## PrepTest 78, Game 3, Setup

### Transcript

This is PT 78 game number 3 and here we're told that there's an antique shop that will be auctioning exactly one antique per day for six consecutive days, one through six. And that's why I'm gonna draw six slots one through six horizontally and list our six variables H, L, M, S, T, V above them. The first rule says that the sundial is not auctioned on June 1st.

So I want a diagram that like this, what's more important than putting it in your list of rules is actually drawing a slash s underneath spot number one. So that's gonna remind us never put an S in one. The second rule says that if H is auctioned earlier than L then the mirror is auctioned earlier than the lamp. So it's a conditional rule, and I'm gonna diagram that like this.

What this means is if you put the H summer before L, then you also have to put the M summer before L. Be careful not to read this backward, this doesn't mean that if you put the M before L, that you also have to put the H before L. In other words, you are allowed, at least based on this rule alone to have this arrangement over here, where M is before L, but the L is actually before H.

The contrapositive of this rule means if M is not before L, then H is not before how L or in other words if L is before M, then L is before H. Now let's take a look at the third rule which says that S is earlier than both M and V. So that one should be an easy one diagram like this, S is before M and V.

And the final rule says that table is earlier than H or earlier than V, but not both. So what this means is if you put the T before H like this, then you're not allowed to have the T before V. And if you put the T before V, then you're not allowed to have the T before H. So what does that actually mean in these two situations?

Well if you're in the T before H situation and you can't put T before V, well, that means V is gonna have to come before T. And if you're in the T before V situation, that means you can't put T before H. So that means H is gonna have to come before T. So that's why this rule over here, you can actually diagram it directly like this. The arrangement is either V-T-H or H-T-V.

So now that we've diagrammed the rules, let's think about deductions that we can make. And whenever we have this or but not both ordering rule, it's a good idea to start by thinking about what the world would look like. What the game would look like, depending on which option we picked depending on which part of the or we were in.

So, for example, if we're in the first world where V is before T is before H, then that means the S has to come before V and the S also has to come before M. So for example, if we're in the world where V is before T is before H, and that means the S would come before the V and also the T and also the H. And let me just go ahead and get rid of this text here for some space. This right here is what it would look like if we're in the V before T before H situation.

You can see that I've just attached the S before M and V over here. But what if we were in the H before T before V situation? Well, that means the S would be extending off of the V, and it would look like this. So this diagram at the bottom here is a combination of rules three and four. And if we're able to make this connection, then we don't need to pay attention to the individual rules anymore.

We could just focus on this larger pair of ordering chains and that means the remaining rules that still have to be applied are rules one and two. So when we go to the questions, basically I will start by thinking about which parts of this or relationship we're in. Are we in the left side, or are we in the right side where H is before T is before V and then, I'll also keep in mind that S can't go first.

And then, also pay attention to our conditionals is, if H is before L then M is before L and if L is before M, then L has to be before H.