[yequalsx]

Assuming the question is serious I'll attempt an answer. It shows an example of how using graph theory one can solve a problem that seemingly at first has nothing to do with graphs. The proof relies on graph theoretic concepts and understanding the proof helps one get a firmer grasp on these concepts. This particular example and this particular experience may not seem helpful but done often enough you'll end up being good at graph theory.

[marlag]

Thx! And I was being serious. For me, one problem of gaining understanding of the domain is, how do I identify that a problem is indeed a graph problem?

[Jtsummers]

Practice, and read about similar puzzles/problems. Track down reprints/scans of Martin Gardner's Mathematical Games and similar publications.

In general, for identifying if something is a graph problem:

1) Through some mapping, (almost?) everything can be transformed into a graph problem of some sort. Now, that's a bit too big a set and ignores the real question.

2) How do I identify that a problem is practically solved with graphs?

Some heuristics (note, these are heuristics, particularly useful for getting started, but not at all absolutes):

Is it discrete? In the given problem from the link we have a finite number of extreme points (ends of each line segment), but an infinite number of points between. Fortunately, thanks to the problem constraint, there are only a few non-endpoints that we are concerned with. So we can safely ignore the infinity of possibilities by only examining this finite set.

Are there transitions between these states that are easily modeled as edges? In the given puzzle, absolutely. And its symmetric so an undirected graph is suitable (sometimes directed graphs would be more suitable or the only applicable solution).

After this, proving various things (like that the generalized statement that all properly constructed puzzles have a solution) will require learning some of the basic properties of graphs, and how to restate the premises (such as constraints on altitude and movement) in a graph-theoretic form. It's easy to intuit the proof now that you have a simplified, but complete, model of the problem, but harder to state it with mathematical certitude. That part really will just require more practice and exposure.

At some point you'll develop sufficient vocabulary in the field that you may not be able to prove it straight off, but you'll know what and where to look for the elements you need to construct a proof like they have.

[tzs]

> Practice, and read about similar puzzles/problems. Track down reprints/scans of Martin Gardner's Mathematical Games and similar publications.

The entire set of the 15 books that collected his "Mathematical Games" columns from Scientific American are available in PDF form on a CD-ROM:

http://www.amazon.com/Martin-Gardners-Mathematical-Games-Gar...

[wolfgke]

> how do I identify that a problem is indeed a graph problem?

Mathematical intuition. To answer the next obvious question of how to buld mathematical intuition: Solve lots of math problems (I've also heard that reading "Pólya - How to Solve it" is supposed to be helpful for this; I can't say anything about it).

[marlag]

"Solve lots of math problems"

If you are the type of person that absolutely loved each second of each lecture that your math professor held in high school, where he/she tried to prove an equation on the black board, and you acctually managed to pay attention for long enough to acctually understand what he was talking about, and you got a real kick out of that newly gained intuition, and you now long for that type of "profound" enlightenment, how would yo go about gaining in mathematical intuition when you are in your 40ies?

[Jtsummers]

Mathematical Games, by Martin Gardner.

EDIT: Distractions abound so I hit submit before forgetting.

Why: Martin Gardner wrote on a variety of mathematical topics in fields such as geometry, graph theory, number theory, combinatorics, topology, and beyond. His writing is very approachable, and well sourced. This will help to develop a base vocabulary across the mathematical fields that you can use for further research and investigation of your own once you find the areas that interest you, along with being delightful reads just for their own sake for the mentally curious and engaged.

[wolfgke]

> how would yo go about gaining in mathematical intuition when you are in your 40ies?

Exactly the same way that you would go if you were in your 20ies: Get the relevant textbooks that are typically recommended by professors and read them (sorry, the textbooks that I can recommend for basic studies in mathematics are all in German (my native language); only for main studies in mathematics I can tell English textbooks).