OI vs Group Theory, Do You Guys Know?

[ZhiQst] https://www.zhihu.com/question/1957203777686855723 知乎问题：如何解决这种田字形的数字华容道？
[MaBlSS] https://mathstrek.blog/2018/06/12/schreier-sims-algorithm/ Schreier-Sims Algorithm
[LojPr] https://loj.ac/p/177 #177. 生成子群阶数
[LojRe1] https://loj.ac/s/2466950 simple implementation \(O(n^6)\), based on [MaBlSS], solving [LojPr].
[OIwiSS] https://oi-wiki.org/math/algebra/schreier-sims/ OI-wiki Schreier–Sims 算法
[OIwiSS2] https://oi-wiki.org/math/algebra/schreier-sims/#引入 文章 [OIwiSS] 依照计算群论文献的惯例，将群作用记作右作用，这意味着置换的复合由左向右进行
[OIwiSS1] https://oi-wiki.org/math/algebra/schreier-sims/#另一种实现 OI-wiki Schreier–Sims 算法 Knuth 提出的"另一种实现"
[LojRe2] https://loj.ac/s/2467507 an implementation of [OIwiSS1], \(O(n^5)\), solving [LojPr]
[LuoPr] https://www.luogu.com.cn/problem/P5235 洛谷 P5235 [WC2017] 棋盘
[Rich73] https://www.sciencedirect.com/science/article/pii/0095895674900987 Graph puzzles, homotopy, and the alternating group by Richard M Wilson
[Wiki15] https://en.wikipedia.org/wiki/15_puzzle Wikipedia:15 puzzle
[WikiPeb] https://en.wikipedia.org/wiki/Pebble_motion_problems Wikipedia:Pebble motion problems
[LuoRe1] https://www.luogu.com.cn/record/247842971 using the conclusion of [Rich73], solving [LuoPr] in \(\tilde O(qn)\) complexity
[LuoRe2] https://www.luogu.com.cn/record/248208012 using the code snippet from [LojRe2], solving [LuoPr]. The code uses 'edge stack' to categorize edges to VCCs.
[Tom17] https://www.whitman.edu/Documents/Academics/Mathematics/2017/Howe.pdf Two Approaches to Analyzing the Permutations of the 15 Puzzle by Tom Howe
[BlYHX] https://yhx-12243.github.io/OI-transit/records/hdu4702%3Bloj177.html answer key to [LojPr]
[LojEvi] https://loj.ac/p/2285/files chess2.in consists theta0 graph. So the questions setter must be aware of the conclusion of [Rich73].
[Rob22] https://faculty.etsu.edu/beelerr/fifteen-supp1.pdf The Fifteen Puzzle and Related Puzzles (Supplemental Material for Intro to Modern Algebra) by Robert A. Beeler

Miscellaneous:
电脑whiteboard里有[Rich73]结论的证明推导