The number of digits used to number the pages of a book is twice the number of  pages of the book. If the number of pages of the book is a three-digit number, how many pages does the book have?

To solve this problem, we need to understand that the number of digits used to number the pages of a book is twice the number of pages of the book.

1. First, we know that the number of pages is a three-digit number. This means it is between 100 and 999.

2. We also know that the number of digits used to number the pages is twice the number of pages.

3. If we consider a book with 100 pages, the first 9 pages will use 1 digit, the next 90 pages will use 2 digits, and the last page will use 3 digits. So, a 100-page book will use 91 + 902 + 1*3 = 191 digits.

4. If we consider a book with 200 pages, the first 9 pages will use 1 digit, the next 90 pages will use 2 digits, and the next 101 pages will use 3 digits. So, a 200-page book will use 91 + 902 + 101*3 = 393 digits.

5. We can see that the number of digits used is not twice the number of pages in both cases.

6. By continuing this pattern, we find that a book with 300 pages uses 91 + 902 + 200*3 = 693 digits, which is more than twice the number of pages.

7. Therefore, the number of pages in the book must be between 200 and 300.

8. By testing numbers within this range, we find that a book with 256 pages uses 91 + 902 + 157*3 = 512 digits, which is exactly twice the number of pages.

So, the book has 256 pages.