My car has a 5-digit odometer, which measures the miles since the car was built, and a 3-digit trip meter, which measures the miles since I last set it. Every so often, one or both of the readings is a palindrome. The meters reset to 000 after 999 and to 00000 after 99999.

The current readings are 123 and 12345. Assuming that I do not reset the trip meter, when is the next time both readings will be palindromes?

When was the most recent time both readings were palindromes?

Prove no matter what the mileage and trip meters read, they can eventually be made to both be palindromes without resetting the trip meter.

Note: A palindrome reads the same forwards and backwards, like 262 or 37173.

Moving forward, the next few times the two odometers are both palindromes they have these values:

909, 13131
919, 14141
929, 15151
939, 16161
949, 17171
959, 18181
969, 19191

080, 20302
090, 21312

000, 22222
010, 23232
020, 24242
030, 25252
040, 26262
050, 27272
060, 28282
070, 29292

181, 30403
191, 31413

101, 32323
111, 33333
.
.
.

That should be enough to give someone here the spark they need for solving the proof.

Posted by Erik O. on 2004-06-16 09:42:31