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Rotational Puzzles on Graphs

Rotational Puzzles on Graphs. By Jaap Scherphuis For G4G7, 2006. Two puzzle types. Sliding puzzles:. Twisty Puzzles:. Sliding Puzzles. Richard M. Wilson “Graph Puzzles, Homotopy, and the Alternating Group” J. Combin. Theory (Series B) 16 (1974) 86-96. Sliding Puzzle Graphs.

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Rotational Puzzles on Graphs

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  1. Rotational Puzzles on Graphs By Jaap Scherphuis For G4G7, 2006

  2. Two puzzle types Sliding puzzles: Twisty Puzzles:

  3. Sliding Puzzles Richard M. Wilson “Graph Puzzles, Homotopy, and the Alternating Group” J. Combin. Theory (Series B) 16 (1974) 86-96.

  4. Sliding Puzzle Graphs

  5. Twisty Puzzle Graphs

  6. Question: Given any twisty puzzle, what positions can be achieved? Assumptions: • Puzzle given by a graph, with designated twistable faces. • Every move is a turn of one face, i.e. a cycle. • All puzzle pieces are distinguishable. • Only one type of piece. • Orientations ignored.

  7. Permutation Parities Parity lemma: Let p be an odd permutation in Sn. Then <An,p>=Sn.

  8. From small to large Extension lemma: Let 1≤k≤m≤n. Let c be the cycle (k k+1 k+2 … n). Then <Am,c> contains An.

  9. Two-faced puzzle (14, 1, 14) (1+, 1, 1+)

  10. Other two-faced puzzles

  11. Two exceptional cases (2, 2, 2) (1, 3, 2) Expect: S6, |S6|=720 permutations Actual: PGL2(Z5), |PGL2(Z5)|=120 No 3-cycles

  12. Expanding the exceptions PGL2(Z5) ? S8 Every extension of (2,2,2) or (1,3,2) gives rise to Sn.

  13. Sliding puzzle exceptions “Tricky Six”

  14. Recap • Nearly every rotational graph puzzle allowsat least all even permuations. • Exceptions: • Trivial: Polygon, one face – Cyclic group • Interesting: Two graphs – PGL2(Z5) • Other: No pair of faces with single overlap.

  15. Richard M. Wilson “Graph Puzzles, Homotopy, and the Alternating Group” J. Combin. Theory (Series B) 16 (1974) 86-96. Jaap Scherphuis Jaap’s Puzzle Page https://www.jaapsch.net/puzzles/

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