[CV117] Painting a Cube

Painting a Cube is a puzzle in Professor Layton and the Curious Village.

Puzzle

In front of you sits a blank paper cube that you've decided to paint. You need to paint the cube so that all faces that touch are different colors.

Using three colors of paint, how many ways can you paint the cube so that it satisfies the above condition?

Each painting scheme should be a different pattern, not just the same pattern with the colors rearranged. Also, assume that you can't leave any sides of the square blank.

Hints

Click a Tab to reveal the Hint.

As you know, all cubes have six sides. Because of this structure, every face of the cube touches four others, meaning that only one of the five other faces doesn't touch any given face.

Taking Hint One a step further, in order to paint the cube three colors and have no two connecting faces be of the same color, you should use each color to paint opposing faces.

You need to paint two opposing faces of the cube each color. Count how many different ways there are of doing that and you've solved the puzzle.

Just remember, simply reconfiguring which colors go where doesn't count as an entirely new arrangement.

Solution

Incorrect

Nope!

All rotated and color-swapped versions of a particular painting scheme count as the same pattern. Don't count them as separate patterns when tallying up the total number of possible solutions.

Correct

If you have to paint the cube with three colors, then your only choice is to paint opposing sides of the cube the same color.

As seen in the diagram above, even if you were to change where you used each color, rotating the cube proves that you're really just reusing the same idea of painting opposing sides the same color.

There's only one unique way to color this cube using three paints.

A big thanks to http://professorlaytonwalkthrough.blogspot.com

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Puzzle

Hints

Solution

Incorrect

Correct