The Tube Railway
(
MISCELLANEOUS PUZZLES)
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.
Answer:
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.
