CayleyR: Solving the TopSpin puzzle via cycle intersection

We present cayleyR, an R package for solving permutation puzzles by detecting cycle intersections in Cayley graphs.

The core algorithm performs an iterative bidirectional search: from both the initial and target permutation states, random operation sequences generate cycles in the Cayley graph of the symmetric group Sn; their intersection yields a connecting path.

When no direct intersection is found, a distance-guided bridge selection narrows the gap, and the process repeats.

The package targets the TopSpin(n,k) puzzle, whose state space is a Cayley graph of Sn generated by a cyclic shift and a prefix reversal.

We describe the mathematical framework, the algorithm, and its implementation, which combines a C++ hash-indexed state store with optional Vulkan GPU acceleration.

The software is publicly available on CRAN.