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Answer :
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 |
| a
b e d c |
| a
c d b e |
| a
c e b d |
| a
d e b c |
| a
d c b e |
| b
a e c d |
| b
a e d c |
| b
c d a e |
| b
c e a d |
| b
d c a e |
| b
d e a c |
| e
c a b d |
| e
c b a d |
| e
d a b c |
| e
d b a c |
Transfer the
letters indicating
the bridges from the diagram to the corresponding bridges in the
original illustration and everything will be quite obvious.
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