In case anyone is interested, the answer to the Google Treasure Hunt 2008 robot path question is as follows:

For a board of size RxC, where R is the number of rows and C is the number of columns and P is the number of distinct paths the robot could take, we find that:

By the way, this particular question has little to do with networking or computer science and nothing at all to do with low-level UNIX trivia. Wonder if the subsequent questions will be more pertinent.