The hexominoes

All polyominoes of order 6 — figures built from 6 cells. Edge-joined unit squares of the square lattice — the classic polyform family, home of the twelve pentominoes.

✓ 15/15 verification checks passed

• PASS · all forms have n cells
• PASS · all forms connected
• PASS · fixed forms distinct
• PASS · orbit count: free -> fixed
• PASS · orbit count: free -> one-sided
• PASS · orbit count: one-sided -> fixed
• PASS · free <= one-sided <= fixed
• PASS · canonical form idempotent
• PASS · canonical form group-invariant
• PASS · encoding round-trips
• PASS · free forms expand to the fixed set
• PASS · free forms expand to the one-sided set
• PASS · matches A000105 (free)
• PASS · matches A000988 (onesided)
• PASS · matches A001168 (fixed)

drawing…

Free 35 polyominoes

Two figures are the same free form if one can be turned into the other by translation, rotation or reflection. This is the usual count — the one that gives 12 pentominoes.

One-Sided 60 polyominoes

One-sided forms may be translated and rotated but not flipped over, so a chiral figure and its mirror image are counted separately.

Fixed 216 polyominoes

Fixed forms may only be translated. Every distinct rotation and reflection is counted as its own figure.