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(3): computer solution by Charlie)
That's true, and that same car dealer had better not permute the digits either. But we're allowing that in this puzzle (at least in the last paragraph). I tried to clarify that with the phrase "rotations of the entire wheel set".
Anyway, I was personally surprised to find that once the wheels are positioned for one solution, you can find any of the other possible solutions by simply rotating the whole set, and permuting the wheels. In this sense, the solution is unique.
re(4): computer solution
