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If you write the numbers 1, 2, 3, ..., 10, 11, 12, ..., 99, 100, ..., what is the 1000th digit you write?

 Jun 14, 2020
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Let's use casework.

From 1-9 there are 9x1= 9 digits

From 10-99 there are 90x2= 180 digits

So far we have 180+9=189 digits. Now we have to start breaking it down into smaller cases. 

From 100-299 there are 200x3=600 digits

This is not enough. We need 1000-(600+189)=211 more digits

Since we are in the 3 digit number region, lets divide 211 by 3. This equals 70 with a remainder of 1. We add 70 to 299 to find the last number.

So that means from 1-369, there are 999 digits. The next digit, the 1000th digit, is 3.

 Jun 14, 2020

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