How to prove connecting 3 dots to three squares without overlap is impossible?
The problem of connecting these points using edges is a GraphTheoretic Problem. If you are unfamiliar with graph theory, I suggest readng up on it here.
Specifically, we say that a graph is planar if it can be drawn such that no edges overlap, as you are trying to do. Wagner's theorem states that a graph is planar only if it does not contain $K_5$ or $K_{3,3}$ as a minor.
But the graph you're trying to achieve  connecting every *
with every 0
 is $K_{3,3}$. And every graph is a minor of itself, so the graph you're trying to achieve contains $K_{3,3}$ as a minor, so it is not planar.
You'll probably want to read up on complete graphs to understand this answer. $K_{3,3}$ is the complete bipartite graph on two sets of three vertices.
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OneChillDude
Updated on August 01, 2022Comments

OneChillDude over 1 year
I was given this problem to solve by a professor who promised an A if anyone could solve it. I'm nearly certain it is impossible, because at some point you have too many vertices and inevitably box yourself in, but I would like to know how to prove it.
How can you prove that it is impossible to connect every
*
with every0
without overlapping lines?* * * 0 0 0
EDIT: Professor said you get an A if you solve it, not prove it's impossible. I'm not even in the class, it was my friend's question, I am just curious.

MathMajor over 8 yearsWell, it looks like you will be getting a B then :)

Blue over 8 yearsSo, if I tell you the answer, then do I get the A?

Admin over 8 yearsYour profs is too old school that he/she doesn't even know there 's something called math stackexchage? Half of the class will get A then.

mrf over 8 yearsThe prof only offered an A for a solution, not necessarily for a proof that it's imposible!

achille hui over 8 yearsThere are two proof of this in Chapter 5 of Robin. J. Wilson's book "Introduction to graph theory". One is constructive, the other one uses Euler's formula. If you still are not satisfied, you can look at Chapter 4 of Richard Diestel's book "Graph Theory" (GTM 173) which uses topology to prove the same statement rigorously.

CiaPan over 8 yearsThe drawing requested is a wellknown $K_\{3,3\}$ 'utility graph' en.wikipedia.org/wiki/Water,_gas,_and_electricity which is known to be not planar en.wikipedia.org/wiki/Planar_graph – so the task is impossible.
