An expert question on page numbers for P5–P6, 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
Ming has a storybook numbered from page 1 to the last page.
Each sheet has an odd page on the front and the next even page on the back; if the last page is odd, its back is blank.
His little brother tore out one sheet with page numbers on both sides.
Ming adds up the remaining page numbers and gets 1462.
What is the higher page number on the torn-out sheet?
Show the hints (3)
💡Hint 1: The full total is a little more than 1462. Adding 1 + 2 + …, at which page do you first pass 1462?
💡Hint 2: The torn pages are "odd + the next even". What do you notice when their sum is divided by 4? How big can it be?
💡Hint 3: The book could have 54 or 55 pages. Try each: the torn pages must leave remainder 3 on division by 4, and be no bigger than the last two-sided sheet.
Worked solution
Have a go first, then read the solution step by step.
1.The torn pages are 2k − 1 and 2k, adding to 4k − 1: always remainder 3 on division by 4.
2.The full total = 1462 + the torn pages: above 1462, but by no more than the last two-sided sheet. So the book can only have 54 or 55 pages.
3.With 54 pages: total 1485, torn pages 1485 − 1462 = 23, remainder 3: it works.
4.With 55 pages: total 1540, torn pages 1540 − 1462 = 78, remainder 2: no.
5.So the book has 54 pages and the torn pages add to 23: (23 + 1) ÷ 2 = 12; they are pages 11 and 12.
⚠️Common mistake: using the first page count whose total passes 1462 without checking the torn pages can be "odd + next even".
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