Inspired by the three utilities puzzle from prog_SAHIL I'm now posting a similar puzzle that makes use of the topology of a cup with a handle:
The question is:
How many distinct points can you draw on the surface of this cup, such that it is possible to connect each point with all other points pairwise without any connections crossing each other (or one of the points)?
For example if you have 4 points the connections could look like this:
Answer
This is topologically equivalent to a torus, and you can go up to 7 points:
as shown by this Math.SE answer.
The diagram for this could look for example like this:
In this picture the lines going "under" the square represent connections going through the cup handle and the lines going "over" the square would go along the handle of the cup.
One can also just look up the answer if you know the question asks for the complete graph $K_n$ of degree $n$ with maximum $n$ such that the graph genus $\gamma (K_n)$ is at most $1$. Then
if you take the equation from Wolfram MathWorld $$ \gamma (K_n) = \left\lceil \frac{(n-3)(n-4)}{12} \right\rceil $$ you see that the genus $\gamma (K_n) \le 1$ as long as $n \le 7$.
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