LSAT 31, Game 2 Setup, CD Sale, LSATHacks

This is an explanation of the second logic game from Section I of LSAT Preptest 31, the June 2000 LSAT.

A music store is having a sale on some of its CDs. The store offers jazz, opera, pop, rap, and soul CDs (J, O, P, R, S). Each genre has both new and used CDs (N, U) for a total of ten types of CDs. You must use the rules to determine the possible list of the types of CDs that are on sale.

Game Setup

This is one of the hardest games ever to appear on an LSAT. It’s an in-out grouping game, but it’s complicated by the fact that each type of CD has a new and an old version. 

Sometimes, a sufficient or necessary condition needs both the new and old CD. Sometimes, only one is enough.

As with any other in-out grouping game, it’s possible to make a single diagram and its contrapositive. 

These two diagrams let you solve each question somewhat easily, but the diagrams here are more difficult to set up and understand than on most games.

But without them, the game is practically impossible. So let’s see how to make the diagrams.

The first rule is simple. Used pop is always in, new opera is always out. You can draw that like this:

Or you can put it in a table like this:

I don’t care how you draw it; different diagrams work better for different people. Just don’t forget this rule.

The second rule looks like this:

The rule mentions both types of pop, but I only drew new pop. Why? Used pop is always in. So if a local rule tells you new pop is in, you know this rule is triggered.

You can ignore the third rule for now, since it doesn’t connect with the diagram. Instead, go directly to the fourth rule. (It makes sense to approach the rules out of order in an in-out grouping game, since they can all be joined.)

If we don’t have either type of jazz, then we do have new pop. This leads to both types of soul.

The final rule looks like this:

If we take the contrapositive, we get this:

This next part gets a little tricky. Don’t hesitate to reread it and the rules if it seems strange.

We can join this to our other diagram; sort of. The other diagram ends with both types of soul being in. But either one of them is sufficient to force both types of rap out. So we can connect the two, but we’ll need to add an “or” to the arrow to remind us that only one of the two souls is needed to force rap out.

This is easy to explain in person, but harder to explain in a book. If you have Pn, you have both souls. But if you have either soul, you have neither type of rap. So Pn leads to the a sufficient condition for having no rap.

But you don’t want to think you need both souls out to have no rap. That’s why there is an “or” above the arrow. Putting all of the rules on one diagram makes things hard to forget. 

Now we can add the third rule to the diagram. Both types of jazz leads to no rap:

So one type of soul being in, or both types of jazz being in forces rap out. That’s why there’s an “or” between the two arrows.

The complete diagram can be confusing; these symbols are like a new language. But trust me, it’s a powerful language. Once you learn it, even a hard game like this becomes easy. 

If you’re not 100% clear on how I built the diagram and what it means, try drawing it on your own. It should start to make sense.

Taking the contrapositive of the main diagram

You can take the contrapositive of a long chain of rules by flipping everything around, negating everything, and changing “and” to “or” and vice versa.

Once you think you have the correct contrapositive diagram (try it on your own first, as an exercise), double-check it against the rules. An error here can be fatal. Here’s the original again:

Here is the contrapositive:

Notice that the order of everything is reversed, and the lines through the variables are switched. The “and’s” became “or’s”.

So if either type of rap is in, we know two things:

1. All soul is out
2. New pop is out
3. One type of jazz is in.
4. One type of jazz is out.

The main diagram (and the game) are a bit weird

There is some weirdness between S and P on the contrapositive diagram. Again, this is because we combined a rule with “and” and a rule with “or” (the two rules with soul). It’s messy, but it lets us get everything on one diagram.

I put the “or” above the arrow between the S’s and P to show that having one type of soul out is enough to force P out of the game. That’s what the contrapositive of the second rule says (used pop is always in).

Having one rap in forces both soul out. But having one soul out forces new pop out. This is good to know in case a question tells you one soul is out, but doesn’t mention rap.

Again, if you’re unclear about these diagrams, try to make them yourself before going on to the questions. Match each rule from the setup with the diagrams, and see how everything connects together. 

They’re incredibly powerful, once you understand how they work.

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