(related to Problem: The Monk And The Bridges)

The problem of the Bridges may be reduced to the simple diagram shown in the 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:— $$\begin{array}{c}a b e c d\\ a b e d c\\ a c d b e\\ a c e b d\\ a d e b c\\ a d c b e\\ b a e c d\\ b a e d c\\ b c d a e\\ b c e a d\\ b d c a e\\ b d e a c\\ e c a b d\\ e c b a d\\ e d a b c\\ e d b a c\\ \end{array}$$

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.

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Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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