# Sum of the Digits

December 14, 2020 pm31 1:03 pm

A little math puzzle.

I haven’t posted one of these in a while.

Consider the sum of the digits of three-digit numbers. For example, 311, sum is 5. 420, sum is 6. 911, sum is 11.

Try any or all of these:

- What is the average sum of the digits of a three-digit number?
- What is the most common sum of the digits of a three-digit number?
- How many three-digit numbers have the property that the sum of their digits is 12?

Solutions later this week (or in the comments – up to you!)

8 Comments
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So do yuo define “3 digit number” to have a non- zero leading digit or is 001 and 010 eligible

By 3 digit number I mean natural numbers from 100 to 999.

Great problem, Jonathan!

(a) 1 appears 100 times in the 100’s place and 90 times in each of the other 2, so 1 appears 280 times. This is the same for each of the digits 2-9. Thus the sum of all 900 3-digit #’s is

280(1+2+3+…+9) = 280×45 = 12600 so the average sum = 12600/900 = 14

(b) I’m looking for the sum that has the most permutations. The mean of the digits 0-9 is 4.5 so I considered 5+5+5=15 and 5+5+4=14 and 5+4+4=13. If I counted correctly, 13 and 15 produce 69 possible 3-digit #’s and 14 produced 70 results. But I could be off!

(c) 12 resulted in 66 but I’m careless so who knows!

Please correct my errors!

(a) I put for friends, not necessarily math-y, on facebook. There are several approaches, all lead to the same end.

(b) was a lark, I hadn’t thought about it, and now I have a gut feeling that there is NOT symmetry here, which means I know what I will thinking about when I turn off the lights.

(c) was a homework question (the homework question) for a group of kids who have just been solving for positive numbers and for non-negative numbers… It’s fun teaching a puzzle class.

But symmetry and a normal-type distribution makes so much sense! Largest and smallest sums are 27 (999) and 1 (100) whose mean is 14!! With 2 dice of course the largest sum is 12 and smallest is 2, whose avg is 7, which is the most frequently occurring sum (mode). Thanks again for making me think!

I would have thought that two constraints (first digit can’t be 0, others can, and the digits cannot exceed 9) would break symmetry. I’ll have a go at it.

It is symmetric. Triangular numbers 1 – 9, reflected 27 – 19. In between they are decreased (by 1, 5, 12, 22, 35, 22, 12, 5, 1). Wonder what those numbers correspond to?

They are the number of ways to get a sum of s, with one of the digits equal to 10 or greater (not a valid 3-digit number), maintaining the constraint that the first digit may not be 0.

For a sum of 11 we expect 66, we get 5 less, 61, because the following five are not valid 3-digit numbers:

E00, T10, T01, 1T0, 10T (using E for an eleven digit and T for a ten digit)