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Board Puzzles With Algebra Of Binary Variables
Board puzzles with algebra of binary variables ask players to locate the hidden objects based on a set of clue cells and their neighbors marked as variables (unknowns). A variable with value of 1 corresponds to a cell with an object. Contrary, a variable with value of 0 corresponds to an empty cell—no hidden object. Overview These puzzles are based on algebra with binary variables taking a pair of values, for example, (no, yes), (false, true), (not exists, exists), (0, 1). It invites the player quickly establish some equations, and inequalities for the solution. The Partition of a set, partitioning can be used to reduce the complexity of the problem. Moreover, if the puzzle is prepared in a way that there exists Unique (mathematics), a unique solution only, this fact can be used to eliminate some variables without calculation. The problem can be modeled as Linear program#Integer unknowns, binary integer linear programming which is a special case of integer linear programmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and Notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a disjoint union of the subsets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also collectively exhaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique (mathematics)
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols " ∃!" or "∃=1". For example, the formal statement : \exists! n \in \mathbb\,(n - 2 = 4) may be read as "there is exactly one natural number n such that n - 2 =4". Proving uniqueness The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, ''a'' and ''b'') must be equal to each other (i.e. a = b). For example, to show that the equation x + 2 = 5 has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds: : 3 + 2 = 5. To ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Program
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where this function has the smallest (or largest) value if such a point exists. Linear programs are problems that can be expressed in canonical form as : \begin & \text && \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minesweeper (video Game)
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as ''RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to ''TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says '' Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not remem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tentaizu 4x4 Example
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as '' RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to '' TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says ''Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tentaizu 4x4 Example With Variables
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as '' RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to '' TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says ''Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tentaizu 4x4 Example With Variables Solved Partially
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as '' RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to '' TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says ''Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tentaizu 4x4 Example Solved
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as '' RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to '' TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says ''Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tentaizu 4x4 Example Partitioned
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as '' RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to '' TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says ''Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tentaizu 4x4 Example For Inconsistency
Minesweeper is a logic puzzle video game genre generally played on personal computers. The game features a grid of clickable squares, with hidden "mines" scattered throughout the board. The objective is to clear the board without detonating any mines, with help from clues about the number of neighboring mines in each field. Variants of Minesweeper have been made that expand on the basic concepts, such as ''Minesweeper X'', ''Crossmines'', and ''Minehunt''. Minesweeper has been incorporated as a minigame in other games, such as '' RuneScape'' and ''Minecraft'' 2015 April Fools update. The origin of Minesweeper is unclear. According to '' TechRadar'', the first version of the game was 1990's ''Microsoft Minesweeper'', but ''Eurogamer'' says ''Mined-Out'' by Ian Andrew (1983) was the first Minesweeper game. Curt Johnson, the creator of ''Microsoft Minesweeper'', acknowledges that his game's design was borrowed from another game, but it was not ''Mined-Out'', and he does not re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Search Algorithm
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array. Binary search runs in logarithmic time in the worst case, making O(\log n) comparisons, where n is the number of elements in the array. Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians. Early life and education Born in Hungary into a Jewish family, Halmos arrived in the U.S. at 13 years of age. He obtained his B.A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, and was only 19 when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus; but, after failing his masters' oral exams, he shifted to mathematics, graduating in 1938. Joseph L. Doob supervised his dissertation, titled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
