What is the largest number, all of whose digits are 3 or 2, and whose digits add up to $11$?
I want to find a number with the largest possible number of digits, since that would make it larger than any number with fewer digits. But I also have to make sure that the digits add up to $11$, which is a small sum. So I want to use as many 2s as possible, since they are smaller than 3s. If I use five 2s, that would give me a sum of $10$, which is too small. If I use four 2s and one 3, that would give me a sum of $11$, which is perfect. So the largest number with this property is $32222$, since putting the 3 in the leftmost place makes it larger than any other arrangement.

