The Superduper Supermarket 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 largest integer n you can find such that there must be 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 can possibly make them, there are still going to be n crates with the same number of apples.)
There are 25 different numbers of apples possible in the crates:
If we spread these numbers around as evenly as possible, we will have 5 crates for each of the numbers 120, . . . , 144. That makes 125 crates.
But there are 128 crates, so there are three more crates to deal with.
If we spread those out too, then there will be six crates with, say, 120, six with 132, and six with 144.
Thus the number we seek is 6.