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Odometer Sum of the Digits (Posted on 2006-01-19) Difficulty: 3 of 5
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?

See The Solution Submitted by Ken Haley    
Rating: 4.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): computer solution | Comment 9 of 11 |
(In reply to re: computer solution by Ken Haley)

Yes, if 013568 is a solution, 902457 is part of the same solution, and is a permutation of one of the given solutions: 024579.

I think what I am saying is that there are 5 * 720 possibilities for the visible number of miles traveled, seen on the dashboard window, being the 5 possible sets of visible digits and the 720 permutations of each set.  It looks like this is only one set of relative rotations, discounting permutations of the wheels involved.


  Posted by Charlie on 2006-01-20 08:52:49
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