Mathematicly Untangeling Untangle. I have a new addiction, I play Untangle  to often, and i am wondering what is the mathematics behind it.
some free games: (but be warned highly addictive) 
Javascript: 
http://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/untangle.html
Android: 
https://play.google.com/store/apps/details?id=softkos.untanglemeextreme&hl=en
the rules are simple:
You are given a number of points, some of which have lines drawn between them. You can move the points about arbitrarily; your aim is to position the points so that no line crosses another.
in some versions the difficulty is enriched by making 1 to 3 points unmovable.
This simple puzzle gives rise to all kinds of topological questions:


*

*is there an algorhitms to solve them?


Some of the games in the android download I was unable to solve, and that makes me wonder, probably there are unsolvable untangle puzzles (so that in any situation some lines will cross) 


*

*how can you recognize them? 

*what are the minimal cases of unsolvable puzzles? 

*are there fundamentaly different cases of unsolvable puzzles?


Are there any mathematical reports written about these puzzles? 
see now back trying to solve one.....
 A: There are unsolvable puzzles. Untangle gives you a graph, i.e. a set of vertices (points) with edges (lines) between some of them. You are asked to find an embedding of the graph into the two-dimensional plane such that no two edges cross each other. A graph where that is possible is called a planar graph.
Not all graphs are planar - the smallest non-planar graph is the graph containing 5 vertices, and edges between all pairs of vertices. I.e., you start out with a pentagram, and then draw lines between all pairs of corners.
There are various ways to decide whether a graph is planar - check out http://en.wikipedia.org/wiki/Planar_graph and http://en.wikipedia.org/wiki/Planarity_testing
If you read those articles carefully, you'll notice that it doesn't really say that the edges must be lines - it allows edges to be any kind of curve. That, however, doesn't really make a difference. If you find some embedding of a graph into the two-dimensional plane where the edges - if represented as arbitrary curves - don't intersect one another, you can always move the points around such that these curves become straight lines.
A: The theory of force directed drawing algorithms(further theory of Tutte's barycenter method ) is the way to solve it if is planar...if is not you drop in the previous answer.
BtW cool game :)
