The main point is to discover the smallest possible number of Russians
that there could have been. As the enemy opened fire from all
directions, it is clearly necessary to find what is the smallest number
of heads that could form sixteen lines with three heads in every line.
Note that I say sixteen, and not thirty-two, because every line taken by
a bullet may be also taken by another bullet fired in exactly the
opposite direction. Now, as few as eleven points, or heads, may be
arranged to form the required sixteen lines of three, but the discovery
of this arrangement is a hard nut. The diagram at the foot of this page
will show exactly how the thing is to be done.
[Illustration]
If, therefore, eleven Russians were in the positions shown by the stars,
and the thirty-two Turks in the positions indicated by the black dots,
it will be seen, by the lines shown, that each Turk may fire exactly
over the heads of three Russians. But as each bullet kills a man, it is
essential that every Turk shall shoot one of his comrades and be shot by
him in turn; otherwise we should have to provide extra Russians to be
shot, which would be destructive of the correct solution of our problem.
As the firing was simultaneous, this point presents no difficulties. The
answer we thus see is that there were at least eleven Russians amongst
whom there was no casualty, and that all the thirty-two Turks were shot
by one another. It was not stated whether the Russians fired any shots,
but it will be evident that even if they did their firing could not have
been effective: for if one of their bullets killed a Turk, then we have
immediately to provide another man for one of the Turkish bullets to
kill; and as the Turks were known to be thirty-two in number, this would
necessitate our introducing another Russian soldier and, of course,
destroying the solution. I repeat that the difficulty of the puzzle
consists in finding how to arrange eleven points so that they shall form
sixteen lines of three. I am told that the possibility of doing this was
first discovered by the Rev. Mr. Wilkinson some twenty years ago.