Olympiad topic · P2–P6
Page numbers
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.
Try one
★★ MediumA dictionary's pages are numbered from 1 to 167.
How many digits are printed altogether?
- Hint 1: Page numbers have one, two or three digits. How many pages of each?
- Hint 2: How many pages from 10 to 99? And from 100 on?
- 1.Pages 1–9: 9 pages × 1 = 9 digits.
- 2.Pages 10–99: 90 pages × 2 = 180.
- 3.Pages 100–167: 167 − 99 = 68 pages, × 3 = 204.
- 4.9 + 180 + 204 = 393.
- ⚠️Common mistake: thinking pages 10–99 are 89 pages.
Answer: 393 digits
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsCounting pages and digits.
- ★★MediumFree3 question types · unlimited questionsA thicker book, how often a digit appears.
- ★★★HardFree2 question types · unlimited questionsThe pages from the digit count, one sheet torn out.
- ★★★★Expert (competition level)VIP2 question types · unlimited questionsThe torn sheet from a page sum, missing pages whose numbers use the same digits.
Page numbers: what it is and how to solve it
Page numbers: how many digits are printed, how often a digit appears, which pages were torn out.
For example: Pages 1 to 50: how many digits? Pages 1–9 use 9; pages 10–50 are 41 pages of 2 digits, 82; 91 in all.
The key idea: Count in bands: pages 1–9 one digit each (9), 10–99 two each (180), 100–999 three each. A sheet holds two pages: an odd page on the front, the next even page on the back.
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