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PUZZLE #7: PALINDROMES IN THE ODOMETER

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*?

(solution)