original square diagram with the
circular one shown in Figs. 1, 2, and 3 below.
If for the moment we ignore the shading (the purpose of which I shall
proceed to explain), we find that the circular diagram in each case is
merely a simplification of the original square one—that is,
the roads from A lead to B, E, and M in both cases, the roads from L
(Las Vegas) lead to I, K, and S, and so on.
The form below, being circular and symmetrical, answers my purpose
better in applying a mechanical solution, and I therefore adopt it
without altering in any way the conditions of the puzzle.
If such a question as distances from town to town came into the
problem, the new diagrams might require the addition of numbers to
indicate these distances, or they might conceivably not be at all
Now, I draw the
three circular diagrams, as shown,
on a sheet of paper and then cut out three pieces of cardboard of the
forms indicated by the shaded parts of these diagrams.
It can be shown that every route, if marked out with a red pencil, will
form one or other of the designs indicated by the edges of the cards,
or a reflection thereof.
Let us direct our attention to Fig. 1. Here the card is so placed that
the star is at the town T; it therefore gives us (by following the edge
of the card) one of the circular routes from Las Vegas : L, S, R, T, M,
A, E, P, O, J, D, C, B, G, N, Q, K, H, F, I, L.
If we went the other way, we should get L, I, F, H, K, Q, etc., but
these reverse routes were not to be counted.
When we have written out this first route we revolve the card until
M, when we get another different route, at A a third route, at E a
fourth route, and at P a fifth route.
We have thus obtained five different routes by revolving the card as it
But it is evident that if we now take up the card and replace it with
the other side uppermost, we shall in the same manner get five other
routes by revolution.
We therefore see
how, by using the revolving card
in Fig. 1, we may, without any difficulty, at once write out ten
And if we employ the cards in Figs. 2 and 3, we similarly obtain in
each case ten other routes.
routes are all that are possible..