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109-406: The game is played on a square board divided into 20 rows and 20 columns, for a total of 400 squares. There are a total of 84 game tiles, organized into 21 shapes in each of the four colors: blue, yellow, red, and green. The 21 shapes are based on free polyominoes of one to five squares (one monomino, one domino , two trominoes /triominoes, five tetrominoes , and 12 pentominoes ). These shapes are

218-431: A Dragon Quest VIII "slime" controller, a Final Fantasy X-2 "Tiny Bee" dual pistol controller, an Onimusha 3 katana controller, and a Resident Evil 4 chainsaw controller. Optional hardware includes additional DualShock or DualShock 2 controllers, a PS2 DVD remote control , an internal or external hard disk drive (HDD), a network adapter, horizontal and vertical stands, PlayStation or PS2 memory cards ,

327-657: A digital video recorder and DVD burner in addition to playing PS2 games. The device was released in Japan on 13 December 2003, and was the first Sony product to include the XrossMediaBar interface. It did not sell well in the Japanese market and was not released anywhere else. PlayStation 2 users had the option to play select games over the Internet , using dial-up or a broadband Internet connection . The PlayStation 2 Network Adaptor

436-474: A "safe bet" despite not being the "newest or most powerful", noting that the console "yields in-game graphics with more jagged edges". CNET also criticized the DVD playback functionality, claiming that the console's video quality was "passable" and that the playback controls were "rudimentary", recommending users to purchase a remote control. The console's two controller ports and the high cost of its memory cards were also

545-476: A Mayan theme as Rumis , uses pieces made up of unit cubes, in every permutation of three and four cubes in three dimensions. Stefan Kögl created Rumis independently from Bernard Tavitian, and is thus not related. However, Rumis was rebranded into Blokus 3D since the Blokus brand proved stronger than Rumis . There is also a major rule change; instead of being required to place pieces so they touch corner-to-corner,

654-483: A crucial difference in the flavour of the game, because players' pieces may (and usually do) touch after the first move. Even more than with the original game, Blokus Duo is an offense-centered game; it is also a much purer strategy game than the four-player game, since one is not in danger of getting ganged up on by three other players (as sometimes happens with the four-player version). Blokus Trigon uses pieces made up of triangles rather than squares ( polyiamonds ) and

763-532: A decentralized model of online gaming where the responsibility is up to the developer to provide the servers, Sony's moves made online gaming a major selling point of the PS2. In September 2004, Sony revealed a newer, slimmer model of the PlayStation 2. In preparation for the launch of the new models (SCPH-700xx-9000x), Sony stopped making the older models (SCPH-3000x-500xx) to let the distribution channel empty its stock of

872-486: A few weeks prior. There were shortages in more than 1,700 shops in North America on the day before Christmas. In 2010, Sony introduced a TV with a built-in PlayStation 2. The PlayStation 2 continued to be produced until 2013 when Sony announced that it had been discontinued after over twelve years of production—one of the longest lifespans of any video game console. New games for the console continued to be made until

981-423: 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 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

1090-458: A given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them. Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only simply connected polyominoes. There are three common ways of distinguishing polyominoes for enumeration: The following table shows

1199-438: A maximum colour depth of 16.7 million true colours . When accounting for features such as lighting , texture mapping , artificial intelligence , and game physics , the console has a real-world performance of 25 million polygons per second. The PlayStation 2 also features two USB ports, and one IEEE 1394 (Firewire) port for SCPH-10000 to 3900x models only. A hard disk drive can be installed in an expansion bay on

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1308-410: A non-strategic way". Sekkoïa and its distributors manufacture four additional variants of the game. Blokus Duo is for two players only, and uses a smaller board (14 × 14); the pieces are opaque with black and white colors (originally translucent with purple and orange colors). The two starting squares are placed, not in the corner (as in the original Blokus game), but nearer to the centre. This makes

1417-452: 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

1526-487: A piece must be placed such that it touches a face of another piece of the same color. Also, a player placing a piece cannot do so if it would create any empty space underneath any part of the piece. The objective is to build one of four different structures, each with its own placement limitations: the Tower, Wall, Steps, and Pyramid. Players attempt to place their blocks to make sure they have as many squares of their color on top of

1635-416: A point of criticism. The slim model of the PlayStation 2 received positive reviews for its incredibly small size and built-in networking but received criticism for easily overheating due to the exclusion of the original model's built-in fan. The requirement for a separate power adapter was criticized while the top-loading disc drive was noted as being less likely to break compared to the tray-loading drive of

1744-442: A rectangle, unresolved at that time). In 2001 Cristopher Moore and John Michael Robson showed that the problem of tiling one polyomino with copies of another is NP-complete . 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

1853-434: A second console around the time of the original PlayStation's launch in late 1994. At some point during development, employees from Argonaut Games , under contract for semiconductor manufacturer LSI Corporation , were instructed to design a rendering chip for Sony's upcoming console. Jez San , founder of Argonaut, recalled that his team had no direct contact with Sony during the development process. Unbeknownst to him, Sony

1962-457: A set of free pentominoes and a chessboard. Some variants of the Sudoku puzzle use nonomino-shaped regions on the grid. The video game Tetris is based on the seven one-sided tetrominoes (spelled "Tetriminos" in the game), and the board game Blokus uses all of the free polyominoes up to pentominoes. The word polyomino and the names of the various sizes of polyomino are all back-formations from

2071-451: A specific task, integrated onto the same die . These units include a central CPU core, two Vector Processing Units (VPU), a 10-channel DMA unit, a memory controller , and an Image Processing Unit (IPU). There are three interfaces: an input output interface to the I/O processor running at a clock speed of 36.864 MHz, a graphics interface to the graphics synthesiser, and a memory interface to

2180-435: 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

2289-441: 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

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2398-451: A television as long as there is access to mains electricity or a similar power source. These screens can fold down onto the PS2 in a similar fashion to laptop screens. There are many accessories for musical games, such as dance pads for Dance Dance Revolution , In the Groove , and Pump It Up titles and High School Musical 3: Senior Year Dance . Konami microphones for use with

2507-572: 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, which

2616-429: A total of 10,035 software titles had been released worldwide including games released in multiple regions as separate titles. Initial reviews of the PlayStation 2 from 2000 highly acclaimed the console, with reviewers commending its hardware and graphics capabilities, its ability to play DVDs, and the system's backward compatibility with games and hardware for the original PlayStation. Early points of criticism included

2725-478: A tour mode, quick play, and a challenge series. For some time, there was an official online version of Blokus where players all over the world could play the four official variations of Blokus. Mattel discontinued the online game on May 18, 2012, stating it did not meet its playability standards. Gameloft developed and released an official Blokus video game for iPhone , iPod Touch , and iPad in April 2010, featuring

2834-418: A unique ( n + m )-omino. This proves A n A m ≤ A n + m . Using this equation, one can show λ ≥ ( A 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

2943-399: Is λ < 4.5252 . 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

3052-663: Is a home video game console developed and marketed by Sony Computer Entertainment . It was first released in Japan on 4 March 2000, in North America on 26 October 2000, in Europe on 24 November 2000, in Australia on 30 November 2000, and other regions thereafter. It is the successor to the PlayStation , as well as the second installment in the PlayStation brand of consoles. As a sixth-generation console , it competed with Nintendo 's GameCube , Sega 's Dreamcast , and Microsoft 's Xbox . It

3161-498: Is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling . Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in Fairy Chess Review between

3270-402: Is a finite set of prime 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. 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,

3379-471: 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

Blokus - Misplaced Pages Continue

3488-488: Is available for the system to connect to older TVs. The PlayStation 2 has undergone many revisions, some only of internal construction and others involving substantial external changes. The PS2 is primarily differentiated between models featuring the original "fat" case design and "slimline" models, which were introduced at the end of 2004. In 2010, the Sony Bravia KDL-22PX300 was made available to consumers. It

3597-431: 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. 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

3706-506: 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

3815-463: Is distributed on CD-ROM and DVD-ROM; the two formats are differentiated by their discs' bottoms, with CD-ROMs being blue and DVD-ROMs being silver. The PlayStation 2 offered some particularly high-profile exclusive games. Most main entries in the Grand Theft Auto , Final Fantasy , and Metal Gear Solid series were released exclusively for the console. Several prolific series got their start on

3924-476: Is exponentially more accurate as n increases. 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

4033-453: Is faster to generate symmetric polyominoes separately (by a variation of this method) 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 . An improvement on Andrew Conway's method, 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

4142-409: Is played on a hexagonal board. This particular variant is optimized for three players but can be played with 2 to 4 players. The same rules apply, meaning that pieces of the same color are still not allowed to touch each other's edges; however, as it is isometric, a corner touching an edge is allowed. Blokus Giant is a larger version, with the game board being about 570 mm (22 in) square. It

4251-417: 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

4360-722: Is the best-selling video game console of all time , having sold over 160 million units worldwide, nearly triple the combined sales of its competing consoles. Announced in 1999, Sony began developing the console after the immense success of its predecessor. The highlight of the system was its Emotion Engine processor, co-developed by Sony and Toshiba , which was said to be more powerful than personal computers. The PS2 offered backward-compatibility for its predecessor's DualShock controller, as well as its games. The PlayStation 2 received widespread critical acclaim upon release. A total of over 4,000 game titles were released worldwide, with over 1.527 billion copies sold. In 2004, Sony revised

4469-568: The Karaoke Revolution games, dual microphones (sold with and used exclusively for SingStar games), various "guitar" controllers (for the Guitar Freaks series and Guitar Hero series), the drum set controller (sold in a box set (or by itself) with a "guitar" controller and a USB microphone (for use with Rock Band and Guitar Hero series, World Tour and newer), and a taiko drum controller for Taiko: Drum Master . Unlike

Blokus - Misplaced Pages Continue

4578-455: 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. 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 n copies of

4687-558: The Nintendo DS reached 100 million shipments in 4 years and 5 months from its launch. By July 2009, the system had sold 138.8 million units worldwide, with 51 million of those units sold in PAL regions . By 31 March 2012, over 155 million PlayStation 2 units were sold worldwide, and in the same year, Sony officially stopped supplying updated sales numbers of the system. On 29 March 2024, Jim Ryan (CEO of Sony Interactive Entertainment) stated on

4796-431: The multitap accessory. There was, however, some concern about the system's abilities following the lack of anti-aliasing in the two most popular early titles, Ridge Racer V and Tekken Tag Tournament . It was exacerbated for a period of time post-launch amid concerns about the relative lack of new quality software. This situation was eventually turned around following a spate of big and highly acclaimed games in

4905-500: The multitap for PlayStation or PS2, a USB motion camera ( EyeToy ), a USB keyboard and mouse , and a headset . The original PS2 multitap (SCPH-10090) cannot be plugged into the newer slim models. The multitap connects to the memory card slot and the controller slot, and the memory card slot on the slimline is shallower. New slim-design multitaps (SCPH-70120) were manufactured for these models; however, third-party adapters also permit original multitaps to be used. Early versions of

5014-469: 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. In recreational mathematics , challenges are often posed for tiling a prescribed region, or the entire plane, with polyominoes, and related problems are investigated in mathematics and computer science . Puzzles commonly ask for tiling

5123-557: The 2001 holiday season with the release of several blockbuster games that maintained the PS2's sales momentum and held off its newer rivals. Sony also countered the Xbox by securing timed PlayStation 2 exclusives for highly anticipated games such as the Grand Theft Auto series and Metal Gear Solid 2: Sons of Liberty . In a 2002 article analysis of the system up to that point, Edge noted: Now, 18 months on, things are looking much rosier for PS2 owners. Developers are coming to terms with

5232-705: The Classic and Duo versions of the game, local and online multiplayer gameplay, and single player tournament mode. The Gameloft version was also released as a digital download game for the PlayStation 3 in December 2010. All editions of the Gameloft Blokus game were discontinued in January 2014. The most recent officially licensed Blokus app was developed by Magmic , and was available for iPhone, iPod Touch, and iPad. This version included solo play and multiplayer options, integrating

5341-436: The Classic, Travel, and Duo versions of the game. The game received an international release in 2008, under the name Blokus Portable: Steambot Championship . Funkitron developed a PC casual game version of Blokus called Blokus World Tour . Released in December 2007, Blokus World Tour was similar to the board game version of Blokus but also featured 16 AI opponents, music and sound effects, and multiple game modes, including

5450-441: 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. 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

5559-605: The L-tromino, L-tetromino, or P-pentomino into a single larger shape similar to the smaller polyomino from which it was formed. 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, and Livio Zucca shows results for some complicated cases like three different pentominoes. The general problem can be hard. The first compatibility figure for

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5668-692: The Official PlayStation Podcast that the PlayStation 2 had sold over 160 million units worldwide. This statement was not corroborated by Sony directly until nearly eight months later, with this figure appearing on the PlayStation 30th Anniversary website on 26 November 2024. Using homebrew programs, it is possible to play various audio and video file formats on a PS2. Homebrew programs can also play patched backups of original PS2 DVD games on unmodified consoles and install retail discs to an installed hard drive on older models. Homebrew emulators of older computer and gaming systems have been developed for

5777-437: The PS2 could be networked via an i.LINK port, though this had little game support and was dropped. Some third-party manufacturers have created devices that allow disabled people to access the PS2 through ordinary switches, etc. Some third-party companies, such as Joytech , have produced LCD monitor and speaker attachments for the PS2, which attach to the back of the console. These allow users to play games without access to

5886-454: The PS2 remained the only sixth-generation console for over 6 months before it faced competition from new rivals: Nintendo's GameCube and Microsoft's Xbox . Many analysts predicted a close three-way matchup among the three consoles. The Xbox had the most powerful hardware, while the GameCube was the least expensive console, and Nintendo changed its policy to encourage third-party developers. While

5995-551: The PlayStation 2 theoretically had the weakest hardware of the three, it had a head start due to its installed base plus strong developer commitment, as well as a built-in DVD player (the Xbox required an adapter, while the GameCube lacked support entirely). The PlayStation 2's initial games lineup was considered mediocre, with a lack of quality games especially amid the expectations of its hardware capabilities (partly caused due to its complex architecture for developers). This changed during

6104-551: The PlayStation 2, including God of War , Ratchet & Clank , Jak and Daxter , Devil May Cry , Kingdom Hearts , and Sly Cooper . Grand Theft Auto: San Andreas was the best-selling game on the console. Game releases peaked in 2004, but declined with the release of the PlayStation 3 in 2006. The last new games for the console were Final Fantasy XI : Seekers of Adoulin in Asia, FIFA 14 in North America, and Pro Evolution Soccer 2014 in Europe. As of 30 June 2007,

6213-424: The PlayStation, which requires the use of an official Sony PlayStation Mouse to play mouse-compatible games, the few PS2 games with mouse support work with a standard USB mouse as well as a USB trackball . In addition, some of these games also support the usage of a USB keyboard for text input, game control (instead of a DualShock or DualShock 2 gamepad, in tandem with a USB mouse), or both. PlayStation 2 software

6322-615: The back of the console, and is required to play certain games, notably the popular Final Fantasy XI . The system has 4 MB of Video RAM in the form of eDRAM . Software for the PlayStation 2 was distributed primarily on DVD-ROMs , with some titles being published on blue-tinted CD-ROM format. In addition, the console can play audio CDs and DVD films and is backward-compatible with almost all original PlayStation games. The PlayStation 2 also supports PlayStation memory cards and controllers, although original PlayStation memory cards will only work with original PlayStation games and

6431-405: The board and a set of pieces to place into the board following standard Blokus rules. Travel Blokus was released as the smaller version of Blokus Duo, then re-released as Blokus To Go! with pieces that could be snapped into the board for storage. Travel Blokus was later re-released as the smaller version of the original Blokus game, rather than Blokus Duo. Blokus 3D , originally marketed in

6540-513: The colour black as it represents the infinity of the universe. The blue represents the intelligence and life spouting up." —Teiyu Goto reflecting on the PlayStation 2's aesthetics Sony unveiled the PlayStation 2 at the Tokyo Game Show on 20 September 1999. Sony showed fully playable demos of upcoming PlayStation 2 games including Gran Turismo 2000 (later released as Gran Turismo 3: A-Spec ) and Tekken Tag Tournament —which showed

6649-601: The console continued to be released until shortly after the end of its life and support from Sony. Released in 1994, the original PlayStation proved to be a phenomenal worldwide success and signalled Sony's rise to power in the video game industry . Its launch elicited critical acclaim and strong sales; it eventually became the first computer entertainment platform to ship over 100 million units. The PlayStation enjoyed particular success outside Japan in part due to Sony's refined development kits , large-scale advertising campaigns, and strong third-party developer support. By

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6758-500: The console with a smaller, lighter body officially known as the Slimline , popularly called the "PS2 Slim". Even after the release of its successor, the PlayStation 3 , it remained popular well into the seventh generation . It continued to be produced until the beginning of 2013 when Sony finally announced that it had been discontinued after over twelve years of production, one of the longest lifespans of any video game console. New games for

6867-519: The console's graphic abilities and power. The PS2 was launched in March 2000 in Japan, October in North America, and November in Europe. Sales of the console, games and accessories pulled in $ 250 million on the first day, beating the $ 97 million made on the first day of the Dreamcast. Directly after its release, it was difficult to find PS2 units on retailer shelves due to manufacturing delays. Another option

6976-476: The console, named the "Graphics Synthesizer". It has a fillrate of 2.4 gigapixels per second, capable of rendering up to 75 million polygons per second. The GPU also runs with a clock frequency of 147.456 MHz (which is half the clock speed of the Emotion Engine), 4 MB of DRAM is capable of transmitting a display output of 1280 x 1024 pixels on both PAL and NTSC televisions. The PlayStation 2 has

7085-685: The controllers may not support all functions (such as analogue buttons) for PlayStation 2 games. The standard PlayStation 2 memory card has an 8  megabyte (MB) capacity and features MagicGate encryption. There are a variety of non-Sony manufactured memory cards available for the PlayStation ;2, allowing for a memory capacity larger than the standard 8 MB. The PlayStation 2 can natively output video resolutions on SDTV and HDTV from 480i to 480p , and some games, such as Gran Turismo 4 and Tourist Trophy , are known to support up-scaled 1080i resolution. The PlayStation 2 supports

7194-602: The end of 2013, including Final Fantasy XI : Seekers of Adoulin for Japan, and FIFA 14 for North America. The last game to be released on the PlayStation 2 is Pro Evolution Soccer 2014 , which was released in the United Kingdom on 8 November 2013. Repair services for the system in Japan ended on 7 September 2018. The PlayStation 2's main central processing unit (CPU) is the 64-bit R5900-based " Emotion Engine ", custom-designed by Sony and Toshiba . The Emotion Engine consists of eight separate "units", each performing

7303-433: The estimate for the number of fixed polyominoes of size n where λ = 4.0626 and c = 0.3169. 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 exists. In other words, A n grows exponentially . The best known lower bound for λ , found in 2016, is 4.00253. The best known upper bound

7412-466: 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 D 4 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

7521-421: The final quarter of 2001. Later reviews, especially after the launch of the competing GameCube and Xbox systems, continued to praise the PlayStation 2's large game library and DVD playback, while routinely criticizing the PlayStation 2's lesser graphics performance compared to the newer systems and its rudimentary online service compared to Xbox Live . In 2002, CNET rated the console 7.3 out of 10, calling it

7630-466: The first in the A000105 sequence. The standard rules of play for all variations of the game are as follows: Once the game ends, each player counts every square on the piece(s) that they did not place on the board, each counting as a negative (−1) point (e.g., an unplayed tetromino is worth −4 points). A player who played all of their pieces is awarded a 15-point bonus. If the last piece played

7739-439: The following standards: composite video (480i), S-Video (480i), RGB (480i/p), VGA (for progressive scan games and PS2 Linux only), YP B P R component video (which display most original PlayStation games in their native 240p mode which most HDTV sets do not support ), and D-Terminal . Cables are available for all of these signal types; these cables also output analogue stereo audio. Additionally, an RF modulator

7848-444: The gamer, as its debut pricing was less than many standalone DVD players on the market. This made the console a low-cost entry into the home theater market. The success of the PS2 at the end of 2000 caused Sega problems both financially and competitively, and Sega announced the discontinuation of the Dreamcast in March 2001, just 18 months after its successful Western launch. Despite the Dreamcast still receiving support through 2001,

7957-425: The group D 4 . Polyominoes have the following possible symmetries; the least number of squares needed in a polyomino with that symmetry is given in each case: In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows: The following table shows the numbers of polyominoes with n squares, sorted by symmetry groups. Each polyomino of size n +1 can be obtained by adding

8066-568: The internal hard disk drive. It also lacked an internal power supply until a later revision (excluding the Japanese version), similar to the GameCube , and had a modified Multitap expansion. The removal of the expansion bay was criticized as a limitation due to the existence of titles such as Final Fantasy XI , which required the HDD use. Sony also manufactured a consumer device called the PSX that can be used as

8175-562: The lack of online support at the time, its inclusion of only two controller ports, and the system's price at launch compared to the Dreamcast in 2000. PC Magazine in 2001 called the console "outstanding", praising its "noteworthy components" such as the Emotion Engine CPU, 32 MB of RAM, support for IEEE 1394 (branded as "i.LINK" by Sony and "FireWire" by Apple), and the console's two USB ports while criticizing its "expensive" games and its support for only two controllers without

8284-493: The late 1990s Sony had dethroned established rivals Sega and Nintendo in the global video game market. Sega, spurred on by its declining market share and significant financial losses, launched the Dreamcast in 1998 as a last-ditch attempt to stay in the industry. Though Sony has kept details of the PlayStation 2's development secret, Ken Kutaragi , the chief designer of the original PlayStation, reportedly began working on

8393-587: The latter referred to as Bloco. Open-source derivatives based on the same concept of polyominoes as Blokus also exist, such as: Blokish, Blockem, or Pentobi. Of note, Freebloks 3D is a free desktop version whose online multiplayer is cross-compatible with Freebloks for Android smartphones. Lastly, one free, closed-source implementation is Blokee, playable immediately on the Web with other players and/or AI, and requiring no software download nor account setup to play. Polyomino#Enumeration of polyominoes A polyomino

8502-426: The number of cases that need to be checked for duplicates. 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

8611-422: The numbers of polyominoes of various types with n cells. Fixed polyominoes were enumerated in 2004 up to n = 56 by Iwan Jensen, and in 2024 up to n = 70 by Gill Barequet and Gil Ben-Shachar. Free polyominoes were enumerated in 2007 up to n = 28 by Tomás Oliveira e Silva, in 2012 up to n = 45 by Toshihiro Shirakawa, and in 2023 up to n = 50 by John Mason. The above OEIS sequences, with

8720-497: The online-capable Xbox. Coinciding with the release of Xbox Live , Sony released the PlayStation 2 Network Adaptor in late 2002, with several online first-party titles released alongside it, such as SOCOM U.S. Navy SEALs to demonstrate its active support for Internet play. Sony also advertised heavily, and its online model had the support of Electronic Arts (EA); EA did not offer online Xbox titles until 2004. Although Sony and Nintendo both started late, and although both followed

8829-423: The original model. Demand for the PlayStation 2 remained strong throughout much of its lifespan, selling over 1.4 million units in Japan by 31 March 2000. Over 10.6 million units were sold worldwide by 31 March 2001. In 2005, the PlayStation 2 became the fastest game console to reach 100 million units shipped, accomplishing the feat within 5 years and 9 months from its launch; this was surpassed 4 years later when

8938-559: The parallel architecture and fears that the PS2 would suffer the same fate as the Sega Saturn now seem ridiculous. A mixture of hard work, trial and error and middleware tools is also ensuring that development cycles are getting shorter and game quality improving. It's taken much longer than anticipated, but exceptional titles are beginning to come through. Sony cut the price of the console in May 2002 from US$ 299 to $ 199 in North America, making it

9047-561: The plane uses a technique in computer science called backtracking . In Jigsaw Sudokus a square grid is tiled with polyomino-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. These problems have been extensively studied for particular polyominoes, and tables of results for individual polyominoes are available. Klarner and Göbel showed that for any polyomino there

9156-406: The plane. Rawsthorne found that all but 235 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. The study of which polyominoes can tile the plane has been facilitated using

9265-421: The polyomino. Directed polyominoes, column (or row) convex polyominoes, and convex polyominoes 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,

9374-531: The same functionality as its predecessor. However, it includes analogue pressure sensitivity to over 100 individual levels of depth on the face, shoulder and D-pad buttons, replacing the digital buttons of the original. Like its predecessor, the DualShock 2 controller has force feedback , or "vibration" functionality. It is lighter and includes two more levels of vibration . Specialized controllers include light guns ( GunCon ), fishing rod and reel controllers,

9483-457: The same level they did when they were 17. The slogan used was: Live in your world. Play in ours. (Stylized as: "LIVE IN Y [REDACTED] UR W [REDACTED] RLD. PL [REDACTED] Y IN [REDACTED] URS.") Backward compatibility had been widely seen as a desirable feature for consumers since the debut of the first successor video game console, but prior to the PlayStation 2 only one console had featured true backward compatibility (i.e. without

9592-412: The same price as the GameCube and $ 100 less than the Xbox. It also planned to cut the price in Japan around that time. It cut the price twice in Japan in 2003. In 2006, Sony cut the cost of the console in anticipation of the release of the PlayStation 3 . Unlike Sega's Dreamcast, Sony originally placed little emphasis on online gaming during its first few years, although that changed upon the launch of

9701-580: The structure as possible. At the end of the game, the player with the most squares from the top view wins. There have been video games based on Blokus. Irem was the first to develop such a kind of video game, in the form of what is known as Blokus Club with Bumpy Trot . Released in Japan in 2005 for the PlayStation 2 , and one year later for the PlayStation Portable , the game features characters from Steambot Chronicles ( Bumpy Trot in Japan) playing

9810-420: The study of polyominoes and their higher-dimensional analogs (which are often referred to as lattice animals in this literature) is applied to problems in physics and chemistry. Polyominoes have been used as models of branched polymers and of percolation clusters. Like many puzzles in recreational mathematics , polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of

9919-402: The symmetries of D 4 . 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 D 4 may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under

10028-405: The system memory. The Emotion Engine CPU has a clock rate of 294.912 MHz (299 MHz on newer versions) and 6,000 MIPS , with a floating point performance of 6.2 GFLOPS . The system's I/O processor was based on the PlayStation's CPU and was designed to provide full backward compatibility with the PlayStation. The system has 32 MB of RDRAM . The GPU is likewise custom-designed for

10137-421: 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

10246-690: The units. After an apparent manufacturing issue—Sony reportedly underestimated demand—caused some initial slowdown in producing the new unit caused in part by shortages between the time Sony cleared out the old units and the new units were ready. The issue was compounded in Britain when a Russian oil tanker became stuck in the Suez Canal , blocking a ship from China carrying PS2s bound for the UK. During one week in November, British sales totalled 6,000 units—compared to 70,000 units

10355-480: The use of add-ons), the Atari 7800 , due to the added hardware costs and industry concerns that backward compatibility could cause the console to appear to be a merely a new model of its predecessor or lead developers to prefer making games for the predecessor system. Later, Sony added new development kits for game developers and more PS2 units for consumers. The PS2's built-in functionality also expanded its audience beyond

10464-460: The users' Facebook and Game Center friends and allowing them to compete on a tournament leaderboard. The Magmic Blokus game was delisted from the App Store in December 2018. There are also video games based on Blokus, but not officially licensed. For example, a mobile game app named Quadrus, available since January 2014. Websites such as Pentolla.com and LeFun.fun also include a game based on Blokus,

10573-426: 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. Because the general problem of tiling regions of the plane with sets of polyominoes is NP-complete , 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

10682-450: The word domino , a common game piece consisting of two squares. The name domino for the game piece is believed to come from the spotted masquerade garment domino , from Latin dominus . Despite this word origin, in naming polyominoes, the first letter d- of domino is fancifully interpreted as a version of the prefix di- meaning "two", and replaced by other numerical prefixes . PlayStation 2 The PlayStation 2 ( PS2 )

10791-683: The years 1937 and 1957, under the name of "dissection problems." The name polyomino was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 " Mathematical Games " column in Scientific American . Related to polyominoes are polyiamonds , formed from equilateral triangles ; polyhexes , formed from regular hexagons ; and other plane polyforms . Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes , or hypercubes to form polyhypercubes. In statistical physics ,

10900-532: Was Ridge Racer V followed by Geki Kuukan Pro Baseball: At the End of the Century 1999 , Tekken Tag Tournament , Kessen , Dead or Alive 2 and Jikkyo Powerful Pro Baseball 7 . Marketing for the PlayStation 2 reverted to the same tactic used in the early days of the original PlayStation: use 17-year-olds as the target audience , since younger audiences aspire to be teenagers and older audiences enjoy video games at

11009-510: Was a 22" HD-Ready television which incorporated a built-in PlayStation 2. The standard color of the PS2 is matte black. Several variations in color were produced in different quantities and regions, including ceramic white, light yellow, metallic blue (aqua), metallic silver, navy (star blue), opaque blue (astral blue), opaque black (midnight black), pearl white, sakura purple, satin gold, satin silver, snow white, super red, transparent blue (ocean blue), and also Limited Edition color Pink, which

11118-401: Was a monomino, provided that all pieces of own color have been played, the player is awarded a 20-point bonus instead. The player with the highest score wins. Blokus rules also allow for two and three player games. In two-player games, each player takes two colors. In three-player games, either one of the players takes two colors or else "the pieces of the fourth color are placed on the board in

11227-509: Was being developed. Chris Deering , then-president of SCEE recalled that there was a degree of trepidation among Sony leaders to produce a console which would recapture or exceed the success of its predecessor. Sony announced the PlayStation 2 on 2 March 1999. Sega's Dreamcast enjoyed a successful US launch on 9 September of that year; fuelled by a large marketing campaign, it sold over 500,000 units within two weeks. "PlayStation 2's real-time graphics have no limitations. That's why I chose

11336-401: Was designing their own chip in-house and had instructed other companies to design rendering chips merely to diversify their options. By early 1997, the press was reporting that a new PlayStation was being developed and would have backward-compatibility with the original PlayStation, a built-in DVD player , and Internet connectivity. However, Sony continued to officially deny that a successor

11445-480: Was distributed in some regions such as Oceania, and parts of Asia. In September 2004, Sony unveiled its third major hardware revision. Available in late October 2004, it was smaller, thinner, and quieter than the original versions and included a built-in Ethernet port (in some markets it also had an integrated modem ). Due to its thinner profile, it did not contain the 3.5" expansion bay and therefore did not support

11554-406: Was later released by Mattel as Blokus XL . Blokus Junior is targeted at younger children. Like Blokus Duo, it is played by two players on a 14×14 board, but it uses only a subset of the pieces that have simpler shapes. There are 12 unique pieces. Each player gets two of each kind, 24 in total. The game also comes with a set of single-player puzzle sheets, each containing a preset piece position on

11663-449: Was purchasing the console online through auction websites such as eBay , where people paid over a thousand dollars for the console. The PS2 initially sold well partly on the basis of the strength of the PlayStation brand and the console's backward compatibility, selling its entire inventory of 1.4 million units in Japan by 31 March 2000, less than a month after launch. From its launch until the end of 2000 in Japan, its best-selling software

11772-514: Was required for the original models, while the slim models included built-in networking ports. Instead of having a unified, subscription-based online service like Xbox Live as competitor Microsoft later chose for its Xbox console, online multiplayer functionality on the PlayStation 2 was the responsibility of the game publisher and ran on third-party servers. Many games that supported online play exclusively supported broadband Internet access. The PlayStation 2's DualShock 2 controller retains most of

11881-456: Was subsequently improved by Barequet and Shalah to 4.5252. 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

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