5.5 - Splitting Diagrams

Recall from our last lesson that splitting has some nuance—it's the method we use to build worlds, not necessarily the worlds themselves. Moreover, your goal isn't to end up with a split that represents every single solution to the game (even though that happens a lot).

This lesson breaks down the process of splitting diagrams, everything from how we do it to when it might be in your best interest to stop.

We'll cover:

• How to split diagrams
• How to make the most of your first split
• When to stop splitting and start the questions

Let's dive in.

How to Split Worlds

Splitting diagrams isn't too complicated. We'll walk through a game together to get a handle on the process.

Here's Game 1 from the June 2007 PrepTest (a.k.a. Test J):

A company employee generates a series of five-digit product codes in accordance with the following rules:
The codes use the digits 0, 1, 2, 3, and 4, and no others.
Each digit occurs exactly once in any code.
The second digit has a value exactly twice that of the first digit.
The value of the third digit is less than the value of the fifth digit.

Our initial setup probably looks something like this:

We have our pieces. We have our game spots representing each digit of a given product code. Good so far.

Making a Good First Step

Next, we need to make a good first step.

Early on in your games practice, I'd recommend just picking a rule and running with it. This trial-and-error process will help you learn what rules have more impact than others.

In general, it's good to start with low-hanging fruit—things that slot right into a diagram explicitly in the rules and that will always be true regardless of how many splits you make. These rules tend to read something like, "A must go fourth" 

This game doesn't have any rules like that, so where do we start? Let's look at the third and fourth rules.

The third rule says the second digit (so the second game slot) has a value exactly twice that of the first digit (the first game slot). We only have 5 players—0 through 4. Among those players, we have two who can even be double the value of another player in the list: 2 and 4. So right away, we know the second game slot must either be 2 or 4. In such scenarios, we'd also know the value of the first slot, 1 or 2 respectively. That looks pretty solid—two splits solving 40% of the game.

The fourth rule says the third slot must be a lower number than the fifth slot. Hmm. That means 4 can never go third, but anybody else could without breaking that particular rule, so long as the fifth spot had a higher number. Follow me here... if that third spot was, say, 0, literally any of the other four players could end up fifth without breaking the rule. We'd need 4 splits based on 0 going third, 3 more based on 1, 2 more based on 2, and a final one based on 3. That's 10 splits to fully bake in that rule, assuming we started there. Not a great use of our time. But if we start with the third rule, this rule has far fewer outcomes.

Do this kind of thinking upfront before your pen ever hits the paper and you'll make fewer total splits, more legible diagrams, and you'll get to the questions efficiently.

So here's what our diagrams look like after splitting based on the third rule:

Notice, I've scratched out the number 2 from our initial list of players. That's because we've now dealt with all possible use cases for that digit—according to the rules, it's either going to end up in the first position (our second world) or the second position (our first world). We've also fully dealt with the third rule. We can't break it any more, so we don't have to worry about it. This is the essence of making worlds—once you can't break a rule, it no longer affects the game.

Next, let's deal with that fourth rule. To bake it in, we'll have to consider its implications in each world. That is, the rule will apply differently in each world.

Starting up top, we've used 1 and 2, so we're left with 0, 3 and, 4. And since the third digit must be smaller than the fifth, we know the third digit can't be 4. In other words, it must be either 0 or 3. When it's 0, the fifth digit could be either 3 or 4. When it's 3, the fifth digit must be 4. That's one more split of the top world—one where the third digit is 0 and another where it's 3.

How about our bottom world? We used 2 and 4, so we're left with 0, 1, and 3. That means our third digit must be either 0 or 1 (since 3 is larger than both). If the third spot is 0, then the fifth could be either 1 or 3. If the third spot is 1, then the fifth digit will be 3. Again, that's one more split of the bottom world based on the value of the third digit. That's a great use of our time and it will pretty much solve the whole game. So let's do it.

Here's what it looks like:

We end up with four total worlds that encapsulate six total solutions (our first and third worlds incorporate a handle, representing two possibilities each like we learned about earlier).

And that's the whole process. Take things one baby step at a time and you'll soon be a pro at splitting.

When to Stop Splitting

The most common question I get about worlds is where to begin. The second most common is when to stop.

I'll start by saying that most games can be completely solved with just a handful of worlds—maybe four to six. Some, you might make as many as ten. But like most rules of thumb on the LSAT, there will be games that defy convention, and your goal isn't to create a diagram for every single possibility.

Knowing when to stop splitting boils down to when you won't glean a better understanding of the game by creating additional splits. In other words, split until you're left with as many rules as you can accurately make sense of in your head.

For some students, that's going the distance and baking them all into your diagrams. I was that way and still am for the most part. For other students, that means baking in a rule or two and then managing two or three other rules as they answer the questions. If that's you, good on ya.

You have my support to stop splitting when you can look me in the eye and say confidently that you can answer the questions with certainty. If that's not making any additional splits beyond your setup, more power to you. It all comes down to accuracy.

"But, Brandon... I baked in all the rules on [insert game here] but there's still flexibility. What did I do wrong?" Maybe nothing, actually. Some games are just like that—usually the more challenging ones. Personally, I love games like that because just noting the remaining flexibility requires you to pause and consider the implications, ultimately placing you in the driver's seat during the questions.

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That does it for splitting diagrams. Now, go practice what you've learned. When you're done, join us for our final LG chapter on how to review Logic Games.

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Thanks for reading. What would you change about this lesson? Leave me a comment below to help us improve this course for you and future test-takers.