The axiom has been tweaked and twisted so often that perhaps no one knows the exact term, but we all know the definition.

The definition of insanity is…

The definition of stupidity is…

(WAIT! Google confirms that it’s insanity, but you’ve probably heard it as any number of terms)

…doing the same thing over and over again and expecting different results.

Well, on the GMAT the definition changes from time to time, so we’ll add this caveat that applies to problems above the 600 level:

GMAT stubbornness is doing the same thing over and over again and being surprised when it doesn’t always work.

Here’s why – it would be wrong to categorically say that the GMAT is not testing your ability to learn, remember, and apply a process. To a fair extent the GMAT does test exactly that. But that’s not ALL it’s testing. Once you get to above-average level problems (and remember that’s above average in a pool that contains just about exclusively college graduates, so it’s an elite academic group to begin with) the GMAT is testing more than just “can you follow directions” – it’s testing things like “can you think on your feet when the situation changes,” “can you manage uncertainty,” and “can you find innovative ways to solve problems when the tried-and-true process doesn’t work.” And that’s where Mental Agility comes in – the GMAT, at the top end, will punish “one trick ponies” and reward those who can adapt on the fly. Consider an example (and please excuse the ugly in-line math formatting):

If a + 2b = (16 – b^2)/a, what is (a + b)^4?

It’s very easy to become seduced by the (16 – b^2) term, recognizing that as a classic “Difference of Squares” setup to be factored into (4 + b)(4 – b). And with good reason – the Difference of Squares rule is a very important concept and extremely helpful on plenty of GMAT problems. But here it makes the expression even messier – you can’t use it to eliminate or combine anything on the left hand side of the equation (a + 2b). So as much as you may beat your head against the wall trying, you need to find a new outlet. And that you can get by multiplying both sides by a to get rid of the denominator on the right:

a^2 + 2ab = 16 – b^2

Here’s where another common “squares” equation comes in: x^2 + 2xy + y^2 = (x + y)^2. If you can see that as your goal, then you have another outlet; you can add b^2 to both sides and you’ll have a squares equation ready to go:

a^2 + 2ab + b^2 = 16, which then becomes (a + b)^2 = 16. And if (a + b)^2 is 16 and we need (a + b)^4, we can square 16 to get 256.

The bigger lesson here is that it pays to have mental agility – many “hard” GMAT problems look easy in retrospect, as they’re not about grinding out long calculations or employing obscure rules. The range of math concepts tested on the GMAT is finite and (relatively, compared to what you learned in high school) small, but the GMAT makes it difficult by punishing those who don’t see the opportunity to change paths. IF your goal is to “grind” – to find a formula for each question, put your head down, and apply it – you may find some trouble. A few key takeaways from this problem include:

- If the “obvious” process or rule isn’t working after a few steps, take a step back and see if there’s another way
- If an algebra problem asks for a combination of variables (here it’s (a + b)^4) try to find a way to get that combination alone and not necessarily solve for that variable. Most “processes” you know are geared toward solving for individual variables; the GMAT knows that and loves to ask for combinations (like xy or (a + b)).
- As you study, pay attention to which “surprise” techniques you didn’t see at first but ended up being the key to solving a problem. Having that quick reference list to scan through in your mind will pay off. For example, this “squares” rule can be extremely helpful any time you’re asked to solve for a combination of variables squared (like (a + b)^2) *or* for a combination of variables multiplied (like 2ab…that comes from that middle term in a^2 + 2ab + b^2).

Most importantly, recognize that while in life doing the same thing over and over again usually gives you the same results, on the GMAT questions are written specifically to reward those who aren’t afraid to change gears when “what’s always worked before” doesn’t work in this case. On 700+ level problems, insanity just might be doing the same thing over and over again and wondering why you didn’t get the same result; the GMAT rewards mental agility.

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*By Brian Galvin*