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In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.
Derived from the Greek word for '5', and "domino", a pentomino is a polyomino of order 5; that is, a polygon in the plane made of 5 equal-sized squares connected edge to edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes.
A crossword is a word game consisting of a grid of black and white squares, into which solvers enter words or phrases ("entries") crossing each other horizontally ("across") and vertically ("down") according to a set of clues. Each white square is typically filled with one letter, while the black squares are used to separate entries. The first white square in each entry is typically numbered to correspond to its clue.
A cryptic crossword is a crossword puzzle in which each clue is a word puzzle. Cryptic crosswords are particularly popular in the United Kingdom, where they originated, as well as Ireland, the Netherlands, and in several Commonwealth nations, including Australia, Canada, India, Kenya, Malta, New Zealand, and South Africa. Compilers of cryptic crosswords are commonly called setters in the UK and constructors in the US. Particularly in the UK, a distinction may be made between cryptics and quick crosswords, and sometimes two sets of clues are given for a single puzzle grid.
A cryptogram is a type of puzzle that consists of a short piece of encrypted text. Generally the cipher used to encrypt the text is simple enough that the cryptogram can be solved by hand. Substitution ciphers where each letter is replaced by a different letter, number, or symbol are frequently used. To solve the puzzle, one must recover the original lettering. Though once used in more serious applications, they are now mainly printed for entertainment in newspapers and magazines.
Dynamic programming is both a mathematical optimization method and an algorithmic paradigm. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
Verbal arithmetic, also known as alphametics, cryptarithmetic, cryptarithm or word addition, is a type of mathematical game consisting of a mathematical equation among unknown numbers, whose digits are represented by letters of the alphabet. The goal is to identify the value of each letter. The name can be extended to puzzles that use non-alphabetic symbols instead of letters.
Kakuro or Kakkuro or Kakoro is a kind of logic puzzle that is often referred to as a mathematical transliteration of the crossword. Kakuro puzzles are regular features in many math-and-logic puzzle publications across the world. In 1966, Canadian Jacob E. Funk, an employee of Dell Magazines, came up with the original English name Cross Sums and other names such as Cross Addition have also been used, but the Japanese name Kakuro, abbreviation of Japanese kasan kurosu, seems to have gained general acceptance and the puzzles appear to be titled this way now in most publications. The popularity of Kakuro in Japan is immense, second only to Sudoku among Nikoli's famed logic-puzzle offerings.
Answer set programming (ASP) is a form of declarative programming oriented towards difficult search problems. It is based on the stable model semantics of logic programming. In ASP, search problems are reduced to computing stable models, and answer set solvers—programs for generating stable models—are used to perform search. The computational process employed in the design of many answer set solvers is an enhancement of the DPLL algorithm and, in principle, it always terminates.
Mathematics can be used to study Sudoku puzzles to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of clues in a valid puzzle?" and "In what ways can Sudoku grids be symmetric?" through the use of combinatorics and group theory.
An acrostic is a type of word puzzle, related somewhat to crossword puzzles, that uses an acrostic form. It typically consists of two parts. The first part is a set of lettered clues, each of which has numbered blanks representing the letters of the answer. The second part is a long series of numbered blanks and spaces, representing a quotation or other text, into which the answers for the clues fit. In some forms of the puzzle, the first letters of each correct clue answer, read in order from clue A on down the list, will spell out the author of the quote and the title of the work it is taken from; this can be used as an additional solving aid.
The Smithy code is a series of letters embedded, as a private amusement, within the April 2006 approved judgement of Mr Justice Peter Smith on The Da Vinci Code copyright case. The judge publicly admitted that "I can't discuss the judgement, but I don't see why a judgement should not be a matter of fun." It was first broken, in the same month, by Dan Tench, a lawyer who writes on media issues for The Guardian, after he received a series of email clues about it from Justice Smith.
A standard Sudoku contains 81 cells, in a 9×9 grid, and has 9 boxes, each box being the intersection of the first, middle, or last 3 rows, and the first, middle, or last 3 columns. Each cell may contain a number from one to nine, and each number can only occur once in each row, column, and box. A Sudoku starts with some cells containing numbers (clues), and the goal is to solve the remaining cells. Proper Sudokus have one solution. Players and investigators use a wide range of computer algorithms to solve Sudokus, study their properties, and make new puzzles, including Sudokus with interesting symmetries and other properties.
A cross-figure is a puzzle similar to a crossword in structure, but with entries that consist of numbers rather than words, where individual digits are entered in the blank cells. Clues may be mathematical, use general knowledge or refer to other clues.
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.
Camouflage is an American television game show that aired on Game Show Network. Hosted by Roger Lodge, and billed as "the hidden word game where the answer is always right in front of you", Camouflage originally aired for 40 episodes from July 2 to 27, 2007. The show is a word game, with contestants searching for a hidden word or phrase in a string of jumbled letters. The show is produced by Enjoy the Ride Productions in association with McB Entertainment.
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. Conversely, a variable with value of 0 corresponds to an empty cell—no hidden object.
References
1. G.R. Worder, Word Sum Puzzles, 2010