This is another "puzzlah" from Car Talk, on NPR radio. I reproduce it here because I want to emphasize the uniqueness of the solution. It's also very cute, I must say.

My car's odometer measures distance traveled in whole numbers--no tenths of miles--up to 999,999. Last week I was driving along on the freeway and noticed that the last four digits, but not the last five digits, formed a palindrome; that is, it read the same backwards as forwards (e.g., 1221). After one mile went by, though, the last five digits did form a palindrome; and, after one more mile, the middle four digits formed a palindrome. Finally, when the third mile rolled over, all six digits formed a palindrome. What was my mileage at the time I first noticed all these palindromes-- and why?