Digital book
If it takes 852 digits to number all the pages of a book, what is the number of the last page?
Age
11 to 14
Challenge level
Problem
The pages of a book are numbered $1$, $2$, $3$, $\dots$ In total, it takes $852$ digits to number all the pages of the book.
What is the number of the last page?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Pages $1$ to $9$ inclusive require $9$ digits; pages $10$ to $99$ inclusive require $180$ digits. So, in total, $189$ digits are required to number all of the pages before the three-digit page numbers commence with page number $100$.
This leaves $663$ digits, so the last page in the book is the $221^\mbox{st}$ page which has a three-digit number, namely page $320$.