Pentagon Island has one airport located in the town of Winchester. A railroad links the five cities on the island -- Winchester, Bennettsville, Columbus, Flint, and Ventura. The track forms a complete loop, and trains run in both directions (see the diagram below).
When you land at the airport in Winchester, you are given a free pass, for eight trips on the train, between one town and the next. In how many different ways can you make a trip of exactly eight trips, starting and ending in Winchester?
If you are thinking that it is impossible to use all 8 tickets -- here is one possible trip: W - B - W - B - W - B - W - B - W.
Solution to the Problem:
The answer is 70 different trips!
To solve this problem, I drew a tree diagram. I started with W on one end. There are only two cities that can be reached directly from W -- they are B and V. Then from B, you can go to W or C. I continued this for the top half of the tree and then I could see by symmetry that there would be double that number. Here are the thirty-five 8-city trips in the top half of the tree:
W-B-W-B-W-B-W-B-W
W-B-W-B-W-B-W-V-W
W-B-W-B-W-B-C-B-W
W-B-W-B-W-V-W-B-W
W-B-W-B-W-V-W-V-W
W-B-W-B-W-V-F-V-W
W-B-W-B-C-B-W-V-W
W-B-W-B-C-B-C-B-W
W-B-W-B-C-F-C-B-W
W-B-W-V-W-B-W-B-W
W-B-W-V-W-B-W-V-W
W-B-W-V-W-B-C-B-W
W-B-W-V-W-V-W-B-W
W-B-W-V-W-V-W-V-W
W-B-W-V-W-V-F-V-W
W-B-W-V-F-V-W-V-W
W-B-W-V-F-V-W-B-W
W-B-W-V-F-V-F-V-W
W-B-C-B-W-B-W-B-W
W-B-C-B-W-B-W-V-W
W-B-C-B-W-B-C-B-W
W-B-C-B-W-V-W-B-W
W-B-C-B-W-V-W-V-W
W-B-C-B-W-V-F-V-W
W-B-C-B-C-B-W-V-W
W-B-C-B-C-B-C-B-W
W-B-C-B-C-F-C-B-W
W-B-C-F-C-B-W-B-W
W-B-C-F-C-B-W-V-W
W-B-C-F-C-B-C-B-W
W-B-C-F-C-F-C-B-W
W-B-C-F-V-F-C-B-W
w-v-f-c-b-c-f-v-w
w-v-f-c-f-c-f-v-w
w-v-f-c-f-v-f-v-w
w-v-f-c-f-v-w-b-w
w-v-f-c-f-v-w-v-w
w-v-f-v-f-c-f-v-w
w-v-f-v-f-v-f-v-w
w-v-f-v-f-v-w-b-w
w-v-f-v-f-v-w-v-w
w-v-f-v-w-b-c-b-w
w-v-f-v-w-b-w-b-w
w-v-f-v-w-b-w-v-w
w-v-f-v-w-v-f-v-w
w-v-f-v-w-v-w-b-w
w-v-f-v-w-v-w-v-w
w-v-w-b-c-b-c-b-w
w-v-w-b-c-b-w-b-w
w-v-w-b-c-b-w-v-w
w-v-w-b-c-f-c-b-w
w-v-w-b-w-b-c-b-w
w-v-w-b-w-b-w-b-w
w-v-w-b-w-b-w-v-w
w-v-w-b-w-v-f-v-w
w-v-w-b-w-v-w-b-w
w-v-w-b-w-v-w-v-w
w-v-w-v-f-c-f-v-w
w-v-w-v-f-v-f-v-w
w-v-w-v-f-v-w-b-w
w-v-w-v-f-v-w-v-w
w-v-w-v-w-b-c-b-w
w-v-w-v-w-b-w-b-w
w-v-w-v-w-b-w-v-w
w-v-w-v-w-v-f-v-w
w-v-w-v-w-v-w-b-w
w-v-w-v-w-v-w-v-w
Correctly solved by:
| 1. Evelyne Stalzer | Montville, New Jersey |
| 2. James Alarie | University of Michigan -- Flint Flint, Michigan |
| 3. David & Judy Dixon | Bennettsville, South Carolina |
| 4. Jeffrey Gaither | University of Virginia Charlottesville, Virginia |
| Honorable Mention goes to Ana Gonzalez of Winchester, Virginia whose answer was very close. She sent in an enumerated list similar to that above. |