## Counting Up To One Thousand

#### Problem

A boy writes down the counting numbers from 1 to 20 in order and because each of the numbers from 10 to 20 contain two digits he notes that he has actually written down thirty-one digits in total.

- If his list went from 1 to 1000 how many digits would he have written down?
- If one-thousand digits were written down in total, how many complete numbers would have been written down?

#### Solution

For the first part we consider 1-digit, 2-digit, and 3-digit numbers separately:

1 to 9: 9 1 = 9 digits

10 to 99: 90 2 = 180 digits

101 to 999: 900 3 = 2700 digits

Hence he will have written down 9 + 180 + 2700 + 4 = 2893 digits in total.

For the second part we note that 9 + 180 = 189 digits would be written for the numbers 1 to 99. Therefore there would be 1000 189 = 811 more digits written down, and, as 811/3 = 270.333..., this is equivalent to writing down an additional 270 complete 3-digit numbers. Hence he would have written all the numbers from 1 to 369.

However, a more elegant approach is to assume that each of the numbers from 1 to 999 are written as 3-digit numbers; for example, 2 is written as 002 and 73 is written as 073.

Now we note that each of the numbers from 1 to 99 have an unnecessary zero in the first column and each of the numbers 1 to 9 have an unnecessary zero in the second column, making 99 + 9 = 108 digits that would not actually be written down.

That is, this system would be equivalent to writing down a list of 3-digit numbers comprising of 1108 digits in total. But 1108/3 = 369.333..., therefore he would have written all the complete numbers from 1 to 369.