My lousy car has an odometer without 4s — in every position, the counter advances from 3 directly to 5. For example, when it read 000039 I drove one mile and watched it roll over to 000050. Today the odometer reads 002005. How many miles has the car actually traveled?
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Because it’s using 9 digits, the odometer is recording the mileage in base 9, except that its digits 5, 6, 7, 8, and 9 represent the base-9 digits 4, 5, 6, 7, and 8. So the actual (base 10) mileage today is just 2004(9), or 2 × 93 + 4 = 2 × 729 + 4 = 1462.
(From the 2005 American Mathematics Competition.)
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