Hi Nicole,

Let's try some smaller numbers.

It takes 6 digits to number the pages of a book consecutively. How many pages are there?

That one is easy. The digits must be 1, 2, 3, 4, 5 and 6 so the book has 6 pages.

It takes 159 digits to number the pages of a book consecutively. How many pages are there?

The first 9 pages are numbered with the numbers from 1 to 9 and then you start with two digit numbers. Two digit numbers will take you to page 99 so that's 9 pages with a one digit page number and 99 - 9 = 90 pages with a two digit number. The 90 pages from 10 to 99 take two digits each so that's 90 × 2 = 180 digits, add in the 9 one digit pages and you get that a 99 page book uses 180 + 9 = 189 digits. My book only uses 159 digits so it has fewer than 99 pages. In fact I know that its 2 digit page numbers use 159 - 9 = 150 digits. These are 2 digit page numbers so there are 150/2 = 75 pages with a 2 digit page number. Hence the number of pages is 9 + 75 = 84.

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

Penny