A map of downtown Winchester is shown below. Billy lives at the corner of Monmouth and Stewart. Betty lives at the corner of Cameron and Boscawen. Billy decides that he will visit Betty every day after school until he has tried every different route from his house to her house. Billy travels only east and north. How many different routes can Billy take to Betty's house? Boscawen +---+---+---+---O---+ | | | | | | Cork +---+---+---+---+---+ N | | | | | | ^ Clifford +---+---+---+---+---+ | | | | | | | W <--+--> E Cecil +---+---+---+---+---+ | | | | | | | ! Leicester +---+---+---+---+---+ S | | | | | | Monmouth O---+---+---+---+---+ | | | | | | S W B L C K t a r o a e e s a u m n w h d d e t a i d o r r n o u o t g c n n t k o n
Solution:
The answer is 126 different ways.
Using a table similar to Pascal's triangle,
the numbers in the following table show how many
different ways you can get to a particular corner.
Remember, you can only travel north and east.
Boscawen 1---6--21--56--126--+ | | | | | | Cork 1---5--15--35--70---+ N | | | | | | ^ Clifford 1---4--10--20--35---+ | | | | | | | W <--+--> E Cecil 1---3---6--10--15---+ | | | | | | | ! Leicester 1---2---3---4---5---+ S | | | | | | Monmouth 1---1---1---1---1---+ | | | | | | S W B L C K t a r o a e e s a u m n w h d d e t a i d o r r n o u o t g c n n t k o n
Alternate solution
C. W. Larson from Northern Arizona University wrote
that Billy can only take one route to Betty's house
since by traveling only north and east, once he
gets to Betty's house, he can never return to his own!
Correctly solved by:
| 1. Christophe W. Larson | Flagstaff, AZ (Northern Arizona University) |
| 2. Trey Genda | Winchester, VA |