2070:
4092:
276:
260:
2050:, 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).
2111:
245:
38:
215:
230:
4083:
2127:
different pentominoes. 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. Many pairs of polyominoes have been proved incompatible by systematic exhaustion. No algorithm is known for deciding whether two arbitrary polyominoes are compatible.
1491:
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,
1542:
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
2126:
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, and Livio Zucca shows results for some complicated cases like three
2077:
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 and all but four heptominoes tile the plane. It was then established by David Bird that all but 26 octominoes tile the plane. Rawsthorne found that all but 235
1986:
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. Where multiple copies
1465:+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. This method may be used to enumerate either free or fixed polyominoes.
1953:
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.
1829:-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,
1745:
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
2078:
polyominoes of size 9 tile, and such results have been extended to higher area by Rhoads (to size 14) 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.
2135:
In addition to the tiling problems described above, there are recreational mathematics puzzles that require folding a polyomino to create other shapes. Gardner proposed several simple games with a set of free pentominoes and a chessboard. Some variants of the
1987:
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
1511:
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.
784:. 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
1492:
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
1716:
2002:, 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
1816:
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
2043:
rectangles it tiles, such that all other rectangles it tiles can be tiled by those prime rectangles. Kamenetsky and Cooke showed how various disjoint (called "holey") polyominoes can tile rectangles.
2216:, a special kind of polyomino used in number theory to describe integer partitions and in group theory and applications in mathematical physics to describe representations of the symmetric group.
1624:
1476:+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
2046:
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
2272:
2035:, and if so, what rectangles they can tile. These problems have been extensively studied for particular polyominoes, and tables of results for individual polyominoes are available.
31:
1468:
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
324:
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.
743:
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.
330:
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.
2254:, Preface to the First Edition) writes "the observation that there are twelve distinctive patterns (the pentominoes) that can be formed by five connected stones on a
318:) 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.
1519:-omino. However, it is faster to generate symmetric polyominoes separately (by a variation of this method) and so determine the number of free polyominoes by
2085:: 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.
1547:, 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.
192:
Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only
275:
3099:
2020:
1535:. An improvement on Andrew Conway's method, it is exponentially faster than the previous methods (however, its running time is still exponential in
259:
1654:
185:
polyominoes of a given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are
805:
Polyominoes have the following possible symmetries; the least number of squares needed in a polyomino with that symmetry is given in each case:
3308:
2821:
3776:
3074:
2904:
2368:
2282:
3432:
3263:
2102:
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.
4086:
3614:
3237:
2431:
3887:
3066:
1825:-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
777:-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of
3397:
1908:
if its intersection with any vertical line is convex (in other words, each column has no holes). Similarly, a polyomino is said to be
3706:
2492:
2833:
2210:, the mathematical study of random subsets of integer grids. The finite connected components of these subsets form polyominoes.
1845:
Approximations for the number of fixed polyominoes and free polyominoes are related in a simple way. A free polyomino with no
3859:
3010:
773:) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the
3512:
Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes, polyiamonds and polyhexes".
1934:
Directed polyominoes, column (or row) convex polyominoes, and convex polyominoes have been effectively enumerated by area
1577:
163:
3763:
Barbans, Uldis; Cibulis, Andris; Lee, Gilbert; Liu, Andy; Wainwright, Robert (2005). "Polyomino Number Theory (III)". In
2471:
1543:
to determine the minimum number of squares needed to complete the polyomino of the given width. Combined with the use of
2747:
Barequet, Gill; Rote, Gunter; Shalah, Mira. "λ > 4: An
Improved Lower Bound on the Growth Constant of Polyominoes".
2453:"2024 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX) - Counting Polyominoes, Revisited"
103:
1931:, such that every other square can be reached by movements of up or right one square, without leaving the polyomino.
1550:
Although it has excellent running time, the tradeoff is that this algorithm uses exponential amounts of memory (many
1571:
Theoretical arguments and numerical calculations support the estimate for the number of fixed polyominoes of size n
1558:
above 50), is much harder to program than the other methods, and can't currently be used to count free polyominoes.
3483:
Gardner, Martin (August 1965). "Thoughts on the task of communication with intelligent organisms on other worlds".
2997:
2928:
2298:
Gardner, M. (November 1960). "More about the shapes that can be made with complex dominoes (Mathematical Games)".
2069:
1507:
times, once for each starting square. It can be optimized so that it counts each polyomino only once, rather than
3791:
1963:
311:
174:
162:
in this literature) is applied to problems in physics and chemistry. Polyominoes have been used as models of
1461:, squares may be added next to each polyomino in each possible position, and the resulting polyomino of size
244:
3880:
2842:
1897:
303:
2228:, a kind of undirected graph including as a special case the graphs of vertices and edges of polyominoes.
2088:
Several polyominoes can tile larger copies of themselves, and repeating this process recursively gives a
3999:
124:
4091:
3722:
3454:
Gardner, Martin (July 1965). "On the relation between mathematics and the ordered patterns of Op art".
2694:
Jensen, Iwan; Guttmann, Anthony J. (2000). "Statistics of lattice animals (polyominoes) and polygons".
1515:
If one wishes to count free polyominoes instead, then one may check for symmetries after creating each
3814:
3912:
3656:
2713:
2660:
2617:
1970:
a prescribed region, or the entire plane, with polyominoes, and related problems are investigated in
400:
221:
120:
2847:
4028:
3514:
3485:
3456:
3315:
2868:
2146:
is based on the seven one-sided tetrominoes (spelled "Tetriminos" in the game), and the board game
1939:
1837:
obtained an upper bound of 4.65, which was subsequently improved by
Barequet and Shalah to 4.5252.
1648:
The known theoretical results are not nearly as specific as this estimate. It has been proven that
1544:
1520:
762:
155:
108:
2057:
and John
Michael Robson showed that the problem of tiling one polyomino with copies of another is
214:
4115:
4096:
3873:
3764:
3422:
3340:
3172:
2860:
2790:
2729:
2703:
2651:
Conway, Andrew (1995). "Enumerating 2D percolation series by the finite-lattice method: theory".
2633:
2607:
2337:(1990). "Lattice Animals: Rigorous Results and Wild Guesses". In Grimmett, G.; Welsh, D. (eds.).
2315:
2207:
1728:
1532:
229:
86:
3267:
2352:
158:, the study of polyominoes and their higher-dimensional analogs (which are often referred to as
3241:
941:
symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry:
893:
In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows:
841:
symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry:
4024:
3826:
3819:
3772:
3152:
3070:
3040:
2900:
2427:
2364:
2278:
2268:
1877:
Exact formulas are known for enumerating polyominoes of special classes, such as the class of
950:
symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry:
855:
symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry:
792:
95:
84:
is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in
4009:
3664:
3623:
3564:
3523:
3494:
3465:
3373:
3203:
3164:
3131:
3019:
2977:
2944:
2910:
2852:
2800:
2756:
2721:
2676:
2668:
2625:
2551:
2397:
2307:
2195:
2082:
2036:
1975:
1830:
849:
315:
193:
3693:
3404:
3689:
2914:
2680:
2110:
140:
3091:
791:
may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are
3660:
3339:
Kamenetsky, Dmitry; Cooke, Tristrom (2015). "Tiling rectangles with holey polyominoes".
2717:
2664:
2621:
2452:
3527:
3469:
3059:
2311:
1998:
Because the general problem of tiling regions of the plane with sets of polyominoes is
770:
752:
99:
3498:
3378:
3361:
3136:
3119:
2949:
2932:
2725:
2164:
and the names of the various sizes of polyomino are all back-formations from the word
4109:
3569:
3542:
3024:
3001:
2982:
2965:
2864:
2672:
2556:
2539:
2402:
2385:
2334:
2213:
1988:
1946:
178:
70:
3176:
2637:
1849:(rotation or reflection) corresponds to 8 distinct fixed polyominoes, and for large
3427:
2892:
2733:
2003:
1967:
1916:
if its intersection with any horizontal line is convex. A polyomino is said to be
90:
between the years 1937 and 1957, under the name of "dissection problems." The name
3153:"Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity"
1531:
The most modern algorithm for enumerating the fixed polyominoes was discovered by
37:
2081:
The study of which polyominoes can tile the plane has been facilitated using the
3994:
3968:
2225:
2058:
1999:
1971:
1950:
182:
167:
3829:
2805:
2778:
4046:
4004:
3669:
3644:
3393:
3168:
2629:
2467:
2357:
2054:
2047:
2031:
Another class of problems asks whether copies of a given polyomino can tile a
1893:
1846:
1834:
766:
116:
17:
3854:
3843:
4041:
4014:
3989:
3937:
3927:
3922:
3834:
2779:"Improved upper bounds on the growth constants of polyominoes and polycubes"
2493:"Harmonic Magic Square, Enumeration of Polyominoes considering the symmetry"
2255:
2115:
2032:
1992:
1949:
if its area equals its perimeter. An equable polyomino must be made from an
525:
475:
450:
266:
251:
186:
148:
136:
81:
42:
2856:
2708:
2598:
Jensen, Iwan (February 2001). "Enumerations of
Lattice Animals and Trees".
2174:
for the game piece is believed to come from the spotted masquerade garment
1711:{\displaystyle \lim _{n\rightarrow \infty }(A_{n})^{\frac {1}{n}}=\lambda }
334:
The following table shows the numbers of polyominoes of various types with
295:
There are three common ways of distinguishing polyominoes for enumeration:
3208:
3191:
4056:
3973:
3952:
3947:
3942:
3932:
3896:
3743:
3288:
Klarner, D.A.; Göbel, F. (1969). "Packing boxes with congruent figures".
2612:
2509:
2231:
2166:
2089:
1865:-ominoes. Moreover, this approximation is exponentially more accurate as
1551:
600:
575:
550:
500:
307:
282:
144:
132:
62:
2319:
4062:
4051:
3917:
3844:
MathPages – Notes on enumeration of polyominoes with various symmetries
3628:
3609:
425:
236:
128:
3848:
4069:
4036:
3314:. Stanford University Technical Report STAN-CS-73–338. Archived from
2219:
2148:
2142:
2137:
2010:
77:
66:
58:
3002:"The generating function of convex polyominoes: The resolution of a
2760:
2182:. Despite this word origin, in naming polyominoes, the first letter
3809:
3688:. Providence, RI: American Mathematical Society. pp. 205–217.
2795:
2277:(2nd ed.). Princeton, New Jersey: Princeton University Press.
1750:
can be attached to the bottom-left square of any polyomino of size
1454:. This leads to algorithms for generating polyominoes inductively.
820:
mirror symmetry with respect to one of the grid line directions (4)
3345:
2109:
54:
36:
30:"Polyominoes" redirects here. For the book by Solomon Golomb, see
2039:
and Göbel showed that for any polyomino there is a finite set of
1503:
This method ensures that each fixed polyomino is counted exactly
2013:
a square grid is tiled with polynomino-shaped regions (sequence
1385:
668:
3869:
2140:
puzzle use nonomino-shaped regions on the grid. The video game
1938:, as well as by some other parameters such as perimeter, using
1637:= 0.3169. However, this result is not proven and the values of
935:
mirror symmetry with respect to one of the grid line directions
2092:
tiling of the plane. For instance, for every positive integer
2073:
The two tiling nonominoes not satisfying the Conway criterion.
302:
polyominoes are distinct when none is a rigid transformation (
3767:; Demaine, Erik D.; Demaine, Martin L.; Rodgers, Tom (eds.).
2933:"New enumerative results on two-dimensional directed animals"
2822:"A procedure for improving the upper bound for the number of
1450:+1 can be obtained by adding a square to a polyomino of size
3865:
3718:
1857:-ominoes have no symmetries. Therefore, the number of fixed
3645:"Planar tilings by polyominoes, polyhexes, and polyiamonds"
2258:
board ... is attributed to an ancient master of that game".
2015:
1735:, found in 2016, is 4.00253. The best known upper bound is
1426:
1421:
1416:
1411:
1406:
1401:
1396:
1391:
694:
689:
684:
679:
674:
2170:, a common game piece consisting of two squares. The name
963:
The following table shows the numbers of polyominoes with
2899:(2nd ed.). Boston, MA: Academic Press. p. 151.
3586:
Planar
Tilings and the Search for an Aperiodic Prototile
3264:"List of known prime rectangles for various polyominoes"
2524:
2522:
65:
whose cells are squares. It may be regarded as a finite
1806:. Refinements of this procedure combined with data for
3061:
Polyominoes: A guide to puzzles and problems in tiling
32:
Polyominoes: Puzzles, Patterns, Problems, and
Packings
2190:
is fancifully interpreted as a version of the prefix
1861:-ominoes is approximately 8 times the number of free
1657:
1580:
1488:
squares, or may be arranged to count each once only.
3815:
An implementation and description of Jensen's method
3090:
C.B. Haselgrove; Jenifer Haselgrove (October 1960).
2966:"Generating functions for column-convex polyominoes"
2152:
uses all of the free polyominoes up to pentominoes.
1892:
polyomino is different from the usual definition of
1619:{\displaystyle A_{n}\sim {\frac {c\lambda ^{n}}{n}}}
4023:
3982:
3961:
3903:
2468:"Animal enumerations on the {4,4} Euclidean tiling"
823:
mirror symmetry with respect to a diagonal line (3)
3058:
2356:
2065:Tiling the plane with copies of a single polyomino
1710:
1618:
3849:List of dissection problems in Fairy Chess Review
1457:Most simply, given a list of polyominoes of size
897:2 one-sided polyominoes for each free polyomino:
3810:Karl Dahlke's polyomino finite-rectangle tilings
3771:. Wellesley, MA: A.K. Peters. pp. 131–136.
3649:Journal of Computational and Applied Mathematics
2027:Tiling regions with copies of a single polyomino
1896:, but is similar to the definition used for the
1659:
702:Fixed polyominoes were enumerated in 2004 up to
2114:A minimal compatibility figure for the T and W
2106:Tiling a common figure with various polyominoes
1437:Algorithms for enumeration of fixed polyominoes
938:mirror symmetry with respect to a diagonal line
923:1 one-sided polyomino for each free polyomino:
719:Free polyominoes were enumerated in 2007 up to
3684:Niţică, Viorel (2003). "Rep-tiles revisited".
2696:Journal of Physics A: Mathematical and General
2653:Journal of Physics A: Mathematical and General
1562:Asymptotic growth of the number of polyominoes
135:. Polyominoes have been generalized to higher
3881:
1472:be stored in size to enumerate those of size
838:2 fixed polyominoes for each free polyomino:
817:4 fixed polyominoes for each free polyomino:
809:8 fixed polyominoes for each free polyomino:
8:
80:since at least 1907, and the enumeration of
2428:"Series for lattice animals or polyominoes"
1484:times, once from starting from each of its
877:1 fixed polyomino for each free polyomino:
3888:
3874:
3866:
2540:"Counting polyominoes: Yet another attack"
2386:"Counting polyominoes: yet another attack"
733:by Toshihiro Shirakawa, and in 2023 up to
3668:
3627:
3610:"Isohedral Polyomino Tiling of the Plane"
3568:
3377:
3344:
3207:
3135:
3023:
2981:
2970:Journal of Combinatorial Theory, Series A
2948:
2846:
2804:
2794:
2707:
2611:
2555:
2401:
1691:
1681:
1662:
1656:
1604:
1594:
1585:
1579:
726:by Tomás Oliveira e Silva, in 2012 up to
3820:A paper describing modern estimates (PS)
3398:"Hard Tiling Problems with Simple Tiles"
3238:"References for Rectifiable Polyominoes"
3151:E.D. Demaine; M.L. Demaine (June 2007).
2068:
969:
340:
3588:. PhD dissertation, Rutgers University.
2243:
1982:Tiling regions with sets of polyominoes
716:by Gill Barequet and Gil Ben-Shachar.
203:
3421:Petersen, Ivars (September 25, 1999),
2363:. New York: W.H. Freeman and Company.
1927:if it contains a square, known as the
291:Free, one-sided, and fixed polyominoes
76:Polyominoes have been used in popular
3615:Discrete & Computational Geometry
2777:Barequet, Gill; Shalah, Mira (2022).
2772:
2770:
2194:meaning "two", and replaced by other
1813:produce the lower bound given above.
967:squares, sorted by symmetry groups.
795:of fixed polyominoes under the group
285:, colored according to their symmetry
269:, colored according to their symmetry
7:
3423:"Math Trek: Tiling with Polyominoes"
3092:"A Computer Program for Pentominoes"
3045:, MathEducationPage.org, p. 208
2820:Klarner, D.A.; Rivest, R.L. (1973).
1788:. Using this equation, one can show
57:formed by joining one or more equal
27:Geometric shapes formed from squares
4082:
3435:from the original on March 20, 2008
3307:Klarner, David A. (February 1973).
3067:Mathematical Association of America
2234:, its analogue in three dimensions.
98:in 1953, and it was popularized by
3598:Grünbaum and Shephard, section 9.4
3528:10.1038/scientificamerican0875-112
3470:10.1038/scientificamerican1265-100
3309:"A Finite Basis Theorem Revisited"
3190:S.W. Golomb; L.D. Baumert (1965).
2312:10.1038/scientificamerican1160-186
1669:
709:by Iwan Jensen, and in 2024 up to
25:
3744:"Zucca, L., "Triple Pentominoes""
3499:10.1038/scientificamerican0865-96
3120:"Tiling with Sets of Polyominoes"
2872:(PDF of technical report version)
2222:, a board game using polyominoes.
1966:, challenges are often posed for
1731:. The best known lower bound for
4090:
4081:
2131:Polyominoes in puzzles and games
1920:if it is row and column convex.
274:
258:
243:
228:
213:
3725:from the original on 2011-02-22
3608:Keating, K.; Vince, A. (1999).
3396:; Robson, John Michael (2001).
3366:Journal of Combinatorial Theory
3124:Journal of Combinatorial Theory
2834:Canadian Journal of Mathematics
2474:from the original on 2007-04-23
2434:from the original on 2007-06-12
3860:Wolfram Demonstrations Project
3719:"Resta, G., "Polypolyominoes""
3541:Rawsthorne, Daniel A. (1988).
2600:Journal of Statistical Physics
2510:"Counting size 50 polyominoes"
1873:Special classes of polyominoes
1688:
1674:
1666:
1496:squares have been created, an
1:
3379:10.1016/S0021-9800(66)80033-9
3137:10.1016/S0021-9800(70)80055-2
2950:10.1016/S0012-365X(97)00109-X
1881:polyominoes and the class of
45:, including 6 mirrored pairs.
3707:Mireles, J.L., "Polyominoes"
3570:10.1016/0012-365X(88)90081-7
3543:"Tiling complexity of small
3025:10.1016/0012-365X(93)E0161-V
2983:10.1016/0097-3165(88)90071-4
2557:10.1016/0012-365X(81)90237-5
2403:10.1016/0012-365X(81)90237-5
2384:Redelmeier, D. Hugh (1981).
2339:Disorder in Physical Systems
2096:, it is possible to combine
1900:. A polyomino is said to be
926:all symmetry of the square:
912:4-fold rotational symmetry:
903:2-fold rotational symmetry:
880:all symmetry of the square:
865:4-fold rotational symmetry:
826:2-fold rotational symmetry:
181:problems. The most basic is
3360:Golomb, Solomon W. (1966).
3118:Golomb, Solomon W. (1970).
3000:; Fédou, Jean-Marc (1995).
2726:10.1088/0305-4470/33/29/102
2538:Redelmeier, D.Hugh (1981).
1354:
1325:
1296:
1267:
1238:
1209:
1180:
1151:
1122:
1093:
1064:
1035:
666:
643:
620:
595:
570:
545:
520:
495:
470:
445:
420:
395:
372:
115:Related to polyominoes are
4132:
3769:Tribute to a Mathemagician
3057:Martin, George E. (1996).
2806:10.1007/s00453-022-00948-6
2673:10.1088/0305-4470/28/2/011
2341:. Oxford University Press.
1923:A polyomino is said to be
647:
624:
599:
574:
549:
524:
499:
474:
449:
424:
399:
376:
200:Enumeration of polyominoes
29:
4079:
3792:Oxford English Dictionary
3670:10.1016/j.cam.2004.05.002
3643:Rhoads, Glenn C. (2005).
3584:Rhoads, Glenn C. (2003).
3362:"Tiling with Polyominoes"
3290:Indagationes Mathematicae
3169:10.1007/s00373-007-0713-4
3039:Picciotto, Henri (1999),
2749:Communications of the ACM
2355:; Shephard, G.C. (1987).
1995:to sets of polyominoes.
1554:of memory are needed for
1500:-omino has been created.
747:Symmetries of polyominoes
667:
357:
354:
351:
348:
343:
177:, polyominoes raise many
3157:Graphs and Combinatorics
2998:Bousquet-Mélou, Mireille
2929:Bousquet-Mélou, Mireille
2466:Tomás Oliveira e Silva.
1964:recreational mathematics
1821:. By proving that every
1721:exists. In other words,
175:recreational mathematics
151:to form polyhypercubes.
3192:"Backtrack Programming"
2897:Generatingfunctionology
2630:10.1023/A:1004855020556
1958:Tiling with polyominoes
1446:Each polyomino of size
848:(2) (also known as the
2964:Delest, M.-P. (1988).
2857:10.4153/CJM-1973-060-4
2119:
2074:
1898:orthogonal convex hull
1712:
1620:
1527:Transfer-matrix method
189:for calculating them.
127:, formed from regular
61:edge to edge. It is a
55:plane geometric figure
46:
3794:, 2nd edition, entry
3209:10.1145/321296.321300
3006:-differential system"
2588:Redelmeier, section 6
2579:Redelmeier, section 4
2528:Redelmeier, section 3
2124:compatibility problem
2113:
2072:
1991:, by mapping sets of
1762:)-omino. This proves
1754:to produce a unique (
1713:
1621:
322:one-sided polyominoes
173:Like many puzzles in
121:equilateral triangles
40:
3557:Discrete Mathematics
3011:Discrete Mathematics
2937:Discrete Mathematics
2544:Discrete Mathematics
2390:Discrete Mathematics
2359:Tilings and Patterns
2333:Whittington, S. G.;
1940:generating functions
1888:The definition of a
1655:
1645:are only estimates.
1578:
1545:generating functions
1442:Inductive algorithms
102:in a November 1960 "
3765:Cipra, Barry Arthur
3661:2005JCoAM.174..329R
3515:Scientific American
3486:Scientific American
3457:Scientific American
2718:2000JPhA...33L.257J
2665:1995JPhA...28..335C
2622:2001JSP...102..865J
2300:Scientific American
1729:grows exponentially
793:equivalence classes
156:statistical physics
109:Scientific American
3827:Weisstein, Eric W.
3629:10.1007/PL00009442
3196:Journal of the ACM
2709:cond-mat/0007238v1
2269:Golomb, Solomon W.
2208:Percolation theory
2196:numerical prefixes
2120:
2075:
1708:
1673:
1616:
205:Free polyominoes (
131:; and other plane
104:Mathematical Games
87:Fairy Chess Review
47:
4103:
4102:
3962:Higher dimensions
3858:by Karl Scherer,
3778:978-1-56881-204-5
3394:Moore, Cristopher
3076:978-0-88385-501-0
2906:978-0-12-751956-2
2789:(12): 3559–3586.
2702:(29): L257–L263.
2570:Golomb, pp. 73–79
2416:Golomb, chapter 6
2370:978-0-7167-1193-3
2284:978-0-691-02444-8
1699:
1658:
1614:
1567:Fixed polyominoes
1432:
1431:
700:
699:
164:branched polymers
96:Solomon W. Golomb
41:The 18 one-sided
16:(Redirected from
4123:
4095:
4094:
4085:
4084:
4010:Pseudo-polyomino
3890:
3883:
3876:
3867:
3840:
3839:
3798:
3789:
3783:
3782:
3760:
3754:
3753:
3751:
3750:
3740:
3734:
3733:
3731:
3730:
3715:
3709:
3704:
3698:
3697:
3681:
3675:
3674:
3672:
3640:
3634:
3633:
3631:
3605:
3599:
3596:
3590:
3589:
3581:
3575:
3574:
3572:
3538:
3532:
3531:
3509:
3503:
3502:
3480:
3474:
3473:
3451:
3445:
3443:
3442:
3440:
3418:
3412:
3411:
3409:
3403:. Archived from
3402:
3390:
3384:
3383:
3381:
3357:
3351:
3350:
3348:
3336:
3330:
3329:
3327:
3326:
3320:
3313:
3304:
3298:
3297:
3285:
3279:
3278:
3276:
3275:
3266:. Archived from
3259:
3253:
3252:
3250:
3249:
3240:. Archived from
3233:
3227:
3220:
3214:
3213:
3211:
3187:
3181:
3180:
3148:
3142:
3141:
3139:
3115:
3109:
3108:
3096:
3087:
3081:
3080:
3065:(2nd ed.).
3064:
3054:
3048:
3046:
3036:
3030:
3029:
3027:
2994:
2988:
2987:
2985:
2961:
2955:
2954:
2952:
2925:
2919:
2918:
2893:Wilf, Herbert S.
2889:
2883:
2882:
2880:
2879:
2873:
2867:. Archived from
2850:
2830:
2817:
2811:
2810:
2808:
2798:
2774:
2765:
2764:
2744:
2738:
2737:
2711:
2691:
2685:
2684:
2648:
2642:
2641:
2615:
2613:cond-mat/0007239
2606:(3–4): 865–881.
2595:
2589:
2586:
2580:
2577:
2571:
2568:
2562:
2561:
2559:
2535:
2529:
2526:
2517:
2516:
2514:
2506:
2500:
2499:
2497:
2489:
2483:
2482:
2480:
2479:
2463:
2457:
2456:
2449:
2443:
2442:
2440:
2439:
2423:
2417:
2414:
2408:
2407:
2405:
2381:
2375:
2374:
2362:
2353:Grünbaum, Branko
2349:
2343:
2342:
2330:
2324:
2323:
2295:
2289:
2288:
2265:
2259:
2248:
2101:
2095:
2083:Conway criterion
2055:Cristopher Moore
2018:
1976:computer science
1841:Free polyominoes
1801:
1787:
1741:
1717:
1715:
1714:
1709:
1701:
1700:
1692:
1686:
1685:
1672:
1625:
1623:
1622:
1617:
1615:
1610:
1609:
1608:
1595:
1590:
1589:
1521:Burnside's lemma
970:
850:Klein four-group
739:
732:
725:
715:
708:
341:
316:glide reflection
278:
262:
247:
232:
217:
194:simply connected
94:was invented by
21:
4131:
4130:
4126:
4125:
4124:
4122:
4121:
4120:
4106:
4105:
4104:
4099:
4089:
4075:
4019:
3978:
3957:
3899:
3894:
3825:
3824:
3806:
3801:
3790:
3786:
3779:
3762:
3761:
3757:
3748:
3746:
3742:
3741:
3737:
3728:
3726:
3717:
3716:
3712:
3705:
3701:
3683:
3682:
3678:
3642:
3641:
3637:
3607:
3606:
3602:
3597:
3593:
3583:
3582:
3578:
3548:
3540:
3539:
3535:
3511:
3510:
3506:
3482:
3481:
3477:
3453:
3452:
3448:
3438:
3436:
3420:
3419:
3415:
3407:
3400:
3392:
3391:
3387:
3359:
3358:
3354:
3338:
3337:
3333:
3324:
3322:
3318:
3311:
3306:
3305:
3301:
3287:
3286:
3282:
3273:
3271:
3262:Reid, Michael.
3261:
3260:
3256:
3247:
3245:
3236:Reid, Michael.
3235:
3234:
3230:
3221:
3217:
3189:
3188:
3184:
3150:
3149:
3145:
3117:
3116:
3112:
3094:
3089:
3088:
3084:
3077:
3056:
3055:
3051:
3038:
3037:
3033:
2996:
2995:
2991:
2963:
2962:
2958:
2943:(1–3): 73–106.
2927:
2926:
2922:
2907:
2891:
2890:
2886:
2877:
2875:
2871:
2848:10.1.1.309.9151
2828:
2819:
2818:
2814:
2776:
2775:
2768:
2761:10.1145/2851485
2746:
2745:
2741:
2693:
2692:
2688:
2650:
2649:
2645:
2597:
2596:
2592:
2587:
2583:
2578:
2574:
2569:
2565:
2537:
2536:
2532:
2527:
2520:
2512:
2508:
2507:
2503:
2495:
2491:
2490:
2486:
2477:
2475:
2465:
2464:
2460:
2451:
2450:
2446:
2437:
2435:
2425:
2424:
2420:
2415:
2411:
2383:
2382:
2378:
2371:
2351:
2350:
2346:
2332:
2331:
2327:
2297:
2296:
2292:
2285:
2267:
2266:
2262:
2249:
2245:
2241:
2204:
2158:
2133:
2108:
2097:
2093:
2067:
2029:
2014:
1984:
1960:
1945:A polyomino is
1875:
1843:
1811:
1798:
1789:
1786:
1772:
1768:
1763:
1736:
1726:
1687:
1677:
1653:
1652:
1600:
1596:
1581:
1576:
1575:
1569:
1564:
1529:
1444:
1439:
1388: sequence
1032:
1024:
1015:
1014:
1005:
1004:
996:
987:
982:
956:
947:
932:
918:
909:
886:
871:
861:
847:
832:
812:no symmetry (4)
801:
790:
783:
760:
749:
740:by John Mason.
734:
727:
720:
710:
703:
671: sequence
293:
286:
279:
270:
263:
254:
248:
239:
233:
224:
218:
202:
160:lattice animals
69:of the regular
35:
28:
23:
22:
15:
12:
11:
5:
4129:
4127:
4119:
4118:
4108:
4107:
4101:
4100:
4080:
4077:
4076:
4074:
4073:
4066:
4059:
4054:
4049:
4044:
4039:
4033:
4031:
4021:
4020:
4018:
4017:
4012:
4007:
4002:
3997:
3992:
3986:
3984:
3980:
3979:
3977:
3976:
3971:
3965:
3963:
3959:
3958:
3956:
3955:
3950:
3945:
3940:
3935:
3930:
3925:
3920:
3915:
3909:
3907:
3901:
3900:
3895:
3893:
3892:
3885:
3878:
3870:
3864:
3863:
3851:
3846:
3841:
3822:
3817:
3812:
3805:
3804:External links
3802:
3800:
3799:
3784:
3777:
3755:
3735:
3710:
3699:
3676:
3655:(2): 329–353.
3635:
3622:(4): 615–630.
3600:
3591:
3576:
3533:
3522:(2): 112–115.
3504:
3475:
3464:(1): 100–104.
3446:
3413:
3410:on 2013-06-17.
3385:
3372:(2): 280–296.
3352:
3331:
3299:
3280:
3254:
3228:
3215:
3202:(4): 516–524.
3182:
3143:
3110:
3082:
3075:
3049:
3031:
3018:(1–3): 53–75.
2989:
2956:
2920:
2905:
2884:
2841:(3): 585–602.
2812:
2766:
2739:
2686:
2659:(2): 335–349.
2643:
2590:
2581:
2572:
2563:
2550:(2): 191–203.
2530:
2518:
2501:
2484:
2458:
2444:
2418:
2409:
2396:(2): 191–203.
2376:
2369:
2344:
2335:Soteros, C. E.
2325:
2306:(5): 186–201.
2290:
2283:
2260:
2242:
2240:
2237:
2236:
2235:
2229:
2223:
2217:
2211:
2203:
2200:
2157:
2154:
2132:
2129:
2107:
2104:
2066:
2063:
2028:
2025:
2011:Jigsaw Sudokus
1983:
1980:
1959:
1956:
1874:
1871:
1842:
1839:
1809:
1796:
1778:
1770:
1766:
1724:
1719:
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1584:
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1560:
1528:
1525:
1443:
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1414:
1409:
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1311:
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1295:
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1285:
1282:
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1273:
1270:
1266:
1265:
1262:
1259:
1256:
1253:
1250:
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1244:
1241:
1237:
1236:
1233:
1230:
1227:
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1221:
1218:
1215:
1212:
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1207:
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1201:
1198:
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1189:
1186:
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1160:
1157:
1154:
1150:
1149:
1146:
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1140:
1137:
1134:
1131:
1128:
1125:
1121:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1092:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1063:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1034:
1033:
1030:
1025:
1022:
1017:
1012:
1007:
1002:
997:
994:
989:
984:
979:
976:
961:
960:
959:
958:
954:
948:
945:
939:
936:
933:
930:
921:
920:
919:
916:
910:
907:
901:
891:
890:
889:
888:
884:
875:
874:
873:
869:
863:
859:
853:
845:
836:
835:
834:
830:
824:
821:
815:
814:
813:
799:
788:
781:
771:symmetry group
758:
753:dihedral group
748:
745:
698:
697:
692:
687:
682:
677:
672:
665:
664:
661:
658:
655:
652:
649:
646:
642:
641:
638:
635:
632:
629:
626:
623:
619:
618:
615:
612:
609:
606:
603:
598:
594:
593:
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587:
584:
581:
578:
573:
569:
568:
565:
562:
559:
556:
553:
548:
544:
543:
540:
537:
534:
531:
528:
523:
519:
518:
515:
512:
509:
506:
503:
498:
494:
493:
490:
487:
484:
481:
478:
473:
469:
468:
465:
462:
459:
456:
453:
448:
444:
443:
440:
437:
434:
431:
428:
423:
419:
418:
415:
412:
409:
406:
403:
398:
394:
393:
390:
387:
384:
381:
378:
375:
371:
370:
369:without holes
367:
364:
360:
359:
356:
353:
350:
347:
332:
331:
325:
319:
292:
289:
288:
287:
280:
273:
271:
264:
257:
255:
249:
242:
240:
234:
227:
225:
219:
212:
210:
201:
198:
119:, formed from
100:Martin Gardner
26:
24:
18:Free polyomino
14:
13:
10:
9:
6:
4:
3:
2:
4128:
4117:
4114:
4113:
4111:
4098:
4093:
4088:
4078:
4072:
4071:
4067:
4065:
4064:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4034:
4032:
4030:
4026:
4022:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3987:
3985:
3981:
3975:
3972:
3970:
3967:
3966:
3964:
3960:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3911:
3910:
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3808:
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3537:
3534:
3529:
3525:
3521:
3517:
3516:
3508:
3505:
3500:
3496:
3493:(2): 96–100.
3492:
3488:
3487:
3479:
3476:
3471:
3467:
3463:
3459:
3458:
3450:
3447:
3434:
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3380:
3375:
3371:
3367:
3363:
3356:
3353:
3347:
3342:
3335:
3332:
3321:on 2007-10-23
3317:
3310:
3303:
3300:
3295:
3291:
3284:
3281:
3270:on 2007-04-16
3269:
3265:
3258:
3255:
3244:on 2004-01-16
3243:
3239:
3232:
3229:
3225:
3219:
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3205:
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3086:
3083:
3078:
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3063:
3062:
3053:
3050:
3044:
3043:
3042:Geometry Labs
3035:
3032:
3026:
3021:
3017:
3013:
3012:
3007:
3005:
2999:
2993:
2990:
2984:
2979:
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2912:
2908:
2902:
2898:
2894:
2888:
2885:
2874:on 2006-11-26
2870:
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2494:
2488:
2485:
2473:
2469:
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2459:
2454:
2448:
2445:
2433:
2429:
2426:Iwan Jensen.
2422:
2419:
2413:
2410:
2404:
2399:
2395:
2391:
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2257:
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2238:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2214:Young diagram
2212:
2209:
2206:
2205:
2201:
2199:
2197:
2193:
2189:
2185:
2181:
2178:, from Latin
2177:
2173:
2169:
2168:
2163:
2155:
2153:
2151:
2150:
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2139:
2130:
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2100:
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2049:
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2017:
2012:
2007:
2005:
2001:
1996:
1994:
1990:
1981:
1979:
1977:
1973:
1969:
1965:
1957:
1955:
1952:
1948:
1943:
1941:
1937:
1932:
1930:
1926:
1921:
1919:
1915:
1911:
1907:
1906:column convex
1903:
1899:
1895:
1891:
1886:
1885:polyominoes.
1884:
1880:
1872:
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1663:
1651:
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1649:
1646:
1644:
1640:
1636:
1633:= 4.0626 and
1632:
1611:
1605:
1601:
1597:
1591:
1586:
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1574:
1573:
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1199:
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342:
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261:
256:
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246:
241:
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231:
226:
223:
216:
211:
208:
204:
199:
197:
196:polyominoes.
195:
190:
188:
184:
180:
179:combinatorial
176:
171:
169:
165:
161:
157:
152:
150:
146:
142:
138:
134:
130:
126:
122:
118:
113:
111:
110:
105:
101:
97:
93:
89:
88:
83:
79:
74:
72:
71:square tiling
68:
64:
60:
56:
52:
44:
39:
33:
19:
4068:
4061:
3904:
3853:
3833:
3795:
3787:
3768:
3758:
3747:. Retrieved
3738:
3727:. Retrieved
3713:
3702:
3686:MASS selecta
3685:
3679:
3652:
3648:
3638:
3619:
3613:
3603:
3594:
3585:
3579:
3560:
3556:
3550:
3544:
3536:
3519:
3513:
3507:
3490:
3484:
3478:
3461:
3455:
3449:
3437:, retrieved
3428:Science News
3426:
3416:
3405:the original
3388:
3369:
3365:
3355:
3334:
3323:. Retrieved
3316:the original
3302:
3293:
3289:
3283:
3272:. Retrieved
3268:the original
3257:
3246:. Retrieved
3242:the original
3231:
3223:
3218:
3199:
3195:
3185:
3160:
3156:
3146:
3127:
3123:
3113:
3104:
3098:
3085:
3060:
3052:
3041:
3034:
3015:
3009:
3003:
2992:
2976:(1): 12–31.
2973:
2969:
2959:
2940:
2936:
2923:
2896:
2887:
2876:. Retrieved
2869:the original
2838:
2832:
2823:
2815:
2786:
2783:Algorithmica
2782:
2755:(7): 88–95.
2752:
2748:
2742:
2699:
2695:
2689:
2656:
2652:
2646:
2603:
2599:
2593:
2584:
2575:
2566:
2547:
2543:
2533:
2504:
2487:
2476:. Retrieved
2461:
2447:
2436:. Retrieved
2421:
2412:
2393:
2389:
2379:
2358:
2347:
2338:
2328:
2303:
2299:
2293:
2273:
2263:
2251:
2246:
2191:
2187:
2183:
2179:
2175:
2171:
2165:
2161:
2159:
2147:
2141:
2134:
2123:
2121:
2098:
2087:
2080:
2076:
2052:
2045:
2040:
2030:
2008:
2004:backtracking
1997:
1985:
1961:
1944:
1935:
1933:
1928:
1924:
1922:
1917:
1913:
1910:horizontally
1909:
1905:
1901:
1889:
1887:
1882:
1878:
1876:
1866:
1862:
1858:
1854:
1850:
1844:
1826:
1822:
1818:
1815:
1807:
1803:
1794:
1790:
1783:
1779:
1775:
1764:
1759:
1755:
1751:
1747:
1744:
1737:
1732:
1722:
1720:
1647:
1642:
1638:
1634:
1630:
1628:
1570:
1555:
1549:
1541:
1536:
1530:
1516:
1514:
1508:
1504:
1502:
1497:
1493:
1490:
1485:
1481:
1477:
1473:
1469:
1467:
1462:
1458:
1456:
1451:
1447:
1445:
1433:
1027:
1019:
1009:
999:
991:
973:
964:
962:
951:
942:
927:
913:
904:
892:
881:
866:
856:
842:
827:
804:
796:
785:
778:
774:
755:
750:
742:
735:
728:
721:
718:
711:
704:
701:
344:
335:
333:
327:
321:
299:
294:
206:
191:
172:
159:
153:
114:
107:
106:" column in
91:
85:
75:
50:
48:
4087:WikiProject
3995:Polydrafter
3969:Polyominoid
3905:Polyominoes
3830:"Polyomino"
3226:, chapter 8
3224:Polyominoes
3163:: 195–208.
2274:Polyominoes
2252:Polyominoes
2226:Squaregraph
2116:pentominoes
2059:NP-complete
2000:NP-complete
1989:undecidable
1972:mathematics
1951:even number
1869:increases.
1740:< 4.5252
1533:Iwan Jensen
900:no symmetry
366:with holes
304:translation
267:pentominoes
252:tetrominoes
183:enumerating
168:percolation
139:by joining
117:polyiamonds
82:pentominoes
43:pentominoes
4047:Snake cube
4005:Polyiamond
3749:2023-04-20
3729:2010-07-02
3325:2007-05-12
3296:: 465–472.
3274:2007-05-11
3248:2007-05-11
2915:0831.05001
2878:2007-05-11
2796:1906.11447
2681:0849.05003
2478:2007-05-06
2438:2007-05-06
2048:half plane
1993:Wang tiles
1914:row convex
1902:vertically
1847:symmetries
767:symmetries
648:dodecomino
625:undecomino
355:one-sided
312:reflection
283:hexominoes
250:Five free
187:algorithms
170:clusters.
149:hypercubes
137:dimensions
4116:Polyforms
4042:Soma cube
4015:Polystick
3990:Polyabolo
3938:Heptomino
3928:Pentomino
3923:Tetromino
3897:Polyforms
3835:MathWorld
3563:: 71–75.
3439:March 11,
3346:1411.2699
3130:: 60–71.
2865:121448572
2843:CiteSeerX
2826:-ominoes"
2162:polyomino
2160:The word
2156:Etymology
2033:rectangle
1894:convexity
1793:≥ (
1706:λ
1670:∞
1667:→
1602:λ
1592:∼
1552:gigabytes
526:heptomino
476:pentomino
451:tetromino
237:trominoes
235:Two free
220:One free
145:polycubes
133:polyforms
125:polyhexes
92:polyomino
51:polyomino
4110:Category
4057:Hexastix
3974:Polycube
3953:Decomino
3948:Nonomino
3943:Octomino
3933:Hexomino
3723:Archived
3553:<10)"
3547:-ominoes
3433:archived
3222:Golomb,
3177:17190810
3107:: 16–18.
2931:(1998).
2895:(1994).
2638:10549375
2472:Archived
2432:Archived
2320:24940703
2271:(1994).
2250:Golomb (
2232:Polycube
2202:See also
2090:rep-tile
2053:In 2001
1925:directed
1883:directed
1802:for all
1774:≤
663:505,861
640:135,268
601:decomino
576:nonomino
551:octomino
501:hexomino
377:monomino
308:rotation
281:35 free
265:12 free
209:=2 to 6)
143:to form
129:hexagons
63:polyform
4063:Tantrix
4052:Tangram
4029:puzzles
4000:Polyhex
3918:Tromino
3855:Tetrads
3694:2027179
3657:Bibcode
2734:6461687
2714:Bibcode
2661:Bibcode
2618:Bibcode
2180:dominus
2037:Klarner
2019:in the
2016:A172477
1947:equable
1853:, most
1831:Klarner
1480:-omino
1427:A142886
1422:A144553
1417:A056878
1412:A056877
1407:A006747
1402:A006748
1397:A006746
1392:A006749
761:is the
695:A001168
690:A000988
685:A000104
680:A001419
675:A000105
660:126,759
617:36,446
426:tromino
338:cells.
166:and of
78:puzzles
59:squares
4097:Portal
4070:Tetris
4037:Blokus
3983:Others
3913:Domino
3796:domino
3775:
3692:
3175:
3100:Eureka
3073:
2913:
2903:
2863:
2845:
2732:
2679:
2636:
2367:
2318:
2281:
2220:Blokus
2188:domino
2176:domino
2172:domino
2167:domino
2149:Blokus
2143:Tetris
2138:Sudoku
1968:tiling
1918:convex
1890:convex
1879:convex
1835:Rivest
1629:where
1359:62,878
1330:16,750
986:mirror
981:mirror
657:58,937
651:63,600
637:33,896
634:16,094
628:17,073
592:9,910
567:2,725
401:domino
363:total
358:fixed
222:domino
67:subset
4025:Games
3408:(PDF)
3401:(PDF)
3341:arXiv
3319:(PDF)
3312:(PDF)
3173:S2CID
3095:(PDF)
2861:S2CID
2829:(PDF)
2791:arXiv
2730:S2CID
2704:arXiv
2634:S2CID
2608:arXiv
2513:(PDF)
2496:(PDF)
2316:JSTOR
2239:Notes
2041:prime
1819:twigs
1301:4,461
1272:1,196
978:none
763:group
654:4,663
614:9,189
611:4,460
605:4,655
589:2,500
586:1,248
580:1,285
352:free
349:name
328:fixed
147:, or
141:cubes
53:is a
4027:and
3773:ISBN
3441:2012
3071:ISBN
2901:ISBN
2365:ISBN
2279:ISBN
2122:The
2021:OEIS
1974:and
1929:root
1833:and
1641:and
1386:OEIS
1016:45°
1006:90°
988:45°
983:90°
887:(1).
751:The
738:= 50
731:= 45
724:= 28
714:= 70
707:= 56
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