perplexus.info :: Numbers : Odometer Sum of the Digits

Odometer Sum of the Digits (Posted on 2006-01-19)

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?

Submitted by Ken Haley

Rating: 4.3333 (3 votes)

Solution: (Hide)

013568. The largest sum is 31, and the smallest sum is 23, with a difference of 8. Other than rotations and permutations, this solution is unique. That is, these six digits will appear in some sequence on some row of every solution. Each such solution is obviously just a permutation of the wheels of the above solution.
As Charlie pointed out, there are 5 readings that create a solution as follows:
013568
023578
024579
124679
134689
But a permutation of each of these solutions appears in some row (actually on two rows 180 degrees apart) on the wheel set when it's positioned for the first solution.
The pattern that Steve noticed (reading down the columns of the 5 solutions) is amazing!

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(4): computer solution	Ken Haley	2006-01-20 22:40:15
re(3): computer solution	Charlie	2006-01-20 09:01:40
re(2): computer solution	Charlie	2006-01-20 08:52:49
re: computer solution	Ken Haley	2006-01-20 01:05:23
re: Spolier -- Best solution	Ken Haley	2006-01-20 00:52:40
re(2): computer solution	Eric	2006-01-19 19:54:03
re: computer solution	Steve Herman	2006-01-19 16:12:20
Spolier -- Best solution	Steve Herman	2006-01-19 16:00:09
computer solution	Charlie	2006-01-19 15:02:44
Why is it an "odo" meter?	Leming	2006-01-19 13:29:29
Best I could get playing with it	Jer	2006-01-19 12:39:42

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