Ask HN: What does it mean to have a attack on a problem

3 points | by quietthrow 194 days ago


  • SamReidHughes 194 days ago

    It means you have some path you can take to make progress on the problem. For example, suppose I want to compute the sum for n = 1 to infinity of 1/n^2.

    I can't think of any way to solve it. But one day you discover that you can calculate 1-1/2+1/3-1/4+..., by sticking x^n terms on it: sum(x^n * -(-1)^n/n), differentiating, giving you sum((-x)^(n-1)), recognizing that's a geometric series for 1/(1+x), which clues you in to see if the initial series was a Maclaurin series for log(1+x), and so it is, the sum is log(2).

    This gives you a new attack on the problem: Try parameterizing the series on a variable, like the sum{x^n/n^2}, or sum{1/(nx)^2}, or what-have-you, perform some manipulations, and see what happens.

    If that doesn't work out, another attack is to compute an approximation of the sum on a computer, by adding up the first hundred million terms, and seeing if it approaches a value you recognize. Knowing the actual value might give you clues about the solution technique (if you want to prove the answer).

    Before the age of computers, you might approximate it with the integral 1/x^2 from k+0.5 to infinity, with the first k terms added. Or you might develop general techniques to approximate the sum f(n) more exactly, using the derivatives of f(x) to adjust for the errors that basic integral approximation gave.

    If you find out that it's approximately X, then maybe the series is related to some other formula that is known to give you X. Then by performing some manipulations, you might show the two are equivalent. That's another attack on the problem.

    And then any time you see a formula with 1/n^2's in it... maybe you could somehow connect that to the series.

    • quietthrow 194 days ago

      Thanks for the answer with the examples. I am not much of a math person so am not following the math example but I think I (at least) directionally get what you are saying.

      Follow up question:

      Based on what you are saying does it mean that forming different hypothesis on ways to solve the problem can be considered an attack on the problem OR it has to be more than a hypothesis whereby you might have a general approach to solving that class of problem but the detailed steps for solving a particular instance of that problem needs to be figured out and tested?

  • muzani 194 days ago

    It's more that you have something to start on.

    We can tackle problems like building a Mars colony. We can make rockets, we can do life support, we can scout the area, calculate trajectories, figure out how to make rockets that land.

    However, we can't do time travel because there's nothing to start on.

    • hieloz 194 days ago

      Richard Hamming wrote that serious scientists should have a list of problems that they would like to work on, but they can't work on all of them at the same time, and some are off the active list because there is no attack available. An attack is "a good starting place, some reasonable idea of how to begin."

      This is an article that will help you understand "attack" .