For example, even though the rules of Mancala are straightforward, mathematicians analyze the game using combinatorial game theory.

Within combinatorial game theory it is usually called the nim-sum, as will be done here.

For example, Mancala is a mathematical game, because mathematicians can study it using combinatorial game theory, even though no mathematics is necessary in order to play it.

( Misère Sprouts is perhaps the only misère combinatorial game that is played competitively in an organized forum.

In the book Winning Ways the authors show how to treat Snakes and Ladders as an impartial game in combinatorial game theory even though it is very far from a natural fit to this category.

In combinatorial game theory, the Sprague – Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber.

A longstanding question in combinatorial game theory asks whether there is a game of Beggar-My-Neighbour that goes on forever.

Games like nim also admit of a rigorous analysis using combinatorial game theory.

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 mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of nim heaps.

* Partisan game, in combinatorial game theory

Classic combinatorial search problems include solving the eight queens puzzle or evaluating moves in games with a large game tree, such as reversi or chess.

In combinatorial game theory, a misère game is one played according to the " misère play condition "; that is, a player unable to move wins.

* Perhaps the only misère combinatorial game played competitively in an organized forum is Sprouts.

Aspects of combinatorics include counting the structures of a given kind and size ( enumerative combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ( as in combinatorial designs and matroid theory ), finding " largest ", " smallest ", or " optimal " objects ( extremal combinatorics and combinatorial optimization ), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems ( algebraic combinatorics ).

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory.

Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties.

In 1969 the Society for Industrial and Applied Mathematics established the George Pólya Prize, given alternately in two categories for " a notable application of combinatorial theory " and for " a notable contribution in another area of interest to George Pólya.

The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties.

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial.

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element.

As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well.

In digital circuit theory, combinational logic ( sometimes also referred to as combinatorial logic ) is a type of digital logic which is implemented by boolean circuits, where the output is a pure function of the present input only.

The first systematic study was given by Walther von Dyck, student of Felix Klein, in the early 1880s, laying the foundations for combinatorial group theory.

It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology ( a precursor to algebraic topology ) and abstract algebra ( theory of modules and syzygies ) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In his other work Littlewood collaborated with Raymond Paley on Littlewood – Paley theory in Fourier theory, and with Cyril Offord in combinatorial work on random sums, in developments that opened up fields still intensively studied.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

Enumerative combinatorics is the most classical area of combinatorics, and concentrates on counting the number of certain combinatorial objects.

Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

An alternate approach, used in many microprocessors, is to use PLAs or ROMs ( instead of combinatorial logic ) mainly for instruction decoding, and let a simple state machine ( without much, or any, microcode ) do most of the sequencing.

In combinatorial mathematics, a Steiner system ( named after Jakob Steiner ) is a type of block design, specifically a t-design with λ = 1 and t ≥ 2.

The name combinatorial search is generally used for algorithms that look for a specific sub-structure of a given discrete structure, such as a graph, a string, a finite group, and so on.

The term combinatorial optimization is typically used when the goal is to find a sub-structure with a maximum ( or minimum ) value of some parameter.

The travelling salesman problem ( TSP ) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science.

In practice, it is often possible to achieve substantial improvement over 2-opt without the combinatorial cost of the general 3-opt by restricting the 3-changes to this special subset where two of the removed edges are adjacent.

Tits demonstrated how to every such group G one can associate a simplicial complex Δ = Δ ( G ) with an action of G, called the spherical building of G. The group G imposes very strong combinatorial regularity conditions on the complexes Δ that can arise in this fashion.

As the complexity of the problem increases, a combinatorial proof can become very sophisticated.

A very closely related topic is geometric group theory, which today largely subsumes combinatorial group theory, using techniques from outside combinatorics besides.

Bose did not very much like the idea, and he used to spend most of the time at home working on combinatorial problems using Galois fields.

This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and ( decorated ) edges stand for the description of the gluing, hence the name.