Really good question. Whenever you're picking numbers, your goal should to be to "Play Devil's Advocate" - try to find two different answers to the question so that you can prove the statement insufficient. The first set of numbers you pick will always give you one answer which is usually the "obvious" answer, and then from there on out your only goal should be to get a different answer. For example, if you were given something like:

Is the product jkmn = 1?

(1) jk/mn = 1

Your first set of numbers will probably give you "no" (maybe you try j = 8, k = 1, m = 4, n = 2, so that the statement jk/mn = 1 becomes (8 * 1)(4 * 2) = 1 - clearly then the product of 8 * 1 * 4 * 2 is not going to be 1).

From there, your ONLY goal is to try to find numbers that give you the answer "Yes" because you already have a "No" answer. It's a total waste to try really similar numbers (e.g. all positive integers, like 6, 1, 2, and 3) because you'll probably get really similar results. So you want to think "what would it take to get the product equal to 1 so I get a "Yes"?". And that should prompt you to think about fractions:

2 and 1/2, 4 and 1/4

Or about all values being the same:

1, 1, 1, and 1

Either of those would give you your "Yes" and then once you have that you've proven that it's insufficient information and can move on.

Now, that example isn't insanely hard *but* here's what I like about it - it models the thought process of picking numbers in the mindset of "what numbers don't they think I'll think of?" And I really liked that almost immediately when I started practicing DS way back when - some people hate the notion of "shoot they tricked me into not thinking about fractions" but I always liked the idea that the testmaker had challenged me to think of a set of numbers that would go counter to the "obvious." So trying, for example, "hey what if all the variables were the same?" appealed to me - it was that idea of trying to "break" the conventional wisdom.

So you asked how to improve the skill (not just get a list of types of numbers), and I'd suggest two things:

1) Imagine the testmaker sitting across the table from you and think of number picking like a game (chess, poker, any kind of strategy game) where you're matching wits. What numbers does he not want you to think about?

2) Train yourself to look for clues as to what numbers to pick. When you see > 0 or < 0, that means try positive/negative/zero. When you see inequalities, try using the biggest or smallest number they'll let you pick. Again think of it as a game - here they said that u(u+v) is not equal to zero, so which variable should you immediately say "hey they didn't say that THIS couldn't be zero?"- you have to try v = 0 because they left that door open to you. Whenever they specify something about some but not all variables, try that thing for the other variables.

Ultimately if you're that comfortable with algebra you'll probably do a lot of algebra on test day. But in practice, see if you can - even after you've done the algebra to get the answer if you go that way - train yourself to "beat" the testmaker at his/her own game of hiding the number(s) that give "the other" answer.

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