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Fibonacci Nim
Fibonacci nim is a mathematical subtraction game, a variant of the game of nim. Players alternate removing coins from a pile, on each move taking at most twice as many coins as the previous move, and winning by taking the last coin. The Fibonacci numbers feature heavily in its analysis; in particular, the first player can win if and only if the starting number of coins is not a Fibonacci number. A complete strategy is known for best play in games with a single pile of counters, but not for variants of the game with multiple piles. Rules and history Fibonacci nim is played by two players, who alternate removing coins or other counters from a pile. On the first move, a player is not allowed to take all of the coins, and on each subsequent move, the number of coins removed can be any number that is at most twice the previous move. According to the normal play convention, the player who takes the last coin wins. The game was first described by Michael J. Whinihan in 1963, crediting i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singapore Coins In A Stack
Singapore (), officially the Republic of Singapore, is a sovereign island country and city-state in maritime Southeast Asia. It lies about one degree of latitude () north of the equator, off the southern tip of the Malay Peninsula, bordering the Strait of Malacca to the west, the Singapore Strait to the south, the South China Sea to the east, and the Straits of Johor to the north. The country's territory is composed of one main island, 63 satellite islands and islets, and one outlying islet; the combined area of these has increased by 25% since the country's independence as a result of extensive land reclamation projects. It has the third highest population density in the world. With a multicultural population and recognising the need to respect cultural identities of the major ethnic groups within the nation, Singapore has four official languages: English, Malay, Mandarin, and Tamil. English is the lingua franca and numerous public services are available only in Englis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtraction Game
In combinatorial game theory, a subtraction game is an abstract strategy game whose state can be represented by a natural number or vector of numbers (for instance, the numbers of game tokens in piles of tokens, or the positions of pieces on board) and in which the allowed moves reduce these numbers., "Subtraction games", pp. 83–86. Often, the moves of the game allow any number to be reduced by subtracting a value from a specified ''subtraction set'', and different subtraction games vary in their subtraction sets. These games also vary in whether the last player to move wins (the normal play convention) or loses ( misère play convention). Another winning convention that has also been used is that a player who moves to a position with all numbers zero wins, but that any other position with no moves possible is a draw. Examples Examples of notable subtraction games include the following: * Nim is a game whose state consists of multiple piles of tokens, such as coins or matchsticks, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book '' Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the '' Fibonacci Quarterly''. Applications of Fibonacci numbers includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Play Convention
A normal play convention in a game is the method of determining the winner that is generally regarded as standard. For example: *Preventing the other player from being able to move *Being the first player to achieve a target *Holding the highest value hand *Taking the most card tricks In combinatorial game theory, the normal play convention of an impartial game In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference betw ... is that the last player able to move is the winner. By contrast " misère games" involve upsetting the convention and declaring a winner the individual who would normally be considered the loser. Gaming {{Game-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oregon State University
Oregon State University (OSU) is a public land-grant, research university in Corvallis, Oregon. OSU offers more than 200 undergraduate-degree programs along with a variety of graduate and doctoral degrees. It has the 10th largest engineering college in the nation for 2022. Undergraduate enrollment for all colleges combined averages close to 32,000, making it the state's largest university. Out-of-state students make up over one-quarter of undergraduates and an additional 5,500 students are engaged in graduate coursework through the university. Since its founding, over 272,000 students have graduated from OSU. It is classified among "Doctoral Universities – Very high research activity". Chartered as a land-grant university initially, OSU became one of the four inaugural members of the Sea Grant in 1971. It joined the Space Grant and Sun Grant research consortia in 1991 and 2003, respectively, making it the first public university and one of just four in total to attain me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeckendorf Representations 89px
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Zeckendorf may refer to: * Edouard Zeckendorf, Belgian mathematician known for Zeckendorf's theorem * William Zeckendorf, Sr (1905-1976), American real estate developer * William Zeckendorf, Jr. (1929-2014), real estate developer * Zeckendorf Towers, a condominium in New York City * Zeckendorf, Bavaria. a town near Bamberg, Bavaria. * Louis Zeckendorf, American pioneer * Zeckendorf v. Steinfeld, a case decided by the Supreme Court of the United States The Supreme Court of the United States (SCOTUS) is the highest court in the federal judiciary of the United States. It has ultimate appellate jurisdiction over all U.S. Federal tribunals in the United States, federal court cases, and over Stat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeckendorf's Theorem
In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of ''one or more'' distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. More precisely, if is any positive integer, there exist positive integers , with , such that :N = \sum_^k F_, where is the th Fibonacci number. Such a sum is called the Zeckendorf representation of . The Fibonacci coding of can be derived from its Zeckendorf representation. For example, the Zeckendorf representation of 64 is :. There are other ways of representing 64 as the sum of Fibonacci numbers : : : : but these are not Zeckendorf representations because 34 and 21 are consecutive Fibonacci numbers, as are 5 and 3. For any given positive integer, its Zeckendor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impartial Game
In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. Furthermore, impartial games are played with perfect information and no chance moves, meaning all information about the game and operations for both players are visible to both players. Impartial games include Nim, Sprouts, Kayles, Quarto, Cram, Chomp, Subtract a square, Notakto, and poset games. Go and chess are not impartial, as each player can only place or move pieces of their own color. Games such as poker, dice Dice (singular die or dice) are small, throwable objects with marked si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sprague–Grundy Theorem
In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim. The Grundy value or nim-value of any impartial game is the unique nimber that the game is equivalent to. In the case of a game whose positions are indexed by the natural numbers (like nim itself, which is indexed by its heap sizes), the sequence of nimbers for successive positions of the game is called the nim-sequence of the game. The Sprague–Grundy theorem and its proof encapsulate the main results of a theory discovered independently by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nim-sum
In mathematics, the nimbers, also called ''Grundy numbers'', are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with ''nimber addition'' and ''nimber multiplication'', which are distinct from ordinal addition and ordinal multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering. Nimbers have the characteristic that their Left and Right options are identical, following a certain schema, and that they are their own negatives, such that a positive ordinal may be added to another positive ordinal using nimber addition to find an ordinal of a lower value. The minimum excludant operation is applied to sets of nimbers. Uses Nim Nim is a game in which two players take turns removing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
