An expert question on most and least for P5–P6, 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 maths game:
15 is written on the board.
You split 15 into whole numbers that add up to it, all different from each other, then multiply them together.
What is the biggest product possible?
Show the hints (3)
💡Hint 1: Is a 1 any use? Should the parts be far apart or as close together as possible?
💡Hint 2: Start at 2 and take 2, 3, 4, … until adding the next number would go past 15.
💡Hint 3: 2 + 3 + 4 + 5 = 14, 1 short. Where should the 1 go so the parts stay different and as close as possible?
Worked solution
Have a go first, then read the solution step by step.
1.No 1s: adding a 1 to the biggest part keeps the sum and makes the product bigger: 2 + 3 + 1 gives 6, but 2 + 4 gives 8.
2.The parts can skip at most one number: if two numbers x < y are skipped, change the part x − 1 into x and the part y + 1 into y. The sum stays the same, and x × y beats (x − 1) × (y + 1), so the product grows. For example 3 and 6 skip 4 and 5; 4 and 5 instead: 3 × 6 = 18, 4 × 5 = 20.
3.So take consecutive numbers from 2: 2 + 3 + 4 + 5 = 14, and adding 6 would pass 15. That leaves 15 − 14 = 1.
4.Add the extra 1 one each to the biggest numbers: 2, 3, 4, 5 becomes 2, 3, 4, 6, still all different.
5.The biggest product: 2 × 3 × 4 × 6 = 144.
⚠️Common mistake: using lots of 3s (the parts must differ), or starting 1 + 2 + 3 + …, wasting a 1.
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