KA-0095The extremal principle
5 points, difficulty 3 of 3Level 5–6about 150sOne hundred apples are packed into twelve boxes. What is the largest number n for which you can always be sure that some box holds at least n apples?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Look at the fullest box.
2The strategy
Suppose every box held fewer than n. Work out the largest total that would allow, and find the smallest n making that total fall short of 100. Then build a packing that reaches exactly n.
3The full solution
If every box held at most 8 the total would be at most 96, short of 100 - so some box holds at least 9. And 9 is achievable: four boxes of 9 and eight of 8 make exactly 100.
Solution
The reliable way
Both halves are needed. First the bound: if every box held 8 or fewer, the total would be at most 12 x 8 = 96, which is less than 100 - so some box must hold at least 9. Then the example: four boxes of 9 and eight boxes of 8 add to 36 + 64 = 100, and no box holds more than 9. So 9 is guaranteed and 10 is not. The transferable idea: an always question needs a bound and a matching example, and only the two together pin the answer down.
The elegant way
The fullest box holds at least the average rounded up, which is 100 divided by 12 rounded up - that is 9.
Why this is on the test: Looking at the extreme object rather than a typical one is the whole method, and the answer is the average rounded up rather than down.