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?

8 is the "best" that we can do.

Proof:
If the odometer has no nines in it, then the next reading is different by +6.

If the odometer reading has one nine in it, then the next reading is higher by -4 (ie, it is lower by 4).

If  the odometer reading has two nines in it, then the next reading is 14 lower, so the best definitely requires avoiding two nines in the same number.

Cycling through our ten numbers, then, we go -4 six times and +6 four times, and then we are  back where we are started.  Since we cannot avoid having two - 4's in a row, 8 is the "best" that we can do.

This is achieved by any cycle that avoids -4 three times in a row, and avoids +6 twice in a row.  Further, -4 -4 +6 -4 -4 would lead to a 10 differential.  So, it needs to be
 -4 -4 +6 -4 +6 -4 -4 +6 -4 +6.

In other words, we must avoid having a reading with 9 come up three times in a row, or 4 times out of 5 consecutive readings.

In still other words, our reading can't have three numbers next to each other (mod 10), and can't have 4 numbers which are within 5 consecutive.
 
Our starting number can't have a 0 1 2 in it, because this would leave to 3 consecutive 9's, so the minimum reading starts with 0 1 3.  But the next digit can't be a 4, because this would give 4 nines out of 5 consecutive readings.  So the minimum reading starts with 0 1 3 5.  6 is safe as a next digit, and the last digit cannot be 7 but it can be 8.  So the minimum reading is 0 1 3 5 6 8.

Edited on January 19, 2006, 4:03 pm

Posted by Steve Herman on 2006-01-19 16:00:09