Parity Property of Hexagonal Sliding Puzzles.

Area of Science:

• Combinatorics
• Discrete Mathematics
• Computational Topology

Background:

• Sliding puzzles, like the 15 Puzzle, are modeled using puzzle graphs to understand solvability.
• Previous research on square puzzles established parity properties determining solvability.
• Hexagonal puzzles present more complex parity behaviors influenced by board shape and holes.

Purpose of the Study:

• To analyze the puzzle graphs of hexagonal sliding puzzles with varying shapes and numbers of holes.
• To identify conditions for solvability in hexagonal sliding puzzles.
• To extend the understanding of puzzle graph properties beyond square configurations.

Main Methods:

• Combinatorial analysis of hexagonal puzzle graphs.
• Investigation of parity properties specific to hexagonal, triangular, and parallelogram boards.
• Development of a solvability criterion incorporating parity and tile placement.

Main Results:

• Hexagonal sliding puzzles with three or more holes on sufficiently large boards are generally solvable.
• A solvability criterion for puzzles with two or more holes was established, involving parity and "tight corner" tile positions.
• More complex parity properties were observed in hexagonal puzzles compared to square ones.

Conclusions:

• The study reveals that hexagonal sliding puzzles exhibit richer solvability conditions than their square counterparts.
• Understanding hexagonal puzzle graphs offers insights into the configuration spaces of mechanical puzzles.
• The findings contribute to the combinatorial and topological understanding of tiling and sliding puzzles.