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 computer solution by Charlie)

So, Charlie, are you saying there are 5 unique solutions...not counting permutations and rotations of the entire wheel set?  If so, look again.

Let me make sure I'm clear.  If 013568 is a solution, 902457 is part of the same solution (just read the previous row on the odometer), and so on.  And so are all possible permutations of each of those readouts (by rearranging the wheels without rotating them with respect to each other).

Posted by Ken Haley on 2006-01-20 01:05:23