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It’s become a bit of a running joke in my classes that I say, “The GMAT is a game of [a thing].” Every time I say it, I make it sound like I’ve revealed the hidden key to GMAT mastery:
“The GMAT is a game of recognizing patterns.”
“The GMAT is a game of specificity.”
“The GMAT is a game of making inferences.”
Of course, all of these are true, so I stand by them. Here’s another:
“The GMAT is a game of considering possibilities.”
For instance, in Critical Reasoning questions, noticing alternative possible causes for some event is often a big part of the right answer. If a person comes in soaking wet and holding an umbrella, and I conclude that it’s raining, you must think, “Hold up, isn’t it also possible that he’s a lunatic who constantly pours water on himself and carries umbrellas because he thinks they protect him from Zombie Martians?” To which I’d say, “Touché, that totally seems like a possibility.”
This skill is also tested on certain Data Sufficiency questions. A very great many of these problems, when you boil them down, have very little to do with ‘math’—they’re about noticing possibilities given restraints. Sure, sometimes there are some hidden equations you can set up or properties you can use—but often there’s a more straightforward approach. We call it ‘testing cases.’
I always talk about the GMAT testing cases process in my classes, but I’ve realized lately how murky it still is for many students. They have a general sense of what to do (‘choose numbers and try to get different answers’), but they often get crossed up or confused as they work through a problem, so I wanted to diagram the clear, repeatable process for testing cases. I give you:
THE GMAT TESTING CASES PROCESS FLOWCHART OF ZEUS OR SOME OTHER ANCIENT DEITY OF YOUR CHOOSING:
*FOR QUESTION BOXES, A RED ARROW IS ‘NO’ AND A GREEN ARROW IS ‘YES’*
This might look hellish, but if you practice and get used to it, this is a simple, logical approach for all testing cases situations (one note: on some geometry problems, it’s possible you won’t be testing ‘numbers’ but testing ‘shapes.’ The process is the same).
Most students, I notice, tend to move pretty well through the top of the chart—this makes sense, it’s the simplest part. Based on where I notice students tend to trip up, here are a few big guidelines to keep in mind:
1) The rules must be followed. Always check that the numbers you’ve chosen follow the rules in the statement(s) and question stem. You are never trying to pick numbers that break these rules. Never. Never, ever, ever, ever, never, never, ever, try to break the rules. Sing it, Taylor: “We, are never ever ever, [trying to break the rules].”
2) When you pick numbers that happen to break the rules, remember that you haven’t proven anything. All you should do at that point is throw that case out as if it were Meryl Streep’s most recent Oscar. Then, decide if you want to try new numbers or presume sufficiency.
3) After you’ve gotten one answer to your main question, there’s no point choosing numbers that you know will get you the same answer. For instance, if the main question is: “Is p an integer?” and I’ve tested p = 4 and it’s followed all my rules (so I have a case where “yes, p is an integer”), then there’s no reason for me to test p = 10. It doesn’t matter if it follows my rules or not, even if it does, it gives me another “yes, p is an integer” case. So I’ve wasted time on it. If you wanted to test p = 10, it’s possibly because you thought you were trying to break the rules instead of trying to get a different answer to the questions. See Taylor Swift about trying to pick numbers to break the rules.
Students tend to get most mixed up at that second diamond, when they’re trying to choose numbers that both follow the rules and give a different answer from they one they’ve already gotten. It can be tough to find numbers that do both those things at once, but don’t let that difficulty get exacerbated by a confusion about the process. Choose numbers that you suspect (or know, but pretend it’s just ‘suspect’) will give you a different answer, and check that they follow your rules. If they don’t, try to find different numbers that do follow the rules. But always, first, follow the rules. Like I did in high school.
Let’s work through this with an example. And because I hate myself, let’s make it my least favorite type of question. Try it yourself first, using the flowchart, and then follow my train of thought as I work through it as well.
If x and y are integers, and x + y < 0, is x — y > 0?
1) |x| + |y| > |x|
2) x^y = 1
Step 1: Choose numbers to test. Let’s say x = 2 and y = -3.
Step 2: Do these numbers follow my rules? Well, 2 + -3 < 0, and |2| + |-3| > |2|, so yes.
Step 3: What is the answer to the main question in this case? 2 — (-3) > 0, so this answer is ‘yes.’
Step 4: Can I think of different numbers that might follow the rules and get me a different answer (a ‘no’)? I think so. Let’s try x = 1 and y = 2 (I’m thinking ahead—I know 1 — 2 < 0, so I’ll have a ‘no’ answer under my belt).
Step 5: Do these numbers follow my rules? Ah, nope. 1 + 2 > 0.
Step 6: Throw these numbers out.
Back to step 4: Okay, I get it, if x + y < 0, I can’t have two positives, or that rule won’t be followed. So let’s try two negatives, x = -3 and y = -2.
Step 5: Do these numbers follow my rules? (-3) + (-2) < 0 and |-3| + |-2| > |-3| so yes.
Step 6: What is the answer to my main question? (-3) — (-2) is not > 0, so ‘no.’
Step 7: Is that answer different from the one I got before? Yes.
I have proven insufficiency for statement 1. Let’s move to statement 2.
Step 1: Choose numbers to test. Well, I know anything to the ‘0’ power is 1. Let’s try x = -2 and y = 0.
Step 2: Do these numbers follow my rules? –2 + 0 < 0, and (-2)^0 = 1, so yep.
Step 3: What is the answer to the main question? -2 — (0) < 0, so ‘no.’
Step 4: Can I think of different numbers that might follow the rules and get me a different answer (a ‘yes’)? Well, with y = 0, x will always have to be negative to follow my first rule, and this would get me the same ‘no’ answer. But I also know 1 to any power is also 1. So let’s try x = 1 and y = -2.
Step 5: Do these numbers follow my rules? 1 + (-2) < 0, and 1^(-2) = 1, so yes.
Step 6: What is the answer to the main question? 1 — (-2) > 0, so ‘yes.’
Step 7: Is that answer different from the one I got before? Yes.
I have proven insufficiency for statement 2.
Now we get to take them together. This is when things get really fun.
Since y = 0 is a nice easy case for our second statement, let’s start there again:
Step 1: Choose numbers to test. Let’s go with x = -2 and y = 0.
Step 2: Do these numbers follow my rules? Nope. |-2| + |0| is not > |-2|… Huh. I realize now that y cannot be zero.
Step 3: Throw these numbers out.
Step 1 again: Well, since y can’t be zero, let’s go with x = 1 and y = -2.
Step 2: Does this follow my rules? 1 + -2 < 0, |1| + |-2| > |1|, and 1^(-2) = 1. So yes.
Step 3: What answer to my main question do I get? 1 — (-2) > 0, so ‘yes.’
Step 4: Can I think of different numbers that might follow the rules and get me a different answer (a ‘no’)? Well, I think possibly. Let’s see. x = 1 and y = 4.
Step 5: Do they follow my rules? No. 1 + 4 is not < 0. That’s right, they can’t both be positive.
Step 6: Throw these numbers out. Note that I have NOT shown insufficiency yet.
Back to step 4: Erm…well… How about 1 and -4? Eh, hold up, that’s just going to get me another ‘yes’ because 1 — (-4) > 0. If x = 1, then y has to be negative… This will always give me a ‘yes.’ Could x be negative 1? Well, yes, if y is even… Okay. x = -1, y = …can’t be zero… can’t be two or greater because (-1) + (2) > 0…
So y = -2? Oh, no, (-1) — (-2) > 0…this would be another ‘yes’ answer to my main question…
I can’t think of any other numbers. I’m going to just say that together we have sufficiency.
And that is correct.
This was a challenging example, I thought. I needed to see what cases were possible based on my knowledge of number properties, but I couldn’t just use the properties alone (perhaps you could, in which case, I bow to you). There are several rules to keep track of. The key is that I knew exactly what I was doing at every step of the way, and exactly what I would need to do next based on the possible outcomes. I was always working with a single goal in mind: to choose cases that both follow my rules AND result in two different answers to the main question.
Also notice towards the end, I was able to intuit more which numbers would break the rules and which numbers would only give me the same answer I’d already gotten. As you work through the problem, if you’ve used good logic and process, your understanding of what the numbers you’re thinking of will do should become more instinctive.
Obviously, you won’t write out ‘step 1, step 2, etc…’ on the GMAT. But some organization and structure can help. I often make a testing cases table. The one for this problem looks like this:
The bits in red are some of the things that are happening in my head as I work. Also, usually I’d cross out cases that broke one of my rules, but for clarity here, I left them un-crossed. And notice that sometimes I know what the answer to my main question will be as soon as I pick the numbers, but nonetheless, for the sake of the flowchart, I act like I don’t, and I check to make sure the rules are followed before I allow myself to consider that answer.
There are a few more advanced habits and skills to use during the GMAT testing cases process, but before you think about these, you have to make sure you’re 100% clear on the process itself. So crack open your OG and work through a few DS problems specifically following that flowchart until the process is ingrained.
After all, the GMAT is a game of process.
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Reed Arnold is a Manhattan Prep instructor based in New York, NY. He has a B.A. in economics, philosophy, and mathematics and an M.S. in commerce, both from the University of Virginia. He enjoys writing, acting, Chipotle burritos, and teaching the GMAT. Check out Reed’s upcoming GMAT courses here.