We explore how hexagonal polyominoes (polyhexes) are counted by A000228 and how their boundaries can be encoded as words in a free group. From coloring arguments to Conway–Lagarias’s group-theoretic tiling framework, tiling becomes an algebraic operation with conjugacy and normal-subgroup structure. We also look at homomorphisms like a signed-area map that detects obstructions and discuss how symmetry and boundary invariants of polyhexes connect to the counting sequence, offering insights for number-theory-minded students.

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