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?

It took me a few minutes to get the odometer out of my car, and then I realized it was digital. . .

With 6 starting numbers, and 10 choices, the most even distribution would have starting numbers 10/6 = 1.67 apart. Since we are bound by integers:

0 x 1.67 = 0
1 x 1.67 = 1.67 (round to 2)
2 x 1.67 = 3.33 (round to 3)
3 x 1.67 = 5
4 x 1.67 = 6.67 (round to 7)
5 x 1.67 = 8.33 (round to 8)

Using starting numbers: 0,2,3,5,7, and 8 I got totals ranging from 23 to 31. This gives a difference between max and min of 8.

I found the same solution for dropping the minimum result {0,2,3,5,7,8). (Read as: I could not improve the first answer.) However with 23 occuring twice, the difference is still 8.

Posted by Leming on 2006-01-19 13:29:29