 # How Do You Handle Minimum/Maximum Questions in Games?

Here's what I do with minimum or maximum number questions (I'll start with minimums, then we'll look at maximums):

I do not try to think about what the minimum might be. I don't try to start from the smallest number. Instead, I start from my prior work, and I do not lose sight of my placeholders.

Here's the process, using PrepTest 34, Game 4, Q 21 as an example:

1. Eliminate options based on placeholders. Here, for example, we know from our placeholders that we'll never have fewer than 2 doctors at Souderton, so we can get rid of (A) and (B) straightaway.

2. Use prior work to eliminate other bad options. For example, we've seen as few as 3 doctors in the answer to Q 20, so we can eliminate (E).

3. Finally, start from the smallest number you've ever seen in prior work, then ask whether you can make that number even smaller. For example, if you had tried out answer choice (C) in Q 23 before answering Q 21, then you would know that it's possible to have only two doctors at Souderton. On the other hand, if you hadn't done that question yet (or if you got it without trying out (C)), then you'd start with the smallest number you'd seen before. There are only 3 doctors (N, K, and O) in Q 20, so I'll start from my work for that question.

Because I physically (visually) use my placeholders, I can see that moving K or O out will profit me nothing; I'd have to replace them with P or J, respectively. So if I want fewer than 3 doctors, I need to move N out. If I can do that successfully (i.e., without breaking any rules or adding any more doctors), then I'll know the smallest number possible is 2.

And we can! If we move N out, that takes O out. If O is out, then J is in (so now we have _ J | _ _ N O)

J kicks out K, and when K is out, P is in (giving us P J | K _ N O)

Finally, when P is in, L is out, so we can have as few as two doctors (P and J) at Souderton, while everyone else is at Randsborough.

[Note: if you wanted, you could skip steps 1 and 2 and do the entire question using only step 3. Sometimes, though, you'll get all the way to the right answer by using only steps 1 and 2, so I've included them here because they're not much work to do].

Maximum questions (let's use this same game, and pretend that Q 21 had asked for the maximum number of doctors at Randsborough, and that the test writers had given us options (A) 6; (B) 5; (C) 4; (D) 3; and (E) 2):

1. Eliminate options based on placeholders. Here, for example, we know from our placeholders that we'll always have at least 2 doctors at Souderton, so we can never have more than 4 at Randsborough, and we can get rid of (A) and (B) straightaway.

2. Use prior work to eliminate other bad options. For example, we've seen as many as 3 doctors in the answer to Q 20, so we can eliminate (E).

3. Finally, start from the largest number you've ever seen in prior work, then ask whether you can make that number even bigger. For example, if you had tried out answer choice (C) in Q 23 before answering Q 21, then you would know that it's possible to have only two doctors at Souderton, leaving a maximum of 4 at Randsborough. On the other hand, if you hadn't done that question yet (or if you got it without trying out (C)), then you'd start with the largest number you'd seen before. There are 3 doctors (J, L, and P) in Q 20, so I'll start from my work for that question.

Because I physically (visually) use my placeholders, I can see that moving K or O over to Randsborough will profit me nothing; I'd have to trade P or J to Souderton for them. So if I want more than 3 doctors, I need to move N to Randsborough. If I can do that successfully (i.e., without breaking any rules or losing any more doctors), then I'll know the largest number possible is 4. And, of course, we can move N from Souderton to Randsborough, in exactly the same process we used above.

That's it! That's the process.