Polyomino

There are three common ways of distinguishing polyominoes for enumeration:[6][7]

• free polyominoes are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another (pieces that can be picked up and flipped over). Translating, rotating, reflecting, or glide reflecting a free polyomino does not change its shape.
• one-sided polyominoes are distinct when none is a translation or rotation of another (pieces that cannot be flipped over). Translating or rotating a one-sided polyomino does not change its shape.
• fixed polyominoes are distinct when none is a translation of another (pieces that can be neither flipped nor rotated). Translating a fixed polyomino will not change its shape.

The following table shows the numbers of polyominoes of various types with n cells.

As of 2004, Iwan Jensen has enumerated the fixed polyominoes up to n = 56 – approximately 6.915×10³¹.[8]

Free polyominoes were enumerated in 2007 up to n = 28 by Tomás Oliveira e Silva,[9] in 2012 up to n = 45 by Toshihiro Shirakawa,[10] and in 2023 up to n = 50 by John Mason.[11]

The above OEIS sequences, with the exception of A001419, include the count of 1 for the number of null-polyominoes; a null-polyomino is one that is formed of zero squares.

The dihedral group D₄ is the group of symmetries (symmetry group) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of D₄. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the fewer distinct fixed counterparts it has. Therefore, a free polyomino that is invariant under some or all non-trivial symmetries of D₄ may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D₄.

Polyominoes have the following possible symmetries;[12] the least number of squares needed in a polyomino with that symmetry is given in each case:

• 8 fixed polyominoes for each free polyomino:
  • no symmetry (4)
• 4 fixed polyominoes for each free polyomino:
  • mirror symmetry with respect to one of the grid line directions (4)
  • mirror symmetry with respect to a diagonal line (3)
  • 2-fold rotational symmetry: C₂ (4)
• 2 fixed polyominoes for each free polyomino:
  • symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: D₂ (2) (also known as the Klein four-group)
  • symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: D₂ (7)
  • 4-fold rotational symmetry: C₄ (8)
• 1 fixed polyomino for each free polyomino:
  • all symmetry of the square: D₄ (1).

In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows:

• 2 one-sided polyominoes for each free polyomino:
  • no symmetry
  • 2-fold rotational symmetry: C₂
  • 4-fold rotational symmetry: C₄
• 1 one-sided polyomino for each free polyomino:
  • all symmetry of the square: D₄
  • mirror symmetry with respect to one of the grid line directions
  • mirror symmetry with respect to a diagonal line
  • symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: D₂
  • symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: D₂.

The following table shows the numbers of polyominoes with n squares, sorted by symmetry groups.

[13]

Each polyomino of size n+1 can be obtained by adding a square to a polyomino of size n. This leads to algorithms for generating polyominoes inductively.

Most simply, given a list of polyominoes of size n, squares may be added next to each polyomino in each possible position, and the resulting polyomino of size n+1 added to the list if not a duplicate of one already found; refinements in ordering the enumeration and marking adjacent squares that should not be considered reduce the number of cases that need to be checked for duplicates.[14] This method may be used to enumerate either free or fixed polyominoes.

A more sophisticated method, described by Redelmeier, has been used by many authors as a way of not only counting polyominoes (without requiring that all polyominoes of size n be stored in size to enumerate those of size n+1), but also proving upper bounds on their number. The basic idea is that we begin with a single square, and from there, recursively add squares. Depending on the details, it may count each n-omino n times, once from starting from each of its n squares, or may be arranged to count each once only.

The simplest implementation involves adding one square at a time. Beginning with an initial square, number the adjacent squares, clockwise from the top, 1, 2, 3, and 4. Now pick a number between 1 and 4, and add a square at that location. Number the unnumbered adjacent squares, starting with 5. Then, pick a number larger than the previously picked number, and add that square. Continue picking a number larger than the number of the current square, adding that square, and then numbering the new adjacent squares. When n squares have been created, an n-omino has been created.

This method ensures that each fixed polyomino is counted exactly n times, once for each starting square. It can be optimized so that it counts each polyomino only once, rather than n times. Starting with the initial square, declare it to be the lower-left square of the polyomino. Simply do not number any square that is on a lower row, or left of the square on the same row. This is the version described by Redelmeier.

If one wishes to count free polyominoes instead, then one may check for symmetries after creating each n-omino. However, it is faster[15] to generate symmetric polyominoes separately (by a variation of this method)[16] and so determine the number of free polyominoes by Burnside's lemma.

The most modern algorithm for enumerating the fixed polyominoes was discovered by Iwan Jensen.[17] An improvement on Andrew Conway's method,[18] it is exponentially faster than the previous methods (however, its running time is still exponential in n).

Both Conway's and Jensen's versions of the transfer-matrix method involve counting the number of polyominoes that have a certain width. Computing the number for all widths gives the total number of polyominoes. The basic idea behind the method is that possible beginning rows are considered, and then to determine the minimum number of squares needed to complete the polyomino of the given width. Combined with the use of generating functions, this technique is able to count many polyominoes at once, thus enabling it to run many times faster than methods that have to generate every polyomino.

Although it has excellent running time, the tradeoff is that this algorithm uses exponential amounts of memory (many gigabytes of memory are needed for n above 50), is much harder to program than the other methods, and can't currently be used to count free polyominoes.

Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n

${\displaystyle A_{n}\sim {\frac {c\lambda ^{n}}{n}}}$

where λ = 4.0626 and c = 0.3169.[19] However, this result is not proven and the values of λ and c are only estimates.

The known theoretical results are not nearly as specific as this estimate. It has been proven that

${\displaystyle \lim _{n\rightarrow \infty }(A_{n})^{\frac {1}{n}}=\lambda }$

exists. In other words, A_n grows exponentially. The best known lower bound for λ, found in 2016, is 4.00253.[20] The best known upper bound is λ < 4.5252.[21]

To establish a lower bound, a simple but highly effective method is concatenation of polyominoes. Define the upper-right square to be the rightmost square in the uppermost row of the polyomino. Define the bottom-left square similarly. Then, the upper-right square of any polyomino of size n can be attached to the bottom-left square of any polyomino of size m to produce a unique (n+m)-omino. This proves A_nA_m ≤ A_n+m. Using this equation, one can show λ ≥ (A_n)^1/n for all n. Refinements of this procedure combined with data for A_n produce the lower bound given above.

The upper bound is attained by generalizing the inductive method of enumerating polyominoes. Instead of adding one square at a time, one adds a cluster of squares at a time. This is often described as adding twigs. By proving that every n-omino is a sequence of twigs, and by proving limits on the combinations of possible twigs, one obtains an upper bound on the number of n-ominoes. For example, in the algorithm outlined above, at each step we must choose a larger number, and at most three new numbers are added (since at most three unnumbered squares are adjacent to any numbered square). This can be used to obtain an upper bound of 6.75. Using 2.8 million twigs, Klarner and Rivest obtained an upper bound of 4.65,[22] which was subsequently improved by Barequet and Shalah to 4.5252.[21]

Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no symmetries (rotation or reflection) corresponds to 8 distinct fixed polyominoes, and for large n, most n-ominoes have no symmetries. Therefore, the number of fixed n-ominoes is approximately 8 times the number of free n-ominoes. Moreover, this approximation is exponentially more accurate as n increases.[12]

Exact formulas are known for enumerating polyominoes of special classes, such as the class of convex polyominoes and the class of directed polyominoes.

The definition of a convex polyomino is different from the usual definition of convexity, but is similar to the definition used for the orthogonal convex hull. A polyomino is said to be vertically or column convex if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be horizontally or row convex if its intersection with any horizontal line is convex. A polyomino is said to be convex if it is row and column convex.[23]

A polyomino is said to be directed if it contains a square, known as the root, such that every other square can be reached by movements of up or right one square, without leaving the polyomino.

Directed polyominoes,[24] column (or row) convex polyominoes,[25] and convex polyominoes[26] have been effectively enumerated by area n, as well as by some other parameters such as perimeter, using generating functions.

A polyomino is equable if its area equals its perimeter. An equable polyomino must be made from an even number of squares; every even number greater than 15 is possible. For instance, the 16-omino in the form of a 4 × 4 square and the 18-omino in the form of a 3 × 6 rectangle are both equable. For polyominoes with 15 squares or fewer, the perimeter always exceeds the area.[27]

Puzzles commonly ask for tiling a given region with a given set of polyominoes, such as the 12 pentominoes. Golomb's and Gardner's books have many examples. A typical puzzle is to tile a 6×10 rectangle with the twelve pentominoes; the 2339 solutions to this were found in 1960.[29] Where multiple copies of the polyominoes in the set are allowed, Golomb defines a hierarchy of different regions that a set may be able to tile, such as rectangles, strips, and the whole plane, and shows that whether polyominoes from a given set can tile the plane is undecidable, by mapping sets of Wang tiles to sets of polyominoes.[30]

Because the general problem of tiling regions of the plane with sets of polyominoes is NP-complete,[31] tiling with more than a few pieces rapidly becomes intractable and so the aid of a computer is required. The traditional approach to tiling finite regions of the plane uses a technique in computer science called backtracking.[32]

In Jigsaw Sudokus a square grid is tiled with polynomino-shaped regions (sequence A172477 in the OEIS).

Another class of problems asks whether copies of a given polyomino can tile a rectangle, and if so, what rectangles they can tile.[33] These problems have been extensively studied for particular polyominoes,[34] and tables of results for individual polyominoes are available.[35] Klarner and Göbel showed that for any polyomino there is a finite set of prime rectangles it tiles, such that all other rectangles it tiles can be tiled by those prime rectangles.[36][37] Kamenetsky and Cooke showed how various disjoint (called "holey") polyominoes can tile rectangles.[38]

Beyond rectangles, Golomb gave his hierarchy for single polyominoes: a polyomino may tile a rectangle, a half strip, a bent strip, an enlarged copy of itself, a quadrant, a strip, a half plane, the whole plane, certain combinations, or none of these. There are certain implications among these, both obvious (for example, if a polyomino tiles the half plane then it tiles the whole plane) and less so (for example, if a polyomino tiles an enlarged copy of itself, then it tiles the quadrant). Polyominoes of size up to 6 are characterized in this hierarchy (with the status of one hexomino, later found to tile a rectangle, unresolved at that time).[39]

In 2001 Cristopher Moore and John Michael Robson showed that the problem of tiling one polyomino with copies of another is NP-complete.[40][41]

The two tiling nonominoes not satisfying the Conway criterion.

Tiling the plane with copies of a single polyomino has also been much discussed. It was noted in 1965 that all polyominoes up to hexominoes[42] and all but four heptominoes tile the plane.[43] It was then established by David Bird that all but 26 octominoes tile the plane.[44] Rawsthorne found that all but 235 polyominoes of size 9 tile,[45] and such results have been extended to higher area by Rhoads (to size 14)[46] and others. Polyominoes tiling the plane have been classified by the symmetries of their tilings and by the number of aspects (orientations) in which the tiles appear in them.[47][48]

The study of which polyominoes can tile the plane has been facilitated using the Conway criterion: except for two nonominoes, all tiling polyominoes up to size 9 form a patch of at least one tile satisfying it, with higher-size exceptions more frequent.[49]

Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a rep-tile tiling of the plane. For instance, for every positive integer n, it is possible to combine n² copies of the L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed.[50]

A minimal compatibility figure for the T and W pentominoes.

The compatibility problem is to take two or more polyominoes and find a figure that can be tiled with each. Polyomino compatibility has been widely studied since the 1990s. Jorge Luis Mireles and Giovanni Resta have published websites of systematic results,[51][52] and Livio Zucca shows results for some complicated cases like three different pentominoes.[53] The general problem can be hard. The first compatibility figure for the L and X pentominoes was published in 2005 and had 80 tiles of each kind.[54] Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.