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The Page Numbers Brain Teaser
A printer numbers every page of a book, starting at page 1. Altogether the printer uses exactly 192 digits. For example, page 7 uses one digit and page 23 uses two digits. How many pages does the book have?
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What's the puzzle?
A printer numbers every page of a book, starting at page 1. Altogether the printer uses exactly 192 digits. For example, page 7 uses one digit and page 23 uses two digits. How many pages does the book have?
What's the answer?
100 pages
How do you work it out?
- Pages 1 to 9 use one digit each: 9 × 1 = 9 digits.
- Pages 10 to 99 are 90 pages using two digits each: 90 × 2 = 180 digits.
- So far that is 9 + 180 = 189 digits, leaving 192 - 189 = 3 digits.
- Three digits is exactly one more page with a three-digit number: page 100.
- The book has 100 pages.
Frequently asked questions
Why is 64 the wrong answer to the page numbers puzzle?
Dividing 192 by 3 gives 64, but that assumes every page number has three digits. In the page numbers puzzle, pages 1 to 9 use one digit and pages 10 to 99 use two, so the digits must be counted group by group.
What skill does the page numbers puzzle practise?
The page numbers puzzle from LoveMaths Tutoring UK practises place value and systematic counting: sorting numbers by how many digits they have, then working out how many numbers fall into each group.
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