# Misc Solution: Sum of Digit-Sums between one and a million

What is the sum of the digit sums between one and one-million?

## Solution

The solution can be calculated quite quickly without the help of a computer. The trick is similar to the summation of natural numbers:

$\sum\limits_{i=1}^n i = \frac{1}{2}n(n+1)$

The way this works is by writing down all natural numbers from $$1$$ to $$n$$ and add a second column where the numbers are reversed:

 1 n 2 n-1 3 n-2 ... ... n-1 2 n 1

Summing every row always gives $$n+1$$. How often? Exactly $$n$$ times. Dividing by two results in the given formula. Coming back to the original problem, we can do a similar thing. We start with 0 up to 999999 to include all 6 digits numbers. The number one-million will be treated separately.

 0 999999 1 999998 2 999997 ... ... 999999 0

What we see is that every row has a sum of $$999999$$ and as such a digit sum of $$6\cdot 9=54$$. We have exactly one million of such digit sums, and doubled the result as in the sum of natural numbers. We have to divide by two again, which gives $$\frac{1}{2} 1000000 \cdot 54$$. Since we have to include the upper bound of one million, we add its digit sum, which is 1 of course. All in all, the sum of all digit sums between one and one-million is 27000001.

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