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The Tube Railway

The above diagram is the plan of an underground railway. 
The fare is uniform for any distance, so long as you do not go twice along any portion of the line during the same journey. 
Now a certain passenger, with plenty of time on his hands, goes daily from A to F. 
How many different routes are there from which he may select? 
For example, he can take the short direct route, A, B, C, D, E, F, in a straight line; or he can go one of the long routes, such as A, B, D, C, B, C, E, D, E, F. 
It will be noted that he has optional lines between certain stations, and his selections of these lead to variations of the complete route. Many readers will find it a very perplexing little problem, though its conditions are so simple

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