Sudoku Cube

Probability Level 5

A large cube consists of 27 monochromatically colored unit cubes. There are 9 different colors and for each color, there are 3 unit cubes of that color.

Now the unit cubes are assembled in the sense of sudoku, i.e. each face of the prepared large cube may contain each color only one-time.

How many sudoku cubes exist?

Hint: Build the cube sequentially starting from the lowest plane!


The answer is 241920.

Andreas Wendler by Andreas Wendler , 63, Germany

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1 solution

Andreas Wendler
Jan 6, 2017

Starting withh the lowest plane we have 9!/4 different possibilities considering rotation's symmetry.

A cube with the color of the middle cube of the first plane must be positioned in a corner of the second plane, there are 4 possibilities. The third cube of that color is then to be found in the diagonal opposite corner of the third plane.

For the colors of both other diagonal cubes in the third plane we get 2*2=4 possibilities.

Now consider that there are 6 positions for each case constructing the cube since each face can be the lowest plane. So we get 9!/4*16/6=241920 different cubes.

An example (ordered 1st to 3rd plane) is:

1 2 3
4 5 6
7 8 9
5 7 8
3 9 2
6 4 1
9 6 4
8 1 7
2 3 5
1 Helpful 0 Interesting 0 Brilliant 0 Confused

Why do we consider rotations of the base, but not of the sides? E.g. The front face is " 789|641|235". If that was the placement on the bottom face, why is that considered a distinct cube?

Calvin Lin Staff - 4 years, 5 months ago

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Starting with the first plane we have a whole count of 9! * 16 cubes as I showed above. These cubes are invariant against tilting since each of the 6 faces can be the first plane and rotation (4 possibilities) so that 6*4=24 different orientations exist.

This gives us: 9! * 16 / 24 = 241920 different sudoku cubes considering all symmetry properties!

Rem.: Please note that for a 2x2x2 cube with 2 cubes trying one color of 4 different ones respectively applying the formula above we get exactly 1 sudoku cube because:

4! * 1 / 24 = 1

It's remarkable that the number of possibilities enormously increases with the addition of 1 cube in each dimension. This behavior is surely explained by the strongly progressive trend of the factorial function.

Andreas Wendler - 4 years, 5 months ago

I don't really follow this solution.

For instance, you say "For the colors of both other diagonal cubes in the third plane we get 2*2=4 possibilities." Why is this true? (In fact, I'm not sure what you mean by diagonal cube.)

Jon Haussmann - 3 years, 10 months ago

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