Artificial Intelligence

Course Highlights This course includes interactive demonstrations which are intended to stimulate interest and to help students gain intuition about how artificial intelligence methods work under a variety of circumstances. Course Description This course introduces students to the basic knowledge representation, problem solving, and learning methods of artificial intelligence. Upon completion of 6.034, students should […]

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Latin square

The “Gamma plus two” method for generating “odd order” magic squares, the“Gamma plus two plus swap” method for generating “singly even order” magicsquares, and Durer’s method for generating “doubly even order” magic squares. By Professor Edward Brumgnach, P.E. City University of New YorkQueensborough Community College In combinatorics and in experimental design, a Latin square is […]

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N-Queens

Construct a magic square nxn (using the numbers 1 to n2) and place n queens only on these cells which contain prime numbers, such that no queen can take any other queen. 1. What is the smallest magic square (n) having solution? 2. Get one solution for the next three larger magic squares (n+1, n+2 […]

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Adventures in Group Theory

Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys David Joyner  5-15-2008 Published on Jan 15, 2016 The vast majority of people who tackle the Rubik’s cube never succeed in solving it without looking up somebody else’s solution. In this video the Mathologer reveals a simple insight that will enable all those […]

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systematically generate all permutations

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting. These differ from combinations, which are selections of some members of a set where order is disregarded. […]

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pentomino

Pentominoes Pentominoes: Puzzles & Solutions PENTOMINOES – An Introduction A pentomino is a plane geometric figure formed by joining five equal squares edge to edge. It is a polyomino with five cells. There are twelve pentominoes, not counting rotations and reflections as distinct. They are used chiefly in recreational mathematics for puzzles and problems.[1] Pentominoes […]

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a handout given to physics students at Harvard

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad […]

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Revenue equivalence theorem

Revenue equivalence is a concept in auction theory that states that given certain conditions, any auction mechanism that results in the same outcomes (i.e. allocates items to the same bidders) also has the same expected revenue. An auction is a special case of a mechanism. In this case, the mechanism takes buyers’ bids and decides […]

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Markov decision processes

Markov decision processes (MDPs), named after Andrey Markov, provide a mathematical framework for modeling decision making in situations where outcomes are partlyrandom and partly under the control of a decision maker. MDPs are useful for studying a wide range of optimization problems solved via dynamic programming andreinforcement learning. MDPs were known at least as early […]

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