A392363 Smallest connected polyiamond containing all free n-iamonds under rigid motions of the triangular lattice.

1, 2, 3, 5, 7, 10, 13, 16, 19, 24, 27, 32, 37, 42

Offset

1,2

Comments

A polyiamond is a connected set of unit equilateral triangles (cells) in the triangular lattice, joined edge-to-edge. Each cell has exactly 3 edge-neighbors. Free n-iamonds are equivalence classes under the full symmetry group of the lattice (the dihedral group D_6 together with all lattice translations); the count of free n-iamonds is A000577(n). A placement of a free n-iamond into a container is a D_6 orientation followed by a lattice translation (dr, dc) with (dr+dc) even; the parity condition ensures the translation maps the tiling onto itself, preserving cell orientation.

Triangular-grid analog of A327094 (square grid). For n >= 4, a(n) is smaller than the square-grid values, consistent with the lower connectivity of the triangular lattice (3 edge-neighbors per cell vs 4 for squares).

Conjecture: lim_{n -> infinity} a(n)/n^2 = 1/5. The ratio a(n)/n^2 decreases from 0.240 at n=10 to 0.214 at n=14 (monotonic in this range). For all n <= 14, a(n) <= ceiling((n^2+2n)/5), with equality for n <= 8 and n=10, and a(n) is 1-3 less for n=9,11,12,13,14.

The optimal containers for n=11 and n=14 have non-contiguous rows: the bottom row consists of isolated cells at scattered positions (e.g., columns 2, 6, 10 for the n=14 container). Row-contiguity heuristics are therefore unsafe for computing this sequence.

a(1)-a(14) were proved using an unconstrained SAT solver (Glucose 4.2 via PySAT) with lazy connectivity cuts (CEGAR). For each n, the solver found a satisfying assignment at k = a(n) and proved unsatisfiability at k = a(n)-1, establishing optimality. Every reported container was additionally verified by two independent geometric verifiers (disjoint code paths) that check containment of all free n-iamonds; optimality follows from the UNSAT certificates.

References

T. R. Dawson, Problem, Fairy Chess Review, Vol. 5, No. 4, 1942.

Links

Table of n, a(n) for n=1..14.

Peter Exley, Solver code, data, and figures, GitHub.

The Problemist, Fairy Chess Review volumes archive (Vol. 5 covers 1942-1945).

Puzzle Zapper, Polyomino Common Superforms.

Example

For n=4, a(4)=5: contains all 3 free tetraiamonds.
For n=5, a(5)=7: contains all 4 free pentaiamonds.
For n=11, a(11)=27: first n with a non-contiguous row in the optimal container.
For n=14, a(14)=42: contains all 26166 free 14-iamonds.

Crossrefs

Cf. A000577 (free n-iamonds), A327094 (square-grid analog), A394840 (smallest polyiamond with n-cell hole), A352029 (number of minimum-size square containers).

Sequence in context: A072388 A101433 A225645 * A302334 A024180 A183137

Adjacent sequences: A392360 A392361 A392362 * A392364 A392365 A392366

Keyword

nonn,hard,more

Author

Peter Exley, Apr 06 2026

Status

approved