MILLION DIGIT SUM

Let's start simple.

Write the digits from 1 to a million as

$1, 2, 3, 4, \dots \dots ,999996, 999998, 999999, 1000000$

What I do is that I add a $0$ before $1$

Hence our sequence now looks like

$0, 1, 2, 3, \dots \dots, 999996, 999997, 999998, 999999, 1000000$

Then I add the numbers in pairs in this way -

$\begin{aligned} 999999 + 0 &= 999999 \\ 999998 + 1 &= 999999 \\ 999997 + 2 &= 999999 \\ . . . \end{aligned}$

In this way, we see that we get the same number $999999$ as the sum of the pairs. Note that this method does not change the sum of the digits of the numbers since we never add any two digits whose sum is greater than $9$ .

Now the number $999999$ has been repeated half a million times so the sum of all of the digits of the numbers from $1$ to $999999$ is $9 \times 6 \times 500000 = 27,000,000$ .

But the $1$ in $1000000$ hasn't been added. After adding it we get the sum as $27,000,001$ .

There are a million numbers between 000,000 and 999,999 (including) with a total of 6 million digits with an average value of 4.5, so that the sum of all digits is $6,000,000\times 4.5 =27,000,000$ . We pick up an additional 1 from 1,000,000, for a grand total of $\boxed{27,000,001}$ .