I have got this task at high-school math-contest seminar.
The theme is graphs.
Let us have $n \in \mathbb{N}$ and the cube $ n \times n \times n$. An ant can go over a diagonal of little cubes, but he can't turn at the intersection of two little diagonals.
Is it true that if in is odd then an ant can't visit every little facet (all of $6 n \times n$) exactly once? I didn't find the path for $n = 1,3$
I don't see a particular method to solve this task, because you can't treat facets as vertices. You can't treat them as edges also. It isn't an Euler or Hamilton cycle. I don't see an invariant that should be saved as well. The graph is planar.
from Hot Weekly Questions - Mathematics Stack Exchange
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