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The Tube Railway

There are 640 different routes.

A general formula for puzzles of this kind is not practicable.
We have obviously only to consider the variations of route between B and E.
Here there are nine sections or "lines," but it is impossible for a train, under the conditions, to traverse more than seven of these lines in any route. In the following table by "directions" is meant the order of stations irrespective of "routes."
Thus, the "direction" BCDE gives nine "routes," because there are three ways of getting from B to C, and three ways of getting from D to E.
But the "direction" BDCE admits of no variation; therefore yields only one route.

 2 two-line directions of 3 routes — 6 1 three-line " " 1 " — 1 1 " " " 9 " — 9 2 four-line " " 6 " — 12 2 " " " 18 " — 36 6 five-line " " 6 " — 36 2 " " " 18 " — 36 2 six-line " " 36 " — 72 12 seven-line " " 36 " — 432 —— Total 640

We thus see that there are just 640 different routes in all, which is the correct answer to the puzzle.

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