[Math] How to find out the number of ways to solve Instant Insanity

combinationscombinatoricspermutations

Problem : We are given 4 cubes. The 6 faces of every cube are variously colored – Blue, Green, Red or White. Stack the cubes on top of another in such a way that no color appears twice on any of the four sides of this column.

The book says that by a trial-and-error method, a person can try 41,472 ways.
It is given as : 41,472 = 3 * 24 * 24 * 24

How did they do it? I know that by using proper Permutations and Combinations, one can deduce the number of ways. Can anyone please guide me through this?

Best Answer

I am puzzled by the absence of information on how the cubes are actually coloured.

  • what colour is used for the remaining faces?

  • are the colours applied the same for all dices (like standard dice have the dots painted in the same relative arrangement)?

  • how are the colours distributed? Are there all blue cubes?

To compensate for this lack of information, I came up with this story:

Assuming that we use a simple bot without camera to perform the manual labour of stacking the dice.

There are 6 ways to choose the front face, times 4 orientations = 24 configurations.

This applies to four cubes and gives $$ 24 \times 24 \times 24 \times 24 $$

trial configurations.

The control mechanism has colour detection, and it inspects only the four sides.

However

  • The validity of the pattern along the sides is the same (either matching the criterion or not) if we rotate the column as a whole in one of four ways, reduction by factor 4 is possible.

  • The validity of the pattern is the same, if we turn the whole column upside down or not, reduction by factor 2 is possible.

So we would try to provide a software update to the bot, which just tries $$ 3 \times 24 \times 24 \times 24 $$ times and is guaranteed to stack up a valid configuration, if it exists.

Update: I found a page about this puzzle Instant Insanity, the cubes are coloured indeed by $4$ colours applied to the 6 faces. It turns out that the $4^6 = 4096$ ways to colour $6$ faces with $4$ colours reduce to $240$ possibilites if one considers rotations. (Link) The producers of this puzzle pick a specific set of $4$ out of the $240$ possible cubes which has exact one solution.

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