A set of building blocks contains a number of wooden cubes. 
The six faces of each cube are painted, each with a single 
color, in such a way that no two adjacent faces have the 
same color.  Given that only five different colors have 
been used and that no two of the blocks are identical in 
their colorings, what is the maximum number of blocks there 
can be in the set? 
 
 

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                  THE SOLUTION
 

The answer is 55.

The maximum number of blocks in the set is 55.  If only three
of the five available colors are used, then opposite faces of
a block must have the same color.  Thus by symmetry there is
only one way in which a block can be painted with any three
given colors, and there are 10 different ways in which three
colors can be chosen.

If four colors are used, then two pairs of opposite faces
must each have the same color.  By symmetry it doesnt matter
which way around the other two faces are painted.  The colors
for the two pairs of matching faces can be chosen in ten
different ways, and the other two colors can then be chosen
in three ways, giving an overall total of 30 combinations.

Finally, if five colors are used then just one pair of opposite
faces will have the same color.  The remaining four colors can
be arranged in three different ways, so using five colors gives
a total of 5 x 3 = 15 combinations.

The maximum number of blocks in the set is
therefore 10 + 30 + 15 = 55.