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 Question from Chungus: It takes 852 digits to number the pages of a book consecutively. How many pages are there?

Hi Chungus,

There are 9 one digit pages, pages 1 to 9. Next look at 2 digit numbers. They go from 10 to 99 so that is 90 two digit numbers. Each contains 2 digits so when you get to 99 you have used $9 + 2 \times 90 = 189$ digits.

If we apply a similar argument to three digit numbers we will see that we have gone too far so we need to look at three digit numbers in smaller chunks. From 100 to 199 there are 100 three digit numbers. Since each has 3 digits that's $3 \times 100 = 300$ digits so when we get to page 199 we have used $9 + 2 \times 90 + 3 \times 100 = 489$ digits.