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In mathematics, a Diophantine equation is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. 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 statement that asserts the equality of two expressions, which are connected by 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 equality is an equation.
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers or defined as generalizations of the integers.
A crossword is a word puzzle that usually takes the form of a square or a rectangular grid of white- and black-shaded squares. The game's goal is to fill the white squares with letters, forming words or phrases, by solving clues, which lead to the answers. In languages that are written left-to-right, the answer words and phrases are placed in the grid from left to right ("across") and from top to bottom ("down"). The shaded squares are used to separate the words or phrases.
A cryptic crossword is a crossword puzzle in which each clue is a word puzzle in and of itself. Cryptic crosswords are particularly popular in the United Kingdom, where they originated, Ireland, Israel, the Netherlands, and in several Commonwealth nations, including Australia, Canada, India, Kenya, Malta, New Zealand, and South Africa. In the United States, cryptics are sometimes known as "British-style" crosswords. Compilers of cryptic crosswords are commonly called "setters" in the UK.
Dynamic programming is both a mathematical optimization method and a computer programming method. 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.
Sudoku is a logic-based, combinatorial number-placement puzzle. In classic sudoku, the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 subgrids that compose the grid contain all of the digits from 1 to 9. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a single solution.
Sudoku puzzles can be studied mathematically 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. 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, with individual digits being entered in the blank cells. The numbers can be clued in various ways:
Camouflage is a 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. Contrary, a variable with value of 0 corresponds to an empty cell—no hidden object.
Exit is an American game show on Syfy that premiered on June 4, 2013.
A spiral puzzle is a word puzzle in the form of a spiral. The puzzle is formed of a long chain of letters, which spell out a list of clued words when read in either direction. The solutions to the left are entered in the boxes from 1 to 100. The responses to the right provide another set of words that must be entered from 100 to 1. The final solution can consist of a number of boxes, sometimes in colored horizontal, vertical or diagonal rows.
References
1. G.R. Worder, Word Sum Puzzles, 2010