George lives at the corner of 1st Street and 1st Avenue. Each afternoon George visits his girlfriend, Stella, who lives at the corner of 8th Street and 8th Avenue. He varies his route but always walks 1400 meters (it is 100 meters between adjacent Streets and between adjacent Avenues). Sometimes he stops at one of the corner pubs marked as P in the diagram.

On May 1st 2006, after several months of these visits, Stella wants to get married and says to George, "If you don't propose marriage to me soon, I am going to propose to you." George replies, "Starting tomorrow I will walk a different route to your house and when I have used all the different possible routes, you can propose to me." "OK," says Stella, "but you must never go through an intersection with a pub at one of its corners." "It's a deal" replies George.

On what date will Stella be able to propose to George?

The number of ways of getting to any intersection is the sum of the ways of getting to the intersection on the left (west) plus the number of ways of getting to the intersection below (to the south), except at the start there is one way of being located; on the eastern and southern borders there is only one way; and at the four pub locations there are zero valid ways of getting there. That results in this tableau:

 1    8   15   15  120  435 1065 2130

 1    7    7    0  105  315  630 1065

 1    6    0   35  105  210  315  435

 1    5   15   35   70  105  105  120

 1    4   10   20   35   35    0   15

 1    3    6   10   15    0    7   15

 1    2    3    4    5    6    7    8

 1    1    1    1    1    1    1    1

indicating there are 2130 ways of getting from 1st and 1st to 8th and 8th.

So there are 2130 valid (not passing a pub intersection and not going out of one's way) ways of getting from George's house to Stella's house.

2130 days after May 1, 2006, is February 29, 2012. 

Posted by Charlie on 2005-12-26 15:16:55