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I have never been a topology buff and don’t have much idea about it. Hence, on the onset this question might seem silly. I would be talking about two games here which I played when I was a kid:

Game No. 1: As a kid I used to play these thread games which comprised of wearing a circular loop of thread around the fingers and thumb of both the hands and then sort of “weaving out” various shapes and patterns with this loop of thread. A few images of this are shown below:

enter image description here enter image description here

Game No.2: I was once asked by my uncle to draw the following pattern without lifting my pencil and I remember trying for whole few months figuring out a solution for this but to no avail. The pattern is depicted in the image below:

enter image description here

  Now, my question is can I relate the above two games with topology in the sense that is there a way that I can prove mathematically that all the different patterns that I form using the looped thread are topologically the same and that it is impossible in Game 2 to draw the pattern without lifting my pencil. Also, is it possible that if the string game is played with more than one string , say $2$ or $3$ Strings, then can the patterns formed by say $2$ strings be topologically same with that of $3$ strings or a single string.

Since I have never studied topology in much detail I would appreciate if the answer could be a bit comprehensive with complete mathematical details.

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    The second game can be prove to be impossible by graph theory if you are interested – 2017-02-13
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    Yes I would very much be interested in proving it by graph theory – 2017-02-13
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    @naveendankal I mentioned the graph theory solution in my answer. Eulerian paths are characterized by the vertex degree parities. (It sounds complicated but it is not.) – 2017-02-13
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    Thanks , just saw the link you have provided in your answer. – 2017-02-15
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    Having read the title, I suspected that the question will be about playing the Banach--Mazur game in Polish spaces with children who are unaware of Borel determinacy. – 2017-02-16
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    In a loop each point has points to the left and right or points going "in" to point and points going "out" of the point. So every vertex must connect an even number of lines. So game 2 can not be done in game 1 unless you have two *un*looped strings. Even paths that can be traced (but require a starting and ending point) with any odd vertices can not be replicated with looped string. – 2017-02-16
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    For the second, if we call each intersection point a vertex, then you can draw the pattern if and only if there are exactly two vertices that have an odd number of lines coming out of them or no vertices have an odd number of lines coming out of them. – 2017-02-16

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The first case seems like an application of knot theory, and we should expect all possible patterns to be equivalent to the "unknot" (a loop.) Knot theory can indeed be considered a topological field. Any sort of weaving you would do with your hands would be an acceptable deformation of the original loop since you aren't cutting it and you can't force part of the string to pass through another part.

The second one concerns the existence of an Eulerian path in a graph, so I would put it more in graph theory. Graph theory is also very related to topology (in fact if you visit the Eulerian path link, you'll find how it is in the roots of topology.) The existence of four vertices with odd degree precludes an Eulerian path.

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The existence of vertices with odd degree (number of edges that has an extreme in the vertex above) preclude an Eulerian Cycle and the proof is next (contrapositive): Let guess there is a Eulerian cycle and follow the next algorithm: on the picture drawn, follow the path and in each step delete the edge marked. As is a Eulerian cycle (or for the game rules) we have to end in the same vertex we started, we'll call it $v$. First consider any vertex different from $v$, let say $u$. As we follow the algorithm, each time we pass on $u$, we use an edge to get in and other to get out (both different since game rules) so at the end we've erased an even quantity of edges. Then, without considering first and last edge in the path, we can treat $v$ with the same sense, so now counting them, $v$ also has even degree.

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    What's a Euclidean path? Is this a Jungian slip? – 2017-02-16
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    Sorry @AsafKaragila I meant Eulerian cycle; [enter link description here https://en.wikipedia.org/wiki/Eulerian_path is a definition – 2017-02-16
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    I know that you did. This is why I referred to a Freudian slip as a Jungian slip. – 2017-02-16