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The problem is I'm looking on shortest path between points problem and the intuition tells me that the shortest path between points happens when paths don't cross. It's a step one. Then for all sub-paths we're connecting nearest points (I didn't thought much about that part). The problem is that I cannot prove that the shortest path is when there is no crosses. The couple of possible ways here:

  1. Continue reading the book, but I don't like this way because I left something not known and that feeling undermines my thrive to gain knowledge, because I feel that I left something unsolved.
  2. Blindly assume that I'm right (because of no other options).
  3. Read the solution of the problem, so I will know the exact answer. If it will be what I guessed, hopefully there will be a proof of why it's that. But what if what I guessed is not the right answer, I will never know, why.
  4. Ask Here if the shortest path between points is when connections between each pairs doesn't cross between each other and why it is that. This is good, as I will get thorough explaination and I will know is it right or wrong. On the other hand, in this way, I won't think much of this problem, thus I won't get knowledge deep inside.

So how to deal with this. How do mathematitions or whomever else deal when they face a problem they don't know answer but has a strong belief that answer is exactly that?

Clarification: I want to find the answer by myself because college experience told me that only learning ready answers will only give me mechanical and short-term knowledge. Only solving problems by myself will give me really deep understanding of the problem and the solution.

UPD: Problem I was talking is a travelling salesman problem.

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Is this Travelling salesman problem? – Johan Larsson Oct 31 '12 at 16:19
Yes, exactly that. – Smart Wannabe Oct 31 '12 at 16:20
It would be good to specify (in your question) that you're interested in the travelling salesman problem (done), and solving it, but want only hints... – amWhy Oct 31 '12 at 16:25
The generic problem is defined over graphs with arbitrary edge weights. It is not warranted that the "distances" described by the weights are corresponding to a geometrical arrangement in 2D (e.g. given 2 points 1 unit from each other you can specify the third to be 1 unit from one and 10000 from the other). For this reason such intuitive reasoning may not make too much sense for the solution of the generic problem. Other than that its best if you realize that the TSP problem is NP hard and practical solution methods are far from being trivial. – peterm Oct 31 '12 at 16:33
@amWhy well, I'm reading a book about algorithms and this problem as an example. And problem is that author says to think yourselves. So the dilemma is that I don't want to know the answer yet, but I only want to know, is what I'm thinking is right or wrong, but only through a hints. – Smart Wannabe Oct 31 '12 at 16:34

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