A hard question on most and least for P4–P5, 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:
12 is written on the board.
You split 12 into whole numbers that add up to it (as many as you like), then multiply them all together.
What is the biggest product you can get?
Show the hints (3)
💡Hint 1: Is a 1 any use? And is a big part a good idea?
💡Hint 2: Compare: 6 as 3 + 3, or as 2 + 2 + 2. Which gives the bigger product?
💡Hint 3: 12 ÷ 3 = 4. Use as many 3s as you can.
Worked solution
Have a go first, then read the solution step by step.
1.A 1 is no use: multiplying by 1 changes nothing. Adding it to another part makes the product bigger: 3 + 1 gives 3, but 4 gives 4.
2.A part of 5 or more should be split: x into 2 and x − 2, and 2 × (x − 2) is more than x — 7 as 2 + 5 gives 10. 4 = 2 + 2 gives the same product. So only 2s and 3s are needed.
3.Three 2s (sum 6, product 8) lose to two 3s (sum 6, product 9), so use at most two 2s; the rest are 3s.
4.12 ÷ 3 = 4 exactly: 4 threes.
5.The biggest product: 3 × 3 × 3 × 3 = 81.
⚠️Common mistake: splitting into just two parts, like 6 + 6, for only 36.
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