In combinatorial game theory dots and boxes is very close to being an impartial game and many positions can be analyzed using Sprague – Grundy theory.

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.

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.

The first volume introduces combinatorial game theory and its foundation in the surreal numbers ; partizan and impartial games ; Sprague – Grundy theory and misère games.

There are several combinatorial analogs of the Gauss – Bonnet theorem.

In mathematics, a building ( also Tits building, Bruhat – Tits building, named after François Bruhat and Jacques Tits ) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces.

The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and Lipschitz manifolds,, the extension of Atiyah – Singer's signature operator to Lipschitz manifolds, Kasparov's K-homology and topological cobordism.

* The Erdős – Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity.

* The Erdős – Heilbronn conjecture in combinatorial number theory on the number of sums of two sets of residues modulo a prime, proved by J. A.

The initial proposed problem for this project, now called Polymath1 by the Polymath community, was to find a new combinatorial proof to the density version of the Hales – Jewett theorem.

Corresponding to each tiling of the quartic ( partition of the quartic variety into subsets ) is an abstract polyhedron, which abstracts from the geometry and only reflects the combinatorics of the tiling ( this is a general way of obtaining an abstract polytope from a tiling ) – the vertices, edges, and faces of the polyhedron are equal as sets to the vertices, edges, and faces of the tiling, with the same incidence relations, and the ( combinatorial ) automorphism group of the abstract polyhedron equals the ( geometric ) automorphism group of the quartic.

In mathematics, the Hales – Jewett theorem is a fundamental combinatorial result of Ramsey theory, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure ; it is impossible for such objects to be " completely random ".

As a result of this research, Landsberg and Mandelstam discovered the effect of the inelastic combinatorial scattering of light on 21 February 1928 (" combinatorial " – from combination of frequencies of photons and molecular vibrations ).

Many combinatorial algorithms on tableaux are known, including Schützenberger's jeu de taquin and the Robinson – Schensted – Knuth correspondence.

In combinatorial mathematics, the Lubell – Yamamoto – Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by,,, and.

The intersection semilattice determines another combinatorial invariant of the arrangement, the Orlik – Solomon algebra.

A theorem in the Flajolet – Sedgewick theory of symbolic combinatorics treats the enumeration problem of labelled and unlabelled combinatorial classes by means of the creation of symbolic operators that make it possible to translate equations involving combinatorial structures directly ( and automatically ) into equations in the generating functions of these structures.

The Szemerédi – Trotter theorem is a mathematical result in the field of combinatorial geometry.

Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities, which is based on the representation theory of affine Kac – Moody algebras.

John Leslie Britton ( November 18, 1927 – June 13, 1994 ) was an English mathematician from Yorkshire who worked in combinatorial group theory and was an expert on the word problem for groups.