Domineering (also called Stop-Gate or Crosscram ) is a mathematical game that can be played on any collection of squares on a sheet of graph paper . For example, it can be played on a 6×6 square, a rectangle, an entirely irregular polyomino , or a combination of any number of such components. Two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player places tiles vertically, while the other places them horizontally. (Traditionally, these players are called "Left" and "Right", respectively, or "V" and "H". Both conventions are used in this article.) As in most games in combinatorial game theory , the first player who cannot move loses.
25-411: Domineering is a partisan game , in that players use different pieces: the impartial version of the game is Cram . Other than the empty game, where there is no grid, the simplest game is a single box. [REDACTED] In this game, clearly, neither player can move. Since it is a second-player win, it is therefore a zero game . [REDACTED] [REDACTED] This game is a 2-by-1 grid. There
50-529: A Domineering tournament with a $ 500 prize for the winner. This game was played on an 8×8 board. The winner was mathematician Dan Calistrate, who defeated David Wolfe in the final. The tournament was detailed in Richard J. Nowakowski's Games of No Chance (p. 85). A problem about Domineering is to compute the winning strategy for large boards, and particularly square boards. In 2000, Dennis Breuker, Jos Uiterwijk and Jaap van den Herik computed and published
75-568: A number of extra tempos that can be used to put the other side into zugzwang . Partisan games are more difficult to analyze than impartial games , as the Sprague–Grundy theorem does not apply. However, the application of combinatorial game theory to partisan games allows the significance of numbers as games to be seen, in a way that is not possible with impartial games. Mathematical Sciences Research Institute The Simons Laufer Mathematical Sciences Institute ( SLMath ), formerly
100-515: A variety of different areas of mathematical research. There are ten regular members in the SAC, and each member serves a four-year term and is elected by the board of trustees. SLMath hosts some 85 mathematicians and postdoctoral research fellows each semester and holds programs and workshops that draw approximately 2,000 visits by mathematical scientists throughout the year. The visitors come to SLMath to work in an environment that promotes creativity and
125-562: A wider community through the development of human scientific capital, providing postdoctoral training to young scientists and increasing the diversity of the research workforce. The institute also advances the education of young people with conferences on critical issues in mathematics education. Additionally, they host research workshops that are unconnected to the main programs, such as its annual workshop on K-12 mathematics education Critical Issues in Mathematics Education. During
150-445: Is a convention of assigning the game a positive number when Left is winning and a negative one when Right is winning. In this case, Left has no moves, while Right can play a domino to cover the entire board, leaving nothing, which is clearly a zero game. Thus in surreal number notation, this game is {|0} = −1. This makes sense, as this grid is a 1-move advantage for Right. [REDACTED] [REDACTED] [REDACTED] This game
175-417: Is also {|0} = −1, because a single box is unplayable. [REDACTED] [REDACTED] [REDACTED] [REDACTED] This grid is the first case of a choice. Right could play the left two boxes, leaving −1. The rightmost boxes leave −1 as well. He could also play the middle two boxes, leaving two single boxes. This option leaves 0+0 = 0. Thus this game can be expressed as {|0,−1}. This is −2. If this game
200-405: Is not impartial . That is, some moves are available to one player and not to the other, or the payoffs are not symmetric. Most games are partisan. For example, in chess , only one player can move the white pieces. More strongly, when analyzed using combinatorial game theory, many chess positions have values that cannot be expressed as the value of an impartial game, for instance when one side has
225-745: Is not a hot game (also called a cold game ), because each move hurts the player making it, as we can see by examining the moves. Left can move to −1, Right can move to 0 or +1. Thus this game is {−1|0,1} = {−1|0} = − 1 ⁄ 2 . Our 2×3 grid, then, is {2|− 1 ⁄ 2 }, which can also be represented by the mean value, 3 ⁄ 4 , together with the bonus for moving (the "temperature"), 1 + 1 ⁄ 4 , thus: { 2 | − 1 2 } = 3 4 ± 5 4 {\displaystyle \textstyle \left\{2\left|-{\frac {1}{2}}\right.\right\}={\frac {3}{4}}\pm {\frac {5}{4}}} The Mathematical Sciences Research Institute held
250-400: Is played in conjunction with other games, this is two free moves for Right. Vertical columns are evaluated in the same way. If there is a row of 2 n or 2 n +1 boxes, it counts as − n . A column of such size counts as + n . [REDACTED] [REDACTED] [REDACTED] [REDACTED] This is a more complex game. If Left goes first, either move leaves a 1×2 grid, which is +1. Right, on
275-500: The Berkeley Repertory Theater , and co-sponsored a series of mathematics-inspired films with UC Berkeley's Pacific Film Archive for the institute's 20th anniversary. It also created a series of mathematical puzzles that were posted among the advertising placards on San Francisco Muni buses. The Mathical Award is presented to books "that inspire children of all ages to see math in the world around them." Recipients of
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#1732916230295300-774: The China Girls Math Olympiad . The lectures given at SLMath events are recorded and made available for free on the internet. SLMath has sponsored a number of events that reach out to the non-mathematical public, and its Simons Auditorium also hosts special performances of classical music. Mathematician Robert Osserman has held a series of public "conversations" with artists who have been influenced by mathematics in their work, such as composer Philip Glass , actor and writer Steve Martin , playwright Tom Stoppard , and actor and author Alan Alda . SLMath also collaborates with local playwrights for an annual program of new short mathematics-inspired plays at Monday Night Playground at
325-613: The Mathematical Sciences Research Institute ( MSRI ), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, California . It is widely regarded as a world leading mathematical center for collaborative research, drawing thousands of leading researchers from around the world each year. The institute was founded in 1982, and its funding sources include
350-687: The National Science Foundation , private foundations, corporations, and more than 90 universities and institutions. The institute is located at 17 Gauss Way on the Berkeley campus, close to Grizzly Peak in the Berkeley Hills . Given its contribution to the nation's scientific potential, the institute is supported by the National Science Foundation and the National Security Agency . Private individuals, foundations, and nearly 100 Academic Sponsor Institutions, including
375-552: The Berkeley campus. In May 2022, the institute announced that it received an unrestricted $ 70 million gift from James and Marilyn Simons and Henry and Marsha Laufer . In honor of the endowment, MSRI was renamed the Simons Laufer Mathematical Sciences Institute. SLMath is governed by a board of trustees consisting of up to 35 elected members and seven ex-officio members: the director of the institute,
400-656: The deputy director, the Chair of the Committee of Academic Sponsors, the co-Chairs of the Human Resources Advisory Committee and the co-Chairs of the Scientific Advisory Committee (SAC). Unlike many mathematical institutes, SLMath has no permanent faculty or members, and its research activities are overseen by its Scientific Advisory Committee (SAC), a panel of distinguished mathematicians drawn from
425-512: The effective interchange of ideas and techniques. SLMath features two focused programs each semester, attended by foremost mathematicians and postdocs from the United States and abroad; the institute has become a world center of activity in those fields. SLMath takes advantage of its proximity to the Berkeley faculty and to the scientific talent and resources of Lawrence Berkeley National Laboratory ; it also collaborates with organizations across
450-493: The founding director of the institute and Calvin Moore acted as the founding deputy director. Originally located in Berkeley's extension building at 2223 Fulton Street, the institute moved into its current facility in the Berkeley hills on April 1, 1985. The institute initially paid rent to the university for its "Hill Campus" building, but since August 2000, it has occupied the building free of rent, just one of several contributions by
475-528: The nation, including the Chicago Mercantile Exchange , Citadel LLC , IBM , and Microsoft Research . The institute's prize-winning forty-eight thousand square foot building has views of the San Francisco Bay . After 30 years of activity, the reputation of the institute is such that mathematicians make it a professional priority to participate in the institute's programs. SLMath also serves
500-490: The other hand, can move to −1. Thus the surreal number notation is {1|−1}. However, this is not a surreal number because 1 > −1. This is a Game but not a number. The notation for this is ±1, and it is a hot game , because each player wants to move here. [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] This is a 2×3 grid, which is even more complex, but, just like any Domineering game, it can be broken down by looking at what
525-437: The site of Nathan Bullock. Cram is the impartial version of Domineering. The only difference in the rules is that each player may place their dominoes in either orientation. It seems only a small variation in the rules, but it results in a completely different game that can be analyzed with the Sprague–Grundy theorem . Partisan game In combinatorial game theory , a game is partisan (sometimes partizan ) if it
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#1732916230295550-477: The solution for the 8x8 board. The 9x9 board followed soon after some improvements of their program. Then, in 2002, Nathan Bullock solved the 10x10 board, as part of his thesis on Domineering. The 11x11 board has been solved by Jos Uiterwijk in 2016. Domineering is a first-player win for the 2x2, 3x3, 4x4, 6x6, 7x7, 8x8, 9x9, 10x10, and 11x11 square boards, and a second-player win for the 1x1 and 5x5 boards. Some other known values for rectangular boards can be found on
575-673: The summer, the institute holds workshops for graduate students. It also sponsors programs for middle and high school students and their teachers as part of the Math Circles and Circles for Teachers that meet weekly in San Francisco, Berkeley, and Oakland. It also sponsors the Bay Area Mathematical Olympiad (BAMO), the Julia Robinson Mathematics Festival , and the U.S. team of young girls that competes at
600-478: The top mathematics departments in the United States, also provide crucial support and flexibility. Jim Simons , founder of Renaissance Technologies and a Berkeley alumnus, was a long-time supporter of the institute and served on the board of trustees. The institute was founded in September 1982 by three Berkeley professors: Shiing-Shen Chern , Calvin Moore , and Isadore M. Singer . Shiing-Shen Chern acted as
625-426: The various moves for Left and Right are. Left can take the left column (or, equivalently, the right column) and move to ±1, but it is clearly a better idea to split the middle, leaving two separate games, each worth +1. Thus Left's best move is to +2. Right has four "different" moves, but they all leave the following shape in some rotation : [REDACTED] [REDACTED] [REDACTED] [REDACTED] This game
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