A medium question on page numbers for P3–P4, 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
Several sheets in a row have fallen out of an old book.
The first missing page is page 673.
The last missing page uses the same three digits 6, 7, 3 in a different order.
Each sheet has two pages: an odd page on the front, the next even page on the back.
How many sheets fell out?
Show the hints (3)
💡Hint 1: The last missing page is on the back of a sheet. Is it odd or even? Bigger or smaller than 673?
💡Hint 2: List the rearrangements of 6, 7, 3 and keep only the even ones bigger than 673.
💡Hint 3: The last missing page is 736. How many pages from 673 to 736? How many pages to a sheet?
Worked solution
Have a go first, then read the solution step by step.
1.Page 673 is odd, on the front of a sheet; the missing part ends with a whole sheet, so its last page is even and bigger than 673.
2.Other arrangements of 6, 7, 3: 367 (smaller than 673), 376 (smaller than 673), 637 (smaller than 673), 763 (odd), 736 (fits).
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