Counting polyominoes and Klarner's constant

How many shapes can be built from $n$ unit squares joined edge to edge? The counts grow roughly like $4.06^n$, but no formula is known, exact values are known only up to $n=70$, and even the growth rate is known only to lie between 4.0025 and 4.5252.

Problem

A polyomino with $n$ cells is a set of $n$ cells of the square lattice that is connected through shared edges. Counting polyominoes up to translation (fixed polyominoes) gives the sequence $t(n)$: $1, 2, 6, 19, 63, 216, \dots$ for $n=1,2,\dots$ (OEIS A001168). The limit

$$\lambda=\lim_{n\to\infty} t(n)^{1/n}$$

exists (Klarner) and is called Klarner's constant.

Question. How many polyominoes with $n$ cells are there? Concretely: can $t(n)$ be computed in time polynomial in $n$, and what is the exact value of $\lambda$? The same questions for free polyominoes (counted up to rotations and reflections) and for other lattices are also open.

Background

Polyominoes go back to the early 20th century and were popularised by Solomon Golomb and by Martin Gardner's columns in Scientific American. In statistical physics the same objects are called lattice animals, and their counts are used in series expansions for percolation. The question is Problem 37 of The Open Problems Project; the Handbook of Discrete and Computational Geometry lists the polynomial-time question as an unsolved problem [1].

What is known

• Exact counts. Redelmeier (1981) counted fixed polyominoes up to $n=24$. Jensen's transfer-matrix method, which runs in time roughly $3^{n/2}\approx 1.73^n$, reached $n=56$ [1], [2], and Barequet and Ben-Shachar (2024) reached $n=70$ by using bounding boxes on a lattice rotated by $45^\circ$ [3].
• Lower bound. $\lambda>4.00253$ (Barequet, Rote and Shalah, 2016), a rigorous computer-assisted proof that counts polyominoes on twisted cylinders [4].
• Upper bound. $\lambda\le 4.5252$ (Barequet and Shalah, 2022), by composing local configurations called twigs [5]. A 2025 preprint by V. Bui, revised in May 2026, reports $\lambda\le 4.5238$ from a small system of recurrences, with a certificate that can be checked by hand [6]; it is not yet refereed.
• Estimate. Series analysis suggests $\lambda\approx 4.0625696$, an unproved estimate due to Jensen [1].

What would count as progress

• New terms $t(71), t(72),\dots$, computed by a method that also reproduces the known terms; independent recomputation of the terms beyond $n=56$ is useful too.
• A rigorous lower bound above 4.00253, or an upper bound below 4.5252 (below 4.5238 to beat the preprint), with a certificate that can be checked by computer.
• An algorithm for $t(n)$ that is exponentially faster than the transfer-matrix methods, or a polynomial-time algorithm.

References

1. G. Barequet, S. W. Golomb and D. A. Klarner, Polyominoes, chapter 14 in Handbook of Discrete and Computational Geometry, 3rd ed., CRC Press (2017). Preprint
2. I. Jensen, Counting polyominoes: a parallel implementation for cluster computing, Computational Science – ICCS 2003, LNCS, Springer (2003), 203–212. doi:10.1007/3-540-44863-2_21
3. G. Barequet and G. Ben-Shachar, Counting polyominoes, revisited, Proc. ALENEX 2024, 133–143. doi:10.1137/1.9781611977929.10
4. G. Barequet, G. Rote and M. Shalah, $\lambda>4$: an improved lower bound on the growth constant of polyominoes, Comm. ACM 59(7) (2016), 88–95. doi:10.1145/2851485
5. G. Barequet and M. Shalah, Improved upper bounds on the growth constants of polyominoes and polycubes, Algorithmica 84 (2022), 3559–3586. doi:10.1007/s00453-022-00948-6
6. V. Bui, A convolutional approach to bounding the number of polyominoes, preprint (2025). arXiv:2511.00461

Source

Imported from The Open Problems Project, Problem 37, curated by Erik D. Demaine, Joseph S. B. Mitchell and Joseph O'Rourke. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.