Page Numbers
Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?
Age
7 to 11
Challenge level
Problem
Exactly $195$ digits have been used to number the pages in a book.
Image
How many pages does the book have?
Getting Started
Remember, the digits are $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$
Which pages have numbers that have only one digit?
Which pages have two-digit numbers?
Which pages have three-digit numbers?
Student Solutions
Josh of Alameda Middle School, Ampthill, Beds, sends the following solution:
The digits up to page 10 add up to 11.
Then from 11 to 20 they add up to 20, which takes the total to 31.
From 21 to 30 they also add up to 20, which takes the total to 51, and so on.
At 40 the total is 71, at 50 the total is 91, at 60 the total us 111, at 70 the total is 131, at 80 the total is 151, at 90 the total is 171, at 99 the total is 189.
Page 100 makes it 192.
Page 101 makes it 195.
Teachers' Resources
Why do this problem?
This problem would make a good early morning challenge for children to have a go at while you're taking the register, for example. It requires limited knowledge about counting and place value, but is answered most efficiently using some sort of systematic way of working.
Key questions
How many pages have just single-digit page numbers?
How many pages have two-digit page numbers ... etc?