A126140 Maximum order of a polyomino with n cells that tiles a rectangle with congruent copies.

1, 1, 2, 4, 10, 92, 76, 246, 4

Offset

1,3

Comments

The order of a polyomino is defined as the minimum number of congruent copies required to tile a rectangle. The order is undefined if the polyomino cannot tile a rectangle. No example of a non-rectangular polyomino is known for which its order is odd.

From Zachary DeStefano, Feb 16 2026: (Start)

a(10) is known to be at least 138 (K. Dahlke). The main obstacle to determining the exact value of a(10) is deciding if the following polyomino can tile a rectangle.

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This is also the only remaining polyomino of height 2 with unknown order (K. Dahlke).

(End)

References

S. W. Golomb, Polyominoes, second edition, Chapter 8, pp. 97-110, Princeton University Press, 1994.

Links

Table of n, a(n) for n=1..9.

K. Dahlke, 10-omino, Order 138.

K. Dahlke, The Gun Theorem. [Cached copy at the Wayback Machine]

E. Friedman, Polyominoes in Rectangles

M. Garvie and J. Burkardt, A Parallelizable Integer Linear Programming Approach for Tiling Finite Regions of the Plane with Polyominoes, Algorithms, 15 (2022).

M. Reid, Rectifiable polyomino page. [Cached copy at the Wayback Machine]

R. Stanley, What are the most attractive Turing undecidable problems in mathematics?, MathOverflow.

H. Tulleken, Polyominoes 2.2: How they fit together (2019).

Crossrefs

Cf. A126138, A126139, A126141.

Sequence in context: A110073 A090256 A270479 * A326967 A223851 A371621

Adjacent sequences: A126137 A126138 A126139 * A126141 A126142 A126143

Keyword

nonn,hard,more

Author

William Rex Marshall, Dec 19 2006

Extensions

a(7)-a(9) confirmed by Zachary DeStefano, Feb 16 2026

Status

approved