2-7 nested polycube sets

Analysis overview

• Nested polycube sets: Investigating sequences of polycubes from the dicube to the heptacube, where each successive piece is formed by adding one unit cube to the previous piece.
• Symmetry classification: Non-planar pieces are classified according to their three-dimensional symmetries. For planar pieces, we introduce a combinatorial criterion: a planar polycube is considered symmetrical only if its two-dimensional representation has a non-trivial symmetry, either point symmetry (a 180° rotation) or mirror symmetry (reflection). This excludes planar shapes such as the L tetracube while allowing shapes such as the O tetracube.
• Packing analysis: Searching for assemblies of the pieces that fill a rectangular cuboid, and classifying the results as no packing, multiple distinct assemblies, or a unique assembly.
• Symmetry-equivalent solutions: Treating assemblies related by the 48 symmetries of the cube—including rotations and reflections—as equivalent when checking uniqueness.
• Unique all-symmetrical chains: Rechecking the results and identifying four chains whose pieces all satisfy the applicable symmetry criterion and whose packing has one distinct assembly under the 48-symmetry convention.
• Chain-shaped pieces: Extending the analysis beyond growth from one end. Each piece must have a unit-cube adjacency graph with maximum degree two, and each growth transition adds one cube with no more than two face-neighbours.
• Extended growth results: Identifying 196 chains meeting the chain-shaped-piece condition, including 15 with a unique packing under the same symmetry convention; nine of these are additional to the previously identified one-end-growth results.
• Verification: Independently re-enumerating candidate assemblies for the four all-symmetrical unique chains and the 15 chain-shaped unique chains to verify their uniqueness.

Chains with unique solutions to multiple targets

These records have a unique 3×3×3 cube assembly and at least one additional unique target result. For box targets, “1 symmetry class” means all enumerated raw packings reduce to one class under the target box’s rotations/reflections; it does not mean there is only one raw placement.

Modified 3D targets

• chain_001055 — 3×3×3 cube: unique. 2×2×7: sizes 2–7 plus one unit cube; 8 raw packings, 1 symmetry class. 3×3×2: sizes 3–6, omitting the dicube and heptacube; 8 raw packings, 1 symmetry class. 3×3×2 piece IDs: 3-0001, 4-0001, 5-0007, 6-0013.
• chain_009723 — 3×3×3 cube: unique. 2×3×4: dicube and pieces of sizes 4–7, omitting the tricube; 4 raw packings, 1 symmetry class. Piece IDs: 2-0001, 4-0005, 5-0013, 6-0063, 7-0263.
• chain_013203 — 3×3×3 cube: unique. 3×3×2: sizes 3–6, omitting the dicube and heptacube; 8 raw packings, 1 symmetry class. Piece IDs: 3-0002, 4-0007, 5-0015, 6-0059.
• chain_004522 — 3×3×3 cube: unique. 5×2×2: unique packing using pieces of sizes 2–6, omitting the heptacube. Piece IDs: 2-0001, 3-0002, 4-0001, 5-0005, 6-0023. Exact-cover recheck: 8 raw packings, 1 symmetry class under the 16 isometries of the box.
• chain_004523 — 3×3×3 cube: unique. 5×2×2: unique packing using pieces of sizes 2–6, omitting the heptacube. Piece IDs: 2-0001, 3-0002, 4-0001, 5-0005, 6-0023. Its omitted heptacube is 7-0148. Exact-cover recheck: 8 raw packings, 1 symmetry class under the 16 isometries of the box.

Planar tilings

• chain_003559 — unique 5×5 square tiling using sizes 3–7; dicube omitted.
• chain_002803 — unique 9×3 rectangle tiling using all six pieces, sizes 2–7. The two raw arrangements recorded are equivalent by whole-rectangle reflection.
• chain_003552 — unique 4×7 planar tiling with a one-unit-cell void; verified previously.
• chain_003536 — unique 4×6 rectangle tiling omitting the tricube; uses pieces 2-0001, 4-0002, 5-0018, 6-0003, and 7-0001. Four raw placements were found, all equivalent under the rectangle’s rotations/reflections (one symmetry class).

Piece rotations are allowed; individual pieces are not independently reflected. The planar results retain their recorded uniqueness conventions.

Solvability highlights — n/m

The n/m ratio is the number of sequentially solvable assembly classes (n) divided by the total distinct assembly classes (m). Percentages are calculated from the recorded solvability table. The two rotational-move candidates are flagged as proposed exceptions only; they have not been independently verified by rotational-disassembly search.

Zero-solvability cases of interest

1. chain_000950 — 2-0001,3-0001,4-0001,5-0006,6-0017,7-0359; 0/1 (0.00%)
   Candidate rotational-move exception; user-proposed, not independently verified.
2. chain_004713 — 2-0001,3-0002,4-0001,5-0006,6-0017,7-0359; 0/2 (0.00%)
   Candidate rotational-move exception; user-proposed, not independently verified.
3. chain_005036 — 2-0001,3-0002,4-0001,5-0007,6-0105,7-0372; 0/11 (0.00%)
   Zero solvable classes in the recorded n/m results; included as an interesting low-solvability case.

Five lowest solvability percentages

Sorted by ascending percentage, then chain ID for ties. Zero-solvability cases are listed separately above and are excluded from this ranking.

1. chain_001132 — 2-0001,3-0001,4-0001,5-0007,6-0025,7-0337; 2/13 (15.38%)
2. chain_001126 — 2-0001,3-0001,4-0001,5-0007,6-0025,7-0152; 1/4 (25.00%)
3. chain_004787 — 2-0001,3-0002,4-0001,5-0006,6-0087,7-0347; 2/7 (28.57%)
4. chain_001122 — 2-0001,3-0001,4-0001,5-0007,6-0025,7-0103; 5/16 (31.25%)
5. chain_001065 — 2-0001,3-0001,4-0001,5-0007,6-0013,7-0402; 1/3 (33.33%)

Coverage note: n/m values are attached to chains present in the non-planar, valid-cube-packing solvability table. Chains with no valid cube packing are marked N/A; planar-only chains excluded from that audit are marked not assessed. These labels do not imply a new independent recomputation of the underlying ratios.