The Palindromic Odometer

Jan 21, 2008

RAY: This came in from Terry Sare and I didn't even have to change it.

Terry writes, "I was driving on the highway the other day recently and I happened to notice my odometer. Like most odometers nowadays, it shows six digits, in whole miles only -- no tenths of a mile. So, if my car had 300,000 miles, for example, I'd see 3-0-0-0-0-0. And that's all. Until I drove another mile, at which point it would read 3-0-0-0-0-1.

"Now, what I saw that day was very interesting. I noticed that the last 4 digits were palindromic, that is they read the same forwards as backwards. For example, "5-4-4-5" is a palindrome. So, my odometer could have read 3-1-5-4-4-5, with those last four digits, starting with the units, then the tens, then the hundreds, and finally the thousands, being the palindrome.

"One mile later, the last 5 numbers were palindromic. For example, it could have read 3-6-5-4-5-6.

"One mile after that, the middle 4 out of 6 numbers were palindromic. So, the first and last numbers weren't involved in the palindrome, but the middle 4 were palindromic.

"And you ready for this? One mile later, all 6 were palindromic! For example, 2-1-3-3-1-2. "

Pretty good, huh? I thought so. So, here again are the conditions as stated by Terry. "I noticed that the last 4 digits were palindromic. I drove a mile, and the last 5 were palindromic. I drove another mile and the middle 4 were palindromic, and the ends were not involved. And then one mile later, all 6 digits were palindromic."

The question is, what did Terry see on the odometer when he first looked?
RAY: The answer is, 1-9-8, 8-8-8. That's what he first saw when he looked.

TOM: Yeah.

RAY: One mile later it was 1-9-8, 8-8-9, the last five are palindromic. One mile later, 1-9-8, 8-9-0, the middle four are palindromic. And --you're ready for this?

TOM: I'm ready.

RAY: You drive one more mile and it's 1-9-8-, 8-9-1, all six are palindromic. So who's our winner this week?

TOM: The winner is Tim Carnes from Ithaca, New York, and for having his answer selected at random from among the pile of correct answers that we got, Tim is going to get a 26-dollar gift certificate to the Shameless Commerce Division at, with which he can pick up our new 20th anniversary hoodie. It's got our original Car Talk logo on the front with factory installed greasy fingerprints on the pockets, and on the back it says, "20 years and Still Not Canceled."

RAY: Congratulations Tim!

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