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?

I made myself an excel sheet to play with.

It makes sense that the more spread apart the numbers are the more the high and low numbers will be distributed and the more similar the row totals.

The best I could do was to set 5 wheels at 0,2,4,6,8 and the 6th odd or set 1,3,5,7,9 and the 6th even.  The smallest odometer reading would be 012468

Max = 33 Min = 21 Diff = 12

There are 6!*5*2 =7200 different permutations that do this.

(Actually if you cheat and let the wheels be 0, 10/6, 20/6, 30/6, 40/6, 50/6 the totals are 33-25=8)

 

Posted by Jer on 2006-01-19 12:39:42