INSPECTING A MINE.
(
Unicursal and Route Problems)
The diagram is supposed to represent the passages or galleries in a
mine. We will assume that every passage, A to B, B to C, C to H, H to I,
and so on, is one furlong in length. It will be seen that there are
thirty-one of these passages. Now, an official has to inspect all of
them, and he descends by the shaft to the point A. How far must he
travel, and what route do you recommend? The reader may at first say,
"As there are thirty-one passages, each a furlong in length, he will
have to travel just thirty-one furlongs." But this is assuming that he
need never go along a passage more than once, which is not the case.
Take your pencil and try to find the shortest route. You will soon
discover that there is room for considerable judgment. In fact, it is a
perplexing puzzle.
Answer:
Starting from A, the inspector need only travel 36 furlongs if he takes
the following route: A to B, G, H, C, D, I, H, M, N, I, J, O, N, S, R,
M, L, G, F, K, L, Q, R, S, T, O, J, E, D, C, B, A, F, K, P, Q. He thus
passes between A and B twice, between C and D twice, between F and K
twice, between J and O twice, and between R and S twice--five
repetitions. Therefore 31 passages plus 5 repeated equal 36 furlongs.
The little pitfall in this puzzle lies in the fact that we start from an
even node. Otherwise we need only travel 35 furlongs.
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