This puzzle was artfully devised by the yellow man. It is not a matter
for wonder that the representatives of the five countries interested were
bewildered. It would have puzzled the engineers a good deal to construct
those circuitous routes so that the various trains might run with safety.
Diagram 1 shows directions for the five systems of lines, so that no line
shall ever cross another, and this appears to be the method that would
require the shortest possible mileage.
[Illustration]
The reader may wish to know how many different solutions there are to the
puzzle. To this I should answer that the number is indeterminate, and I
will explain why. If we simply consider the case of line A alone, then
one route would be Diagram 2, another 3, another 4, and another 5. If 3
is different from 2, as it undoubtedly is, then we must regard 5 as
different from 4. But a glance at the four diagrams, 2, 3, 4, 5, in
succession will show that we may continue this "winding up" process for
ever; and as there will always be an unobstructed way (however long and
circuitous) from stations B and E to their respective main lines, it is
evident that the number of routes for line A alone is infinite. Therefore
the number of complete solutions must also be infinite, if railway lines,
like other lines, have no breadth; and indeterminate, unless we are told
the greatest number of parallel lines that it is possible to construct in
certain places. If some clear condition, restricting these "windings up,"
were given, there would be no great difficulty in giving the number of
solutions. With any reasonable limitation of the kind, the number would,
I calculate, be little short of two thousand, surprising though it may
appear.