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Maths olympiad question · P4–P5★★★ Hard

Page numbers: olympiad question 6

A hard question on page numbers for P4–P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P2 to P5 papers: how many digits in all, how often a digit appears, and pages torn out.

The question

A printer numbers a book's pages from page 1 to the last page, using 387 digits in all.

How many pages does the book have?

Show the hints (3)
  1. Hint 1: How many digits do pages 1–9 use? And pages 10–99?
  2. Hint 2: The first 99 pages use fewer than 387. The rest go 3 to a page.
  3. Hint 3: Three-digit pages use 387 − 189 = 198 digits: 66 pages.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Pages 1–9 use 9 digits, pages 10–99 use 90 × 2 = 180: 189 in all.
  2. 2.387 − 189 = 198 left, 3 a page: 198 ÷ 3 = 66 pages.
  3. 3.Those 66 pages start at page 100: 99 + 66 = 165.
  4. ⚠️Common mistake: 387 ÷ 3 straight away, forgetting the first 99 pages don't use 3 digits each.

Answer: 165 pages

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