The reader may have noticed that at each end of
the line I give is a four, so that, if we like, we can form a ring
instead of a line.
It can easily be proved that this must always be so.
Every line arrangement will make a circular arrangement if we like to
join the ends.
Now, curious as it may at first appear, the following diagram exactly
represents the conditions when we leave the doubles out of the question
and devote our attention to forming circular arrangements.
Each number, or half domino, is in line with every other number, so
that if we start at any one of the five numbers and go over all the
lines of the pentagon once and once only we shall come back to the
starting place, and the order of our route will give us one of the
circular arrangements for the ten dominoes.
Take your pencil and follow out the following route, starting at the 4:
You have been over all the lines once only, and by repeating all these
figures in this way,
you get an arrangement of the dominoes (without the doubles) which will
be perfectly clear.
Take other routes and you will get other arrangements.
If, therefore, we can ascertain just how many of these circular routes
are obtainable from the pentagon, then the rest is very easy.
Well, the number of different circular routes over
the pentagon is 264.
How I arrive at these figures I will not at present explain, because it
would take a lot of space.
The dominoes may, therefore, be arranged in a circle in just 264
different ways, leaving out the doubles.
Now, in any one of these circles the five doubles may be inserted in 25 = 32
Therefore when we include the doubles there are
264 × 32 = 8,448
different circular arrangements.
But each of those circles may be broken (so as to form our straight
line) in any one of 15 different places.
Consequently, 8,448 × 15 gives 126,720
different ways as the correct answer to the puzzle.
I purposely refrained from asking the reader to
discover in just how many different ways the full set of twenty-eight
dominoes may be arranged in a straight line in accordance with the
ordinary rules of the game, left to right and right to left of any
arrangement counting as different ways.
It is an exceedingly difficult problem, but the
correct answer is 7,959,229,931,520 ways.
The method of solving is very complex.