ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

On numbers and games, by J. H. Conway. Pp ix, 238. £6·50. 1976. SBN 0 12 186350 6 (Academic Press)

Some readers know to play the game of nim well, fewer play a perfect annihilation game, and nobody knows whether there exists an opening move in chess that will guarantee a win for white. These games and many more, belong to the family of combinatorial games, by which we mean the set of all two-player perfect-information games without chance moves and with outcomes lose or win (and sometimes: dynamic tie). The motivation for ONAG may have been, and perhaps was-and I would like to think that it was-the attempt to bridge the theory gap between nim-like and chess-like games. Why is there a gap? Every combinatorial game can be described as a directed graph called game-graph, whose vertices are the game positions, and (u, v) is a directed edge if and only if there is a move from position u to position v. Denote by N the set of all positions from which the Next (first) player can force a win; by P the set of all positions from which the Previous (second) player can force a win; and by T the set of all (dynamic) Tie positions, which are positions from which no player can force a win and therefore both can avoid losing. In an acyclic game-graph there cannot be any tie positions. The N, P, T classification of any game graph R = (V, E) can be determined in 0(\V\ + \E\) steps [8]. For both nim and chess, a finite game-graph can be constructed and the N, P, T classification can be determined. So both games are solvable in principle. If we play nim with n piles, each pile containing at most k tokens, then the game-graph contains (k + \) vertices. Suppose that in (generalized) chess played on an « X « board there are k different pieces. If k is about n/2, then the game-graph of chess contains O (2") vertices. So both game-graphs have exponentially many vertices, and thus both games appear intractable in the usual sense of computational complexity [1, Chapter 10], [14, Chapter 9], namely a computation appears to be required which is asymptotically exponential. From a computational efficiency standpoint, the essential difference between nim and chess is that nim can be viewed as a disjunctive compound (sum) of independent games, namely the individual piles. A disjunctive

http://ieeexplore.ieee.org/iel5/5/31266/01455274.pdf

On numbers and games

Winning ways for your mathematical plays

A thorough revision of a popular text in combinatorial game theory, this second edition reorganizes presentation to make it more widely accessible. The beginning focuses less on technical and more on conceptual material and applications. Still written in a textbook style with supporting evidence and proofs, the authors add many more exercises and examples and implement a two-step approach for some aspects of the material involving an initial introduction, examples, and basic results to be followed later by more technical and abstract results.

Lessons in Play: An Introduction to Combinatorial Game Theory, Second Edition

This paper presents a systemization of representation of multisets and basic operations under multisets, and an overview of the applications of multisets in mathematics, computer science and related areas.

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An overview of the applications of multisets 1

http://www.math.chalmers.se/~hegarty/ID_may21.pdf

Invariant and dual subtraction games resolving the Duchêne-Rigo conjecture

If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,\ldots,n\}$ for $k\ge 4$ and $n$ large.

/pdf/the-structure-of-maximum-subsets-of-1-n-with-no-solutions-to-3aq3cpju5c.pdf

The structure of maximum subsets of {1,...,n} with no solutions to a+b=kc

We study permutations ! of the natural numbers for which the numbers ! (n) are chosen greedily under the restriction that the di!erences ! (n)! n belong to a given (multi)subset M of Z for all n " S, a given subset of N. Various combinatorial properties of such permutations (for quite general M and S) are exhibited and others conjectured. Our results generalise to a large extent known facts in the case M = Z,S = N, where the permutation ! arises in the study of the game of Wytho! Nim.

/pdf/permutations-of-the-natural-numbers-with-prescribed-2zbaux1tt5.pdf

Permutations of the natural numbers with prescribed difference multisets.

The class of Guaranteed Scoring Games (GS) are two-player combinatorial games with the property that Normal-play games (Conway et. al.) are ordered embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS form a partially ordered monoid, via defined Right- and Left-stops over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by 'switching the players'. The order relation between two games is defined by comparing their stops in any  disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013).

/pdf/guaranteed-scoring-games-49kflywmv5.pdf

Guaranteed Scoring Games

Combinatorial scoring games, with the property ‘extra pass moves for a player will do him no harm’, are characterized. The characterization involves an order embedding of Conway’s normal play games, and we call our class the universe of Guaranteed scoring games. Also, we give a theorem for comparing guaranteed games with scores (numbers) which extends Ettinger’s work on Dicot scoring games.

Urban Larsson

Papers

Invariant and dual subtraction games resolving the Duchêne-Rigo conjecture

The structure of maximum subsets of {1,...,n} with no solutions to a+b=kc

Permutations of the natural numbers with prescribed difference multisets.

Guaranteed Scoring Games

Games with guaranteed scores and waiting moves