Hexominoes

Hexominoes are polyomino shapes formed by joining six unit squares edge to edge. There are exactly 35 distinct free hexominoes (when rotations and reflections are treated as equivalent). Unlike pentominoes (12 pieces) or tetrominoes (5 pieces), the large number of hexominoes makes them considerably more challenging for tiling puzzles. Common challenges include fitting subsets of hexominoes into rectangles or other shapes, since the full set of 35 covers 210 squares — an area that is difficult to form into simple rectangles.

Rules

1. You are given a target shape and a set of hexomino pieces (often a selected subset).
2. Place all given pieces into the target shape.
3. Pieces may be rotated and flipped freely.
4. Pieces must not overlap.
5. The entire target shape must be filled with no gaps.

Notable Properties

• The 35 hexominoes cover a total of 210 unit squares (35 x 6).
• 210 does not factor into convenient rectangles: no simple NxM rectangle has area 210 with both dimensions greater than 5. Possible rectangles (3x70, 5x42, 6x35, 7x30, 10x21, 14x15) exist but are often impractical.
• One hexomino has a hole (a 2x3 rectangle with the center two squares missing), making certain tiling problems impossible.
• A common approach is to use subsets of hexominoes or combine them with other polyominoes.

Strategies

• Subset Selection: Choose subsets that tile nicely. For instance, 8 hexominoes (48 squares) can tile a 6x8 rectangle.
• Handle Awkward Pieces Early: Some hexominoes (like the cross-shaped or T-shaped ones) are difficult to place. Position them first.
• Parity Analysis: Color the grid like a checkerboard. Most hexominoes cover 3 black and 3 white squares, but some do not. Check that the parity of your selected pieces matches the target shape.
• Edge Filling: Use long, thin hexominoes along the edges of the target shape.
• Avoid Small Gaps: Never leave an isolated region smaller than 6 squares, as no hexomino can fill it.
• Work Systematically: Fill from one corner, progressing row by row, to keep the open area manageable.

Hexominoes were formally categorized as part of Solomon W. Golomb's comprehensive study of polyominoes, beginning with his 1953 paper and fully detailed in his 1965 book "Polyominoes." Golomb enumerated all 35 free hexominoes and explored their mathematical properties.

The study of hexominoes attracted attention in the recreational mathematics community, particularly through Martin Gardner's "Mathematical Games" column in Scientific American during the 1950s and 1960s. Gardner presented hexomino challenges to his readers, sparking interest in their tiling properties and combinatorial complexity.

Because there are 35 hexominoes (compared to 12 pentominoes), hexomino puzzles are significantly harder, both for human solvers and computers. The computational complexity of hexomino tiling has made them useful as benchmark problems in computer science, particularly for testing backtracking algorithms and constraint solvers. The presence of the "holey" hexomino (with an internal gap) adds an extra layer of difficulty, as it prevents certain configurations.

Hexominoes are less commercially popular than pentominoes due to their complexity, but they have a dedicated following among polyomino enthusiasts and recreational mathematicians. They appear in specialized puzzle books, mathematical competitions, and online puzzle communities. Research into hexomino tiling continues, with questions about which shapes can and cannot be tiled by specific subsets of hexominoes remaining active areas of mathematical investigation.