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Intransitive Dice
A set of dice is intransitive (or nontransitive) if it contains X>2 dice, ''X1'', ''X2'', and ''X3''... with the property that ''X1'' rolls higher than ''X2'' more than half the time, and ''X2'' rolls higher than ''X3'' etc... more than half the time, but where it is not true that ''X1'' rolls higher than ''Xn'' more than half the time. In other words, a set of dice is intransitive if the binary relation – rolls a higher number than more than half the time – on its elements is not transitive. More simply, ''X1'' normally beats ''X2'', ''X2'' normally beats ''X3'', but ''X1'' does not normally beat ''Xn''. It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. This is different in that instead of only "''A'' does not normally beat ''C''" it is now "''C'' normally beats ''A''". Using such a set of dice, one can invent games which are biased in ways th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance. A traditional die is a cube with each of its six faces marked with a different number of dots ( pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Dice may also have other polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their numbers carved out from the material of the dice instead of marked on it. Loaded dice are specifically designed or modified to favor some results over others, for cheating or entertainment purposes. History Dice have been used since before recorded history, and their origin is uncertain. It is hypoth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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GitHub
GitHub () is a Proprietary software, proprietary developer platform that allows developers to create, store, manage, and share their code. It uses Git to provide distributed version control and GitHub itself provides access control, bug tracking system, bug tracking, software feature requests, task management, continuous integration, and wikis for every project. Headquartered in California, GitHub, Inc. has been a subsidiary of Microsoft since 2018. It is commonly used to host open source software development projects. GitHub reported having over 100 million developers and more than 420 million Repository (version control), repositories, including at least 28 million public repositories. It is the world's largest source code host Over five billion developer contributions were made to more than 500 million open source projects in 2024. About Founding The development of the GitHub platform began on October 19, 2005. The site was launched in April 2008 by Tom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quanta Magazine
''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. History ''Quanta Magazine'' was initially launched as ''Simons Science News'' in October 2012, but it was renamed to its current title in July 2013. It was founded by the former ''New York Times'' journalist Thomas Lin, who was the magazine's editor-in-chief until 2024. The two deputy editors are John Rennie and Michael Moyer, formerly of ''Scientific American'', and the art director is Samuel Velasco. In 2024, Samir Patel became the magazine's second editor in chief. Content The articles in the magazine are freely available to read online. ''Scientific American'', ''Wired'', ''The Atlantic'', and ''The Washington Post'', as well as international science publications like '' Spektrum der Wissenschaft'', have reprinted articles from the magazine. In November 2018, MIT Press The MIT Press is the uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Timothy Gowers
Sir William Timothy Gowers, (; born 20 November 1963) is a British mathematician. He is the holder of the Combinatorics chair at the Collège de France, a director of research at the University of Cambridge and a Fellow of Trinity College, Cambridge. In 1998, he received the Fields Medal for research connecting the fields of functional analysis and combinatorics. Education Gowers attended King's College School, Cambridge, as a choirboy in the choir of King's College, Cambridge, King's College choir, and then Eton College as a King's Scholar, where he was taught mathematics by Norman Routledge. In 1981, Gowers won a gold medal at the International Mathematical Olympiad with a perfect score. He completed his PhD, with a dissertation on ''Symmetric Structures in Banach Spaces'' at Trinity College, Cambridge in 1990, supervised by Béla Bollobás. Career and research After his PhD, Gowers was elected to a Junior Research Fellowship at Trinity College. From 1991 until his return to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Voting Paradox
In social choice theory, Condorcet's voting paradox is a fundamental discovery by the Marquis de Condorcet that majority rule is inherently contradiction, self-contradictory. The result implies that it is logically impossible for any voting system to guarantee that a winner will have support from a majority of voters; for example, there can be rock-paper-scissors scenarios where a majority of voters will prefer A to B, B to C, and also C to A, even if every voter's individual preferences are rational and avoid self-contradiction. Examples of Condorcet's paradox are called Condorcet cycles or cyclic ties. In such a cycle, every possible choice is rejected by the electorate in favor of another alternative, who is preferred by more than half of all voters. Thus, any attempt to ground social decision-making in majoritarianism must accept such self-contradictions (commonly called spoiler effects). Systems that attempt to do so, while minimizing the rate of such self-contradictions, ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rock Paper Scissors
Rock, Paper, Scissors (also known by #Names, several other names and word orders) is an Intransitive game, intransitive hand game, usually played between two people, in which each player simultaneously forms one of three shapes with an outstretched hand. These shapes are "rock" (a closed fist: ✊), "paper" (a flat hand: ✋), and "scissors" (a fist with the index finger and middle finger extended, forming a V: ✌). The earliest form of "rock paper scissors"-style game originated in China and was subsequently imported into Japan, where it reached its modern standardized form, before being spread throughout the world in the early 20th century. A simultaneous game, simultaneous, zero-sum game, it has three possible outcomes: a draw, a win, or a loss. A player who decides to play rock will beat another player who chooses scissors ("rock crushes scissors" or "breaks scissors" or sometimes "blunts scissors"), but will lose to one who has played paper ("paper covers rock"); a play of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Nontransitive Game
An intransitive or non-transitive game is a zero-sum game in which pairwise competitions between the strategies contain a cycle. If strategy A beats strategy B, B beats C, and C beats A, then the binary relation "to beat" is intransitive, since transitivity would require that A beat C. The terms "transitive game" or "intransitive game" are not used in game theory. A prototypical example of an intransitive game is the game rock, paper, scissors. In probabilistic games like Penney's game, the violation of transitivity results in a more subtle way, and is often presented as a probability paradox. Examples * Rock, paper, scissors * Penney's game * Intransitive dice * ''Fire Emblem'', the video game franchise that popularized intransitive cycles in unit weapons: swords and magic beats axes and bows, axes and bows beat lances and knives, and lances and knives beat swords and magic See also * Stochastic transitivity Stochastic transitivity models are stochastic versions of the transiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Go First Dice
Go First Dice are a set of dice in which, when rolled together, each die has an equal chance of showing the highest number, the second highest number, and so on. The dice are intended for fairly deciding the order of play in, for example, a board game. The number on each side is unique among the set, so that no ties can be formed. Properties There are three properties of fairness, with increasing strength: * Go-first-fair - Each player has an equal chance of rolling the highest number (going first). * Place-fair - When all the rolls are ranked in order, each player has an equal chance of receiving each rank. * Permutation-fair - Every possible ordering of players has an equal probability, which also ensures it is "place-fair". It is also desired that any subset of dice taken from the set and rolled together should also have the same properties, so they can be used for fewer players as well. Configurations where all die have the same number of sides are presented here, but altern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Freivalds' Algorithm
Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three ''n'' × ''n'' matrices A, B, and C, a general problem is to verify whether A \times B = C. A naïve algorithm would compute the product A \times B explicitly and compare term by term whether this product equals C. However, the best known matrix multiplication algorithm runs in O(n^) time. Freivalds' algorithm utilizes randomization in order to reduce this time bound to O(n^2) with high probability. In O(kn^2) time the algorithm can verify a matrix product with probability of failure less than 2^. The algorithm Input Three ''n'' × ''n'' matrices A, B, and C. Output Yes, if A \times B = C; No, otherwise. Procedure # Generate an ''n'' × 1 random 0/1 vector \vec. # Compute \vec = A \times (B \vec) - C\vec. # Output "Yes" if \vec = (0,0,\ldots,0)^T; "No," otherwise. Error If A \tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Blotto Games
A Colonel Blotto game is a type of two-person constant-sum game in which the players (officers) are tasked to simultaneously distribute limited resources over several objects (battlefields). In the classic version of the game, the player devoting the most resources to a battlefield wins that battlefield, and the gain (or payoff) is equal to the total number of battlefields won. The game was first proposed by Émile Borel in 1921. In 1938 Borel and Ville published a particular optimal strategy (the "disk" solution). The game was studied after the Second World War by scholars in Operation Research, and became a classic in game theory. Gross and Wagner's 1950 research memorandum states Borel's optimal strategy, and coined the fictitious Colonel Blotto and Enemy names. For three battlefields or more, the space of pure strategies is multi-dimensional (two dimensions for three battlefields) and a mixed strategy is thus a probability distribution over a continuous set. The game is a rare ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance. A traditional die is a cube with each of its six faces marked with a different number of dots ( pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Dice may also have other polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their numbers carved out from the material of the dice instead of marked on it. Loaded dice are specifically designed or modified to favor some results over others, for cheating or entertainment purposes. History Dice have been used since before recorded history, and their origin is uncertain. It is hypoth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |