You Derive Me Crazy: Numerical Distributions (LSAT Logic Games Series)
No matter how good you get at Logic Games, finding those difficult inferences will always be a challenge! In our “You Derive Me Crazy” blog series, we’ll take a look at some of the higher-level inferences that repeat on the LSAT, ensuring that you have all the tools necessary to tackle anything the LSAT throws at you on test day
Numbers – if you felt comfortable with them, you’d be taking the GMAT!
I kid. But many of my students do have an aversion to numbers that comes from years of focusing on non-mathematical topics in their undergrad studies.
Unfortunately, some math will help you on certain logic games. Luckily, if you can add and subtract by one, you’re in good shape!
What am I talking about here?
Numerical distributions.
Numerical distributions are a type of inference that tells you how many times a certain element can go, or how many elements can be in each group, or how many elements from each subset you can include in your finalized diagram.
In a Mismatch Ordering game with too few elements, some of those elements are going to have to go more than once (unless slots are left blank, which is rare). For example, if you have 8 laps of a race, but only 3 runners, some of those runners are doing more than one lap. Numerical distributions will help you figure out how many times each element can go.
In an Open Grouping game (i.e. where you don’t know how many elements are in each group), it’d be nice to know, well, how many elements are in each group! Numerical distributions will help you figure this out.
In a Grouping game with element subsets, you might want to know how many elements from each subset you’ll end up with. As an example, imagine a grouping game where we’re figuring out the playoff teams. We know some are from the ACC, some from the SEC, and others from the Big 10. Numerical distributions will help you figure out how many elements from each subset can be included.
But two questions come up: (1) When should I look for them? and (2) How do I figure them out?
(1) When should I look for them?
In general, they’ll show up in Mismatch Ordering (with too few elements), Open Grouping, and Conditional Grouping with Element Subsets. However, there’s a common thread to all of these – they’ll all give you rules that set a minimum or maximum
- Mismatch ordering will tell you everyone has to go at least once, or no one can go consecutively (which means their max is the total number of slots, cut in half).
- Open grouping will tell you a limit to the group size, such as “Twice as many people go to see The Martian as go to see Fantastic Four.”
- Conditional Grouping with Element Subsets will tell you that you can’t have both the University of Florida and Vanderbilt (SEC teams both) in your playoffs, thus limiting the maximum number of SEC teams.
While these are the three most common game types to see this type of inference in, ANY game type that gives you a maximum or a minimum leaves open the possibility for making this type of inference.
(2) How do I figure them out?
This is the tricky part.
Step 1: Figure out what, exactly, you’re trying to figure out. In our examples, we were trying to figure out how many times each racer raced; how many people saw The Martian and Fantastic Four; and how many teams from each division would wind up in the playoffs.
Step 2: Figure out the overall maximum; what is the largest number of times you can get an element to go, or the largest number of elements that can show up in a group.
Step 3: Figure out how many of the numbers can be that large. Then, figure out how many slots/elements are left, and distribute them evenly. You now have your first distribution.
Step 4: Subtract one from the largest number; add it to the smallest number. You now have your second distribution. Keep doing this until you end up repeating a distribution.
Confused? That’s normal! Let’s look at an example.
Let’s say I have a game stating that Matt, Evyn, Kim, and Frank are running a 10-lap race. People are going to have to go more than once. In the rules, we’re told:
1) Each person runs at least one leg of the race
2) No one runs more than 3 legs of the race
We have a minimum (1) and a maximum (3), so we can come up with numerical distributions.
Step 1: (What are we trying to figure out?) – How many legs of the race each person is running. So we’ll end up with 4 numbers (one for each person) that adds up to 10 (the total number of laps).
Step 2: (What’s the overall maximum?) – Easy peasy, they tell us it’s 3.
Step 3: (How many can be at the max?) – In this case, I can have 3 people at the maximum. If all 4 run 3 laps, that’s 12 laps – too many. So my first distribution is:
3 3 3 1 – There are three people who each run 3 laps; the other person runs a single lap
Step 4: (Subtract 1 from the largest, add one to the smallest to get new distributions)
First: 3 3 3 1 (3 is the largest, so subtract one from it; 1 is the smallest, so add 1 to it)
Second: 3 3 2 2 (3 is the largest, so subtract one from it; 2 is the smallest, so add 1 to it)
Third: 3 2 3 2
And we have a repeat! The last two distributions each have 2 3-lappers and 2 2-lappers. We’re done with our distributions!
These distributions always show up in the questions/answers in some way, and the trickiest answers are usually relying on these distributions. By figuring them out up front, you can save yourself the time of trying to work with them later on!
Matt Shinners is a Manhattan Prep instructor based in New York City. After receiving a science degree from Boston College, Matt scored a perfect 180 on his LSAT and holds a J.D. from Harvard Law School. There’s nothing that makes him happier than seeing his students receive the scores they want to get into the schools of their choice. Check out Matt’s upcoming LSAT courses her
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