There is a game I've seen recently (link; note that I have no affiliation with this game) which involves finding a Hamiltonian path connecting a graph of points, as illustrated below.
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The rules are:
- Your starting point is fixed (it is the highlighted dot in the first picture).
- You must form a path connecting all dots on the board, moving only up, down, left or right, and not hitting any dot twice.
- You can't pass through the black squares.
- Typically, the endpoint is fixed as the only dot with valency 1.
Solutions to this game are not unique, but I initially conjectured that they were path homotopic (viewing the paths as embedded in $\mathbb{R}^2$ punctured at the locations of the squares). This turns out to not be true, as the following alternative solution to the above board demonstrates:
Question: Are there any topological invariants to the solutions of this game?
Here are two more boards with multiple solutions for reference:
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