cube, each face of which is composed of six squares, 1.4cm to the side. Imagine you hook the interiors of all the peripheral cubes together so they can slide around each other. You would have an extremely neat little mechanism. And if you had invented it in 1974, you would have the world’s best-selling toy — to date, the Rubik’s Cube has sold more than 400 million copies.
At my house, the family Cube has seen better days. That is because we have a young Labrador. Said Labrador assumes that when everyone leaves the house, we are never coming back and she might as well adjust the decor to her liking. This includes first bringing everything from inside, outside, and then chewing it to little bits as time and energy permit. The Cube recently got her treatment. But I’m happy to say we returned a bit too soon, interrupting her careful disintegration of the world’s most popular toy.
She only managed to remove one small cube.
So, finally, here’s the question:
To calculate the perimeter of a square, you multiply a side by four. To calculate the perimeter of a cube, you multiply a side by 12. But what, now, are the possible perimeters of my family’s irregularly shaped Rubik’s Cube with one cube removed? Does it matter which cube my dog chewed off — the corner, center-side, or center? For the purpose of this puzzle, consider all cubes to be regular, 1.4cm to the side with no space between.