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Combinatorial game theoryCombinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a position evolves through alternating moves, each governed by well-defined rules, with the aim of achieving a specific winning condition.

NimNim is a mathematicalcombinatorial game in which two players take turns removing (or "nimming") objects from distinct heaps or piles. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap or pile. Depending on the version being played, the goal of the game is either to avoid taking the last object or to take the last object.

Computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem.

Abstract strategy gameAbstract strategy games are games like chess, checkers (draughts), and Go. They are typically strategy games with minimal or no narrative theme, an outcome determined mostly by player choice (with minimal or no randomness; deterministic), and in which there is little to no hidden information (also called perfect information games). Pure abstract strategy games are typically governed by simple, explicit rules whose strategic depth arises from combinatorial complexity.

Game treeIn the context of combinatorial game theory, a game tree is a graph representing all possible game states within a sequential game that has perfect information. Such games include chess, checkers, Go, and tic-tac-toe. A game tree can be used to measure the complexity of a game, as it represents all the possible ways that the game can pan out.

Tie (draw)A draw or tie occurs in a competitive sport when the results are identical or inconclusive. Ties or draws are possible in some, but not all, sports and games. Such an outcome, sometimes referred to as deadlock, can also occur in other areas of life such as politics, business, and wherever there are different factions regarding an issue.

MinimaxMinimax (sometimes Minmax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, combinatorial game theory, statistics, and philosophy for minimizing the possible loss for a worst case (maximum loss) scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain.

Tic-tac-toeTic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid, one with Xs and the other with Os. A player wins when they mark all three spaces of a row, column, or diagonal of the grid, whereupon they traditionally draw a line through those three marks to indicate the win.

Maharajah and the SepoysMaharajah and the Sepoys, originally called Shatranj Diwana Shah and also known as the Mad King's Game, Maharajah chess, or Sarvatobhadra ("auspicious on all sides"), is a popular chess variant with different armies for White and Black. It was first played in the 19th century in India. It is a solved game with a forced win for Black.

Two-player gameA two-player game is a multiplayer game that is played by precisely two players. This is distinct from a solitaire game, which is played by only one player.

GameA game is an activity defined by a challenge and a set of rules, typically done for enjoyment, competition, or development. Games may be played informally or in professional competitive settings before audiences. Popular formats include board games, card games, video games, and sports, distinguished from unstructured play by having rules. Scholars have long debated the essence of games, with no single definition achieving universal acceptance.

Proof by exhaustionProof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which a statement is established by dividing the argument into several distinct cases and proving that the statement holds in each case. The cases must be collectively exhaustive, ensuring that all possible situations are considered.