[Illustration]
The problem of the Bridges may be reduced to the simple diagram shown
in illustration. The point M represents the Monk, the point I the
Island, and the point Y the Monastery. Now the only direct ways from M
to I are by the bridges a and b; the only direct ways from I to Y are
by the bridges c and d; and there is a direct way from M to Y by the
bridge e. Now, what we have to do is to count all the routes that will
lead from M to Y, passing over all the bridges, a, b, c, d, and e once
and once only. With the simple diagram under the eye it is quite easy,
without any elaborate rule, to count these routes methodically. Thus,
starting from a, b, we find there are only two ways of completing the
route; with _a, c_, there are only two routes; with a, d, only two
routes; and so on. It will be found that there are sixteen such routes
in all, as in the following list:--
a b e c d b c d a e
a b e d c b c e a d
a c d b e b d c a e
a c e b d b d e a c
a d e b c e c a b d
a d c b e e c b a d
b a e c d e d a b c
b a e d c e d b a c
If the reader will transfer the letters indicating the bridges from the
diagram to the corresponding bridges in the original illustration,
everything will be quite obvious.