It took me an embarrassingly long time to remove the ring from this rigid structure:

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What math could I use to solve similar puzzles? Topology and knot theory seem helpful, but I don't think they can capture the constraints of rigid systems. Could they help even if a general useful theory does not exist?


One can model these kinds of puzzles as configuration spaces which roughly means you have a topological space $F_P$ for the game $P$ such that every point in the space corresponds to a configuration of the puzzle and two points are close if the amount you have to move the puzzle to get from one configuration to another is small. Given a configuration space $F_P$ and a basepoint $x$ (that is, a starting position for the puzzle), if $y$ is considered a solution to the puzzle then $P$ is solvable from the starting position $x$ if and only if $y$ lies in the path component in $F_P$ containing $x$.

Take as an example a movable ball trapped inside a torus (fixed in space) with the same internal radius as that of the ball. The configuration space of this puzzle will depend on if the ball can fit through the 'genus' hole of the torus. If it can, $F_P\cong\mathbb{R}^3\setminus (S^1\times D^2)\sqcup S^1$ where the first disjoint component corresponds to the ball being outside the torus, and the second corresponds to being inside the torus. If the ball can't fit through the 'genus' hole of the torus then $F_P\cong\mathbb{R}^3\setminus D^3\sqcup S^1$ because the center of the ball can only move in an area outside a bounded volume from the geometric center of the torus. These two examples illustrate that these configuration spaces depend on more than the topology of the corresponding objects being manipulated, but also on their rigid structure (I probably could have picked a much easier example but this was the first I thought of).

In both of the above cases, there is no path from a point in the first disjoint component to the second (the circle) and so if the puzzle is 'solved' when the ball can be taken outside of the torus, then there exists no solution if the starting configuration has the ball on the inside of the torus.

The topology of these spaces is rather well studied and has important applications to homotopy theory, the theory of braid groups, motion planning for robotics, and even cryptography.

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