# The $2013$th digit of $1234567891011213141516\ldots$

How do I find the $2013$th digit of the string $12345678910111213141516\ldots$ I still don't get it, how are you suppose to find the exact digit. How did you hint help at all?

• I think it should be $$12345678910111213141516...$$ Note that in your number there's one $\,1\;$ lacking between that zero and that two. – DonAntonio Aug 4 '13 at 15:16

Hint: $\underbrace{123456789}_{9\text{ digits}}\underbrace{101112\dots979899}_{180\text{ digits}}\underbrace{100101102103\dots}_{2013-189\text{ digits}}$

• What does that gotta do w/ anything? isn't that common sense? – Commander Shepard Aug 5 '13 at 0:25
• @CommanderShepard The entire question is common sense. At this level, math is largely about recognizing patterns that seem "obvious", then making them systematic so that you can extend their reach to large problems where naive intuition fails. – Erick Wong Aug 5 '13 at 2:10
• Wtf do you mean at this level? Of course, you're 37! I'm not even in HS yet! – Commander Shepard Aug 5 '13 at 2:17
• @CommanderShepard And it shows. By "this level" I mean at the level of the problem you asked, not how old either of us may be. Higher levels of math demand much creativity and perseverance in addition to the clear thinking needed here. – Erick Wong Aug 5 '13 at 2:42
• Agreed with @ErickWong. – user67258 Aug 5 '13 at 3:28

There are 9 digits from 1 to 9. For the next ones, you have 10 11 12 13 14 15... Notice that the only difference is that there is a "1" preceding each of the digits. Therefore, that makes $9(2)$, or $18$ digits for the 10 - 19. Since we are going from $10$ to $99$, we know that there will be 10 times the amount of digits as there was from $10$ to $19$. That makes $18(10) = 180$ digits.

Counting the first 9, we have found $189$ digits so far. We have $2013-189 = 1824$ more digits until we reach digit number $2013$. These digits come in groups of three, i.e.:

100 101 102 103 104...

By dividing $1824$ by three, we can see how many pairs of three we need to count. The remainder will also tell us which digit in the pair is the correct digit. If we have a remainder of 0, we have the third digit of a three digit pair. If it is two, we must have the second digit in a three digit pair, and so on.

$\frac{1824}{3}$ has a remainder of $0$, so we know that we must be looking at the third digit of a three digit pair, and it must be the $608$th pair.

The $600$th pair would be $699$. So, the $608$th pair would be $707$. The third digit of this is a $7$, which would be the solution.