A medium question on most and least for P2–P4, with a step-by-step solution, diagrams and the answer. Finding the most and the least appears in the Hua Xia Cup from the P1 and P2 first rounds to the P6 papers.
The question
A teacher hands out all 13 books to 3 groups.
Every group gets at least 1, and no two groups get the same number.
What is the smallest number of books the biggest share can be?
Show the hints (2)
💡Hint 1: To keep the biggest share small, should the shares be far apart or as close as possible?
💡Hint 2: Say the biggest share is x; the others can be at most x − 1, x − 2, …. These 3 must add up to at least 13.
Worked solution
Have a go first, then read the solution step by step.
1.If the biggest share were only 5, the shares, all different, would be at most 5, 4, 3: 5 + 4 + 3 = 12, fewer than 13. The books wouldn't all go.
2.With 6 as the biggest it works: for example 3, 4, 6, 3 + 4 + 6 = 13, all different.
3.So the biggest share is at least 6.
⚠️Common mistake: working from 13 ÷ 3, forgetting the shares must differ and can't all be equal.
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