Here is a variant of the three utilities puzzle. Famously, It is impossible to solve it on a plane paper (Euler's theorem can prove that). But can you solve it on a coffee mug?
Connect all three utilities to each of the houses, such that no two lines intersect.
Don't draw on the inside, It's for drinking ofcourse.
Answer
When trying to connect all utilities $A$, $B$ and $C$ to the three houses one realizes that
one needs just 1 "crossing" of lines (see drawing).
This can be done on a topology like a mug (or donut) by having one of the lines going on the handle of the cup and the other one through it.
And this is how it looks like on a cup:
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