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14
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Updating Probabilities
, 2002
"... As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a "naive space", which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR ("coarsening at random") in t ..."
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Cited by 52 (6 self)
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As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a "naive space", which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR ("coarsening at random") in the statistical literature characterizes when "naive" conditioning in a naive space works. We show that the CAR condition holds rather infrequently, and we provide a procedural characterization of it, by giving a randomized algorithm that generates all and only distributions for which CAR holds. This substantially extends previous characterizations of CAR. We also consider more generalized notions of update such as Jeffrey conditioning and minimizing relative entropy (MRE). We give a generalization of the CAR condition that characterizes when Jeffrey conditioning leads to appropriate answers, and show that there exist some very simple settings in which MRE essentially never gives the right results. This generalizes and interconnects previous results obtained in the literature on CAR and MRE.
A hierarchical approach to computer Hex
, 2002
"... Hex is a beautiful game with simple rules and a strategic complexity comparable to that of Chess and Go. The massive gametree search techniques developed mostly for Chess and successfully used for Checkers and a number of other games, become less useful for games with large branching factors like H ..."
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Cited by 24 (0 self)
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Hex is a beautiful game with simple rules and a strategic complexity comparable to that of Chess and Go. The massive gametree search techniques developed mostly for Chess and successfully used for Checkers and a number of other games, become less useful for games with large branching factors like Hex and Go. In this paper, we describe deduction rules, which are used to calculate values of complex Hex positions recursively starting from the simplest ones. We explain how this approach is implemented in HEXYthe strongest Hexplaying computer program, the Gold medallist of the 5th Computer Olympiad in London, August 2000. 2001 Elsevier Science B.V. All rights reserved.
GAMESMAN: A finite, twoperson, perfectinformation game generator
"... This report introduces GAMESMAN, a system for generating graphical parametrizable game applications. Programmers write game modules for a specific game, which when combined with our libraries, compile together to become standalone Xwindow applications as shown in Figure A.1 below. The modules only ..."
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Cited by 3 (2 self)
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This report introduces GAMESMAN, a system for generating graphical parametrizable game applications. Programmers write game modules for a specific game, which when combined with our libraries, compile together to become standalone Xwindow applications as shown in Figure A.1 below. The modules only need contain information about the rules of the game and how the game ends. If the game is smallenough, it may be solved, and the computer can play the role of an oracle, or perfect opponent. This oracle can advise a novice player how to play, and teach the strategy of the game even though none was programmed into the system! If a game is too large to be solved exhaustively, the game programmer can add heuristics to provide an imperfect computer opponent. Finally, the application can provide a useful utility to two human players who are playing each other, since it be a referee who constrains the users moves to be only valid moves, can update the board to respond to the move, and can signal when one of the players has won.
Who Wins Misère Hex?
"... Hex is an elegant and fun game that was first popularized by Martin Gardner [4]. The game was invented by Piet Hein in 1942 and was rediscovered by John Nash at Princeton in 1948. Two players alternate placing white and black stones onto the hexagons of an N × N rhombusshaped board. A hexagon may c ..."
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Cited by 2 (0 self)
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Hex is an elegant and fun game that was first popularized by Martin Gardner [4]. The game was invented by Piet Hein in 1942 and was rediscovered by John Nash at Princeton in 1948. Two players alternate placing white and black stones onto the hexagons of an N × N rhombusshaped board. A hexagon may contain at most one stone. Agameof7 × 7 Hex after three moves. White’sgoalistoputwhitestonesinasetofhexagonsthatconnectthetop and bottom of the rhombus, and Black’s goal is to put black stones in a set of hexagons that connect the left and right sides of the rhombus. Gardner credits Nash with the observation that there exists a winning strategy for the first player in a game of hex. The proof goes as follows. First we observe that the game cannot end in a draw, for in any Hex board filled with white and black stones there must be either a winning path for white, or a winning path for black [1, 3]. (This fact is equivalent to a version of the Brouwer fixed point theorem, as shown by Gale [3].) Since the game is finite, there must be a winning strategy for either the first or the second player. Assume, for the sake of
Carryless arithmetic mod 10
 College Math. J., Special
"... Forms of Nim have been played since antiquity and a complete theory was published as early as 1902 (see [3]). Martin Gardner described the game in one of his earliest columns [7] and returned to it many times over the years ([8]–[16]). Central to the analysis of Nim is Nimaddition. The Nimsum is c ..."
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Forms of Nim have been played since antiquity and a complete theory was published as early as 1902 (see [3]). Martin Gardner described the game in one of his earliest columns [7] and returned to it many times over the years ([8]–[16]). Central to the analysis of Nim is Nimaddition. The Nimsum is calculated by writing the terms in base 2 and adding the columns mod 2, with no carries. A Nim position is a winning position if and only if the Nimsum of the sizes of the heaps is zero [2], [7]. Is there is a generalization of Nim in which the analysis uses the baseb representations of the sizes of the heaps, for b> 2, in which a position is a win if and only if the modb sums of the columns is identically zero? One such game, Rimb (an abbreviation of RestrictedNim) exists, although it is complicated and not well known. It was introduced in an unpublished paper [6] in 1980 and is hinted at in [5]. Despite his interest in Nim, Martin Gardner never mentions Rimb, nor does it appear in Winning Ways [2], which extensively analyzes Nim variants. In the present paper we focus on b = 10, and consider, not Rim10 itself, but the arithmetic that arises if calculations, addition and multiplication, are performed mod 10, with no carries. Along the way we encounter several new and interesting number sequences, which would have appealed
Dear Editor: Word Problems: Applications vs. Mental Manipulatives
"... Thank you for giving place to articles on word problems [1, 2]. I am concerned with word problems very much and would like to say something about them also. First of all, in my opinion, any discussion should include a definition of the subject and in such situations it is better to keep as close as ..."
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Thank you for giving place to articles on word problems [1, 2]. I am concerned with word problems very much and would like to say something about them also. First of all, in my opinion, any discussion should include a definition of the subject and in such situations it is better to keep as close as possible to the exact meaning of the words. I suggest that a nonword problem is a problem, which is formulated using mainly mathematical symbols and only a few special words like “Solve the equation... ” Correspondingly, a word problem is a problem which uses nonmathematical words to convey mathematical meaning. At the K12 level nonword problems tend to be technical exercises, which are necessary, but not exciting. It is only natural that most interesting and nonstandard problems are word problems. It does not mean that all word problems are difficult, but all of them need understanding of the natural language and ability to translate between different modes of representation: in words, in symbols, in images. This is similar to Thomas ’ main idea [2], although I do not quite agree with his treatment of solving equations.
HEX
"... The game of Hex was first invented in 1942 by Piet Hein, a Danish scientist, mathematician, writer, and poet. In 1948, John Nash at Princeton rediscovered the game, which became popular among the math graduate students at Princeton. They called Hex either “Nash ” or ..."
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The game of Hex was first invented in 1942 by Piet Hein, a Danish scientist, mathematician, writer, and poet. In 1948, John Nash at Princeton rediscovered the game, which became popular among the math graduate students at Princeton. They called Hex either “Nash ” or
AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS
, 2005
"... Combinatorial Game Theory is a fascinating and rich theory, based on a simple ..."
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Combinatorial Game Theory is a fascinating and rich theory, based on a simple