The Super Duper Market has 128 crates of apples that have just arrived at the loading dock. Each crate contains at least 120 apples and at most 144 apples. What is the smallest integer n you can find so that there are at least n crates containing the same number of apples? (This means that even if you make the number of apples in each crate as different as you possibly can, there are still going to be n crates with the same number of apples.)
- 4
- 5
- 6
- 24
- 25
Solution
Our strategy is to make the number of apples in each crate as different as possible from the others, that is, to maximize the differences among the boxes and see what has to happen. There are 25 different numbers of apples possible in the crates: 120, 121, 122, and so on up to 144. So let's start with 25 crates, and put 120 apples in the first, 121 in the second, and so on up to 144 in the 25th crate. At this moment, our "smallest integer n" is 0: there are no crates containing the same number of apples (or 1, depending on how you look at it).
But we have more crates to fill. For crates 26 through 50 if we put apples in them as we did before, maximizing diversity, we now have two crates containing 120, 121, and so forth, so our "smallest integer n" is now 2. We continue with crates 51 through 75, 76 through 100, and 101 through 125. Our "smallest integer n" has jumped up to 5. That is, there have to be at least 5 crates with the same number of apples.
BUT! But there are three more crates to account for. Crates 126, 127, and 128 must get their apples too. If we want to keep the duplication down, we put three different numbers of apples in each of the three, say 120 in crate 126, 121 in crate 127, and 122 in crate 128. There are now SIX crates with the same number of apples (indeed, there are three such sixes, not that it matters). Our smallest integer n has come up to 6, which is answer (c).