You are right in pointing out that flexible thinking is an essential part of being able to tackle the harder/unusual problems on the GMAT. In general, this is the type of flow chart that I follow on the exam:
1) Read the problem once and grasp the overall essence of the question. I may not internalize all of the subtleties of the question at this stage. If the question is not clear on the first pass, read again, and perhaps a third time if still not clear.
2) Go back to your mental database and see if the problem fits or has similarities with other problems that I may have dealt in the past.
3) Think of the various strategies that could possibly be applied. Could some statements be simplified? Should I assign variables to different quantities, and rephrase the question algebraically, etc.
4) Take the approach that you believe is best suited, the ability to choose the best approach often rests on how experienced you are with tackling GMAT problems. Be ready to skip to the second best approach if the initial approach leads to a dead end.
5) If two different approaches don't work, it is best to let go the question. There is typically not much time on the GMAT to take a third stab at a problem. In general, it is best to not get stuck on hard problems that are beyond one's reach.
I will further elaborate on these points and answer your specific questions:
1) Attack strategies: Yes, it is absolutely essential to be aware of the possible strategies that could be taken for specific types of problems. Here are some examples:
a) Remainder problems: In case of data sufficiency problems, when the statement is sufficient then there is almost always a direct and clean way to do so, and by that I mean using the remainder algorithm. Let's say you take the general approach and that doesn't lead to sufficiency, then I would quickly use two examples, and see if I can establish a Yes/No, or two distinct values to the specific question. If I am able to do so then I am done.
One could also choose to start examining the statement with examples and try to establish insufficiency. If you get Yes for both the examples, then you can choose to pick a third example, and if you still get a Yes, then it is likely sufficient, although not guaranteed. At this stage, you could choose to reattempt with the general approach, and see if you can establish sufficiency. Obviously, this step depends on whether you can afford to spend time on this step or not. However, if you are able to do so then you know with certainty that your answer is correct. As far as using specific examples, one should select at least three, and one of them should be somewhat obscure(for example, n = 3, 7, and 17, as opposed to n=1,2, and 3). Often the test writers know that students will just pick easy numbers in sequence, and they may make the outcome unique for only those set of values, typically the case with the hardest problems.
And yes, you do have to develop a similar arsenal of attack strategies for other types of problems. The best way to do so is to practice with Official GMAT questions, and internalize the structure of the most common GMAT problems. Once you go through enough of real GMAT problems, you start to see a pattern, the test writers have a very specific style and are very consistent in how they formulate these problems.
Here are two more examples that cement this idea further:
2) Inequalities with three or more variables: In general, one should never use numbers or specific examples to solve a data sufficiency problem that deals with three or more variables. Of course, one could use examples but it would be time consuming and that is simply not the purpose of the test writers. These types of inequalities typically involve the following steps:
a) Rearrange the inequality expression whether that is given in the main stem or the statement. Typically, subtracting the terms and moving them all to one side works. For example, if the question is:
Is ax+by > ay+bx?, then this needs to be rewritten as Is (a-b)(x-y)>0?
b) If all of the variables are positive then rearrange by cross-multiplying. In general, you can't multiply both sides of an inequality by an unknown variable, unless you know whether it is positive or negative. This is a common trap that GMAT relies on.
3) Word Problems with convoluted main stem or statements: Let's say you read one of these problems, and on the first pass it is really not clear what the question is saying, what the relationship between the different quantities is, or the question is asking for a quantity that is some combination of two or more variables. This would almost always require you to translate the statement in to some sort of an algebraic relationship. Only then you should proceed to the statements.
Similar attack strategies exist for other general categories of GMAT problems.
To answer your second question, here are the pointers I look for:
1) If you start with some convoluted expression, then if it starts to simplify(terms in the numerator and denominator cancel, or terms cancel each other by substraction), then you know you are on the right track. GMAT problems are written such that they seem daunting at first, but if you carry out the correct steps, they unravel quickly. They have to, their job is not to test your ability to do arithmetic and algebraic marathons. I would say that is essentially the hallmark of GMAT Problems. I often see non-Official GMAT problems which involve unnecessary algebra and arithmetic, stay away from those.
2) If your algebra and arithmetic gets messy, then you have either made a mistake in one of the operations, look for those. You can either go back and try to catch the mistake(typically hard), or start from scratch. In general, if you make some unusual mistake, say replace x^2 with x, or multiple 7x5 and write 30, then your final answer will almost always not be in the five choices. I use this as a gauge myself, I have ended up with answers that were not in the five choices, and in those cases it is likely some strange mistakes that I made. However, if you make a conceptual mistake, your answer will almost always be in one of the answer choices, typically A or E, although not always.
And finally, I want to emphasize that reading the question carefully is extremely important, in fact read it multiple times. I will often read a question three times to fully understand what they are asking. In case of more basic questions once is sufficient. And finally, before selecting your answer choice, reread what they are asking.
That is all that comes to my mind for now.
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