Number 20, Mary and Paul each wrote down three three-digit numbers using the digits 1 to 9 exactly once. The picture shows an example of how this could be done. Then Mary and Paul ordered their numbers as smallest, middle, and largest. Mary wrote down the largest possible value that the middle number could have, and Paul wrote down the smallest possible value that the middle number could have. What is the difference between their two middle numbers? Pause if you'd like to try and solve it. Let's try and figure out what Mary and Paul could have wrote down for their middle numbers. Mary ordered her numbers from smallest, middle, and largest. She made her middle number as big as possible. Let's see what her number could be. If we start her number off with a 9 in the hundreds place, then her biggest number has to start with an 8, or it has to start with less than a 9, so the biggest digit we could choose is an 8. And we can see that any number that, three-digit number that starts with a 9 is going to be bigger than a three-digit number that starts with an 8. So that means her middle number is bigger than her largest number. That's not possible. So we have to switch these two. And if we switch them, it works. We can see that now her middle number starts with an 8, and her largest number starts with a 9, and that means that they're guaranteed to be, the middle number is guaranteed to be smaller than this largest number. So now we can use whatever digits we want for the remaining two digits of this middle number, right? So let's choose a 7 because that's the largest possible digit that we could choose for the tens place, and then the next largest digit, a 6, for the ones place. And this is going to be the largest possible middle number that she could have. And of course, any other number that's in this smaller one is guaranteed to be smaller because it's starting with, at most, a 5, right? So obviously, any number that starts with a 5 is going to be less than a number that starts with an 8. So now we can use the same sort of logic to figure out Paul's middle number, right? He chose the smallest possible middle number, so we would like to start it with a 1, but if we start it with a 1, and then we start this smallest number with a 2, well, this number is guaranteed to be smaller than this number because it starts with a 1, while this one starts with a 2. So this middle number has to start with a 2, and then if this middle number starts with a 2, the smallest number starts with a 1, and then we can choose whatever other numbers we want, and so we'll choose the next smallest number, which is a 3 for this digit, and then a 4 for this digit. And so this is going to be the smallest possible middle number that Paul could have wrote down. And so now we just need to find the difference between these two numbers, and we can do 6 minus 4 is 2, 7 minus 3 is 4, and 8 minus 2 is 6. So the difference between them is 642, which is answer option A.

three-digit numbers

ascending order

largest middle number

smallest middle number

difference