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The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle, or paradox in logic, probability, and recreational mathematics. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. Historically, it arose as a variant of the necktie paradox. The problem typically is introduced by formulating a hypothetical challenge of the following type:

It seems obvious that there is no point in switching envelopes as the situation is symmetric. However, because you stand to gain twice as much money if you switch while risking only a loss of half of what you currently have, it is possible to argue that it is more beneficial to switch. The problem is to show what is wrong with this argument.

Nomic is a game created in 1982 by philosopher Peter Suber in which the rules of the game include mechanisms for the players to change those rules, usually beginning through a system of democratic voting.

Nomic is a game in which changing the rules is a move. In that respect it differs from almost every other game. The primary activity of Nomic is proposing changes in the rules, debating the wisdom of changing them in that way, voting on the changes, deciding what can and cannot be done afterwards, and doing it. Even this core of the game, of course, can be changed.

The initial ruleset was designed by Peter Suber, and first published in Douglas Hofstadter's column Metamagical Themas in Scientific American in June 1982. The column discussed Suber's then-upcoming book, The Paradox of Self-Amendment, which was published some years later. Nomic now refers to many games, all based on the initial ruleset.

The game is in some ways modeled on modern government systems. It demonstrates that in any system where rule changes are possible, a situation may arise in which the resulting laws are contradictory or insufficient to determine what is in fact legal. Because the game models (and exposes conceptual questions about) a legal system and the problems of legal interpretation, it is named after νόμος (nomos), Greek for "law".

While the victory condition in Suber's initial ruleset is the accumulation of 100 points by the roll of dice, he once said that "this rule is deliberately boring so that players will quickly amend it to please themselves". Players can change the rules to such a degree that points can become irrelevant in favor of a true currency, or make victory an unimportant concern. Any rule in the game, including the rules specifying the criteria for winning and even the rule that rules must be obeyed, can be changed. Any loophole in the ruleset, however, may allow the first player to discover it the chance to pull a "scam" and modify the rules to win the game. Complicating this process is the fact that Suber's initial ruleset allows for the appointment of judges to preside over issues of rule interpretation.

The beer distribution game (also known as the beer game) is a type of gamification that is used to experience typical coordination problems of a supply chain process. It reflects a role-play simulation where several participants play with each other. The game represents a supply chain with a non-coordinated process where problems arise due to lack of information sharing. This game outlines the importance of information sharing, supply chain management and collaboration throughout a supply chain process. Due to lack of information, suppliers, manufacturers, sales people and customers often have an incomplete understanding of what the real demand of an order is. The most interesting part of the game is that each group has no control over another part of the supply chain. Therefore, each group has only significant control over their own part of the supply chain. Each group can highly influence the entire supply chain by ordering too much or too little which can lead to a bullwhip effect. Therefore, the order taking of a group also highly depends on decisions of the other groups.

Siteswap is a numeric juggling notation used to describe or represent juggling patterns. It is also referred to as Quantum Juggling, or The Cambridge Notation. Siteswap may also be used to describe siteswap patterns, possible patterns transcribed using siteswap. Throws are represented by positive integers that specify the number of beats in the future when the object is thrown again: "The idea behind siteswap is to keep track of the order that balls are thrown and caught, and only that." It is an invaluable tool in determining which combinations of throws yield valid juggling patterns for a given number of objects, and has led to previously unknown patterns (such as 441). However, it does not describe body movements such as behind-the-back and under-the-leg. Siteswap assumes that, "throws happen on beats that are equally spaced in time."

For example, a three-ball cascade may be notated "3 ", while a shower may be notated "5 1".

The Hardest Logic Puzzle Ever is a logic puzzle so called by American philosopher and logician George Boolos and published in The Harvard Review of Philosophy in 1996. Boolos' article includes multiple ways of solving the problem. A translation in Italian was published earlier in the newspaper La Repubblica, under the title L'indovinello più difficile del mondo.

It is stated as follows:

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

Boolos provides the following clarifications: a single god may be asked more than one question, questions are permitted to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a fair coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.

A simulation game is "a game that contains a mixture of skill, chance, and strategy to simulate an aspect of reality, such as a stock exchange". Similarly, Finnish author Virpi Ruohomäki states that "a simulation game combines the features of a game (competition, cooperation, rules, participants, roles) with those of a simulation (incorporation of critical features of reality). A game is a simulation game if its rules refer to an empirical model of reality". A properly built simulation game used to teach or learn economics would closely follow the assumptions and rules of the theoretical models within this discipline.

Toyetic is a term referring to the suitability of a media property, such as a cartoon or movie, for merchandising tie-in lines of licensed toys, games and novelties. The term is attributed to Bernard Loomis, a toy development executive for Kenner Toys, in discussing the opportunities for marketing the film Close Encounters of the Third Kind, telling its producer Steven Spielberg that the movie wasn't "toyetic" enough, leading Loomis towards acquiring the lucrative license for the upcoming Star Wars properties.

Mafia (also known as The Werewolves) is a social deduction game, created by Dimitry Davidoff in 1986. The game models a conflict between two groups: an informed minority (the mafiosi or the werewolves), and an uninformed majority (the villagers). At the start of the game, each player is secretly assigned a role affiliated with one of these teams. The game has two alternating phases: first, a night role, during which those with night killing powers may covertly kill other players, and second, a day role, in which surviving players debate the identities of players and vote to eliminate a suspect. The game continues until a faction achieves its win condition; for the village, this usually means eliminating the evil minority, while for the minority this usually means reaching numerical parity with the village and eliminating any rival evil groups.

Nim is a mathematical game of strategy 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.

Variants of Nim have been played since ancient times. The game is said to have originated in China—it closely resembles the Chinese game of 捡石子 jiǎn-shízi, or "picking stones"—but the origin is uncertain; the earliest European references to Nim are from the beginning of the 16th century. Its current name was coined by Charles L. Bouton of Harvard University, who also developed the complete theory of the game in 1901, but the origins of the name were never fully explained.

Nim is typically played as a misère game, in which the player to take the last object loses. Nim can also be played as a normal play game, where the player taking the last object wins. This is called normal play because the last move is a winning move in most games, even though it is not the normal way that Nim is played. In either normal play or a misère game, when the number of heaps with at least two objects is exactly equal to one, the next player who takes next can easily win. If this removes either all or all but one objects from the heap that has two or more, then no heaps will have more than one object, so the players are forced to alternate removing exactly one object until the game ends. If the player leaves an even number of non-zero heaps (as the player would do in normal play), the player takes last; if the player leaves an odd number of heaps (as the player would do in misère play), then the other player takes last.

Normal play Nim (or more precisely the system of nimbers) is fundamental to the Sprague–Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games (see disjunctive sum).

While all normal play impartial games can be assigned a Nim value, that is not the case under the misère convention. Only tame games can be played using the same strategy as misère Nim.

Nim is a special case of a poset game where the poset consists of disjoint chains (the heaps).

The evolution graph of the game of Nim with three heaps is the same as three branches of the evolution graph of the Ulam-Warburton automaton.

At the 1940 New York World's Fair Westinghouse displayed a machine, the Nimatron, that played Nim. From May 11, 1940 to October 27, 1940 only a few people were able to beat the machine in that six week period; if they did they were presented with a coin that said Nim Champ. It was also one of the first ever electronic computerized games. Ferranti built a Nim playing computer which was displayed at the Festival of Britain in 1951. In 1952 Herbert Koppel, Eugene Grant and Howard Bailer, engineers from the W. L. Maxon Corporation, developed a machine weighing 23 kilograms (50 lb) which played Nim against a human opponent and regularly won. A Nim Playing Machine has been described made from TinkerToy.

The game of Nim was the subject of Martin Gardner's February 1958 Mathematical Games column in Scientific American. A version of Nim is played—and has symbolic importance—in the French New Wave film Last Year at Marienbad (1961).

Quantum tic-tac-toe is a "quantum generalization" of tic-tac-toe in which the players' moves are "superpositions" of plays in the classical game. The game was invented by Allan Goff of Novatia Labs, who describes it as "a way of introducing quantum physics without mathematics", and offering "a conceptual foundation for understanding the meaning of quantum mechanics".