A mechanical six-digit car odometer has 6 wheels with the digits 0-9 on each wheel. Imagine taking the odometer out of the car and taking off the cover so you can see all the digits on all the wheels. Each row forms a six digit number. If the first row reads 123456, the next row would read 234567 and so on to the 10th row which would read 012345.

Consider the sum of the digits in each row. Is there a setting of the odometer that results in the sum of each row being the same?

If not, what's the best we can do? Let's define "best" as a setting where difference between the smallest sum and largest sum is minimized. What's the smallest odometer reading that achieves this minimum difference, and what is the difference value?

Finally, if we drop the "smallest odometer reading" requirement, then other than permutations of the wheels and rotations of the entire wheel set, how many distinct solutions are there? Or is this solution unique?

(In reply to re(2): computer solution by Eric)

"013568 is not distinct from 124679"

But if a car dealer sells you a car where the odometer shows 013568, but really had travelled 124679 miles (or 1/10 these), I'd think you should report him to the BBB and regulatory agencies.  The puzzle calls for odometer settings, which depend on what's visible through the window, not just the relative positions of the separate wheels.  I take that as the setting before removing the stack of wheels, taking the cover off.

Posted by Charlie on 2006-01-20 09:01:40