Millionth Digit of the Counting Numbers

Date: 02/26/2001 at 19:58:24
From: Keith Bukovich
Subject: Counting numbers 1,000,000th digit

My friend and I have a disagreement on the answer to this question 
that popped up in class: "A number is formed by writing the counting 
numbers in order: 123456789101112131415... What is the one millionth 
digit in this number?"

She says it is 1 and I thought it might be 0. I looked for a pattern.

      1- 2- 3- 4- 5- 6- 7- 8- 9-10
     11-12-13-14-15-16-17-18-19-20
     21-22-23-24-25-26-27-28-29-30

Every tenth number ends in zero and one million is divisible by ten. 
Then it occurred to me that she might be right, since I was 
considering values, not digits. Still, I can't seem to find the 
pattern for 1 to be the answer. Please let me know who is right. 

Also, if 1 is the answer, what is the pattern?

Thanks for any and all consideration.

Regards,
Keith

Date: 02/26/2001 at 20:31:12
From: Doctor Schwa
Subject: Re: Counting numbers 1,000,000th digit

>      1- 2- 3- 4- 5- 6- 7- 8- 9-10
>     11-12-13-14-15-16-17-18-19-20
>     21-22-23-24-25-26-27-28-29-30
>
>Every tenth number ends in zero and one million is divisible by ten. 
>Then it occurred to me that she might be right, since I was 
>considering values not digits.

That's exactly right. In the first line of your example, it's true 
that the tenth number ends in 0, but the ELEVENTH digit of the list is 
the 0. Then on the next line, you have twenty more digits, so the 31st 
digit is again zero, and so on.


>Still I can't seem to find the pattern for 1 to be the answer. Please 
>let me know who is right. Also, if 1 is the answer, what is the 
>pattern?

The strategy to solve this problem is to count the digits carefully, 
like this (I'll find the ten thousandth digit in the list as an 
example):

     9 digits used up on the one-digit numbers.
     2*90 = 180 digits used up on the two-digit numbers

(so the 189th digit in the list is the last 9 of 99, and the 190th 
digit is the first 1 of 100).

     3 * 900 = 2700 digits used up on the three-digit numbers.

So the 2889th digit is the last 9 of 999, and the 2890th digit is the 
first 1 of 1000. Now, the 2894th digit is the first 1 of 1001, and so 
on; the (2890 + 4n) digit is the first digit of (1000 + n), so the 
9998th digit is when n = 1777. This makes the 10000th digit the third 
digit of 2777, which is a 7.

Similarly the 10001st digit is a 7, and the 10002nd digit is a 2 (the first 
digit of 2778). 

Now keep going with that same kind of reasoning and see if you can 
find the millionth digit in the list.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/