Maths olympiad question · P5–P6★★★★ Expert
Page numbers: olympiad question 9
An expert question on page numbers for P5–P6, with a step-by-step solution 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
Some sheets in a row have been torn from the middle of a storybook.
Each sheet has an odd page on the front and the next even page on the back.
In what is left, the page just before the gap and the page just after it multiply to 2714.
The torn pages add up to 630.
How many sheets were torn out?
Show the hints (3)
- Hint 1: The first torn page must be odd and the last even. So is the page just before the gap odd or even? And the one just after?
- Hint 2: Before the gap is even, after it odd. Split 2714 as even × odd, the odd at least 3 more (at least one sheet in between).
- Hint 3: The possible splits: 2 × 1357, 46 × 59. Work out the sum of the pages between for each, and compare with 630.
Worked solution
Have a go first, then read the solution step by step.
- 1.Each sheet is an odd page and the next even one, so the torn part starts on an odd page and ends on an even one: the page before the gap is even and the one after is odd.
- 2.Write 2714 as even × odd (the odd at least 3 more): 2 × 1357, 46 × 59.
- 3.The torn pages' sum for each (from even + 1 to odd − 1, by (first + last) × count ÷ 2): between 2 and 1357: (3 + 1356) × 1354 ÷ 2 = 920043; between 46 and 59: (47 + 58) × 12 ÷ 2 = 630 ✓.
- 4.Only 46 and 59 fit: pages 47 to 58 were torn, 12 pages, which is 6 sheets.
- 5.Check: 46 × 59 = 2714 ✓; (47 + 58) × 12 ÷ 2 = 630 ✓
- ⚠️Common mistake: giving pages instead of sheets (12 pages is 6 sheets); or splitting 2714 into two odds or two evens — the pages round a gap are always one even, one odd.
Answer: 6 sheets
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