This is an introduction into John Conway's beautiful Combinatorial Game Theory, providing precise statements and detailed proofs for the fundamental parts of his theory. (1) Combinatorial game theory, (2) the GROUP of games, (3) the FIELD of numbers, (4) ordinal numbers, (5) games and numbers, (6) infinitesimal games, (7) impartial games.
Peg solitaire is classically a one-player game played on a grid board containing pegs. The goal of the game is to have a single peg remaining on the board by sequentially jumping with a peg over an adjacent peg onto an empty cell while eliminating the jumped peg. Conway's soldiers is a related game played on $\mathbb{Z}^2$ with pegs initially located on the half-space $y \le 0$. The goal is to bring a peg as far up as possible on the board using peg solitaire jumps. Conway showed that bringing a peg to the line $y = 5$ is impossible with finitely many jumps. Applying Conway's approach, we prov…
We consider the abstract structure of the monoid M of misère impartial game values. Several new results are presented, including a proof that the group of fractions of M is almost torsion-free; a method of calculating the number of distinct games born by day 7; and some new results on the structure of prime games. Also included are proofs of a few older results due to Conway, such as the Cancellation Theorem, that are essential to the analysis but whose proofs are not readily available in the literature. Much of the work presented here was done jointly with John Conway and Dan Hoey, and I dedi…
Friezes with coefficients are maps assigning numbers to the edges and diagonals of a regular polygon such that all Ptolemy relations for crossing diagonals are satisfied. Among these, the classic Conway-Coxeter friezes are the ones where all values are natural numbers and all edges have value 1. Every subpolygon of a Conway-Coxeter frieze yields a frieze with coefficients over the natural numbers. In this paper we give a complete arithmetic criterion for which friezes with coefficients appear as subpolygons of Conway-Coxeter friezes. This generalizes a result of our earlier paper with Peter Jo…
We propose a unifying additive theory for standard conventions in Combinatorial Game Theory, including normal-, misère- and scoring-play, studied by Berlekamp, Conway, Dorbec, Ettinger, Guy, Larsson, Milley, Neto, Nowakowski, Renault, Santos, Siegel, Sopena, Stewart (1976-2019), and others. A game {\em universe} is a set of games that satisfies some standard closure properties. Here, we reveal when the fundamental game comparison problem, ``Is $G\su H$?'', simplifies to a constructive `local' solution, which generalizes Conway's foundational result in ONAG (1976) for normal-play games. This ha…
We study a $2 \times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $\mathrm{PSL}(2,\mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
Temperature of combinatorial games have been long studied since when Conway established the modern combinatorial game theory, and there are several variations of the concepts. In this article, we focus on one of the classical versions of temperature, and give precise proofs to the fundamental claims on temperature, namely, the existence, order-preservation, and homomorphism. Besides, a general relationship between the value and the thermograph of a game is investigated.
We explain the notion of "$q$-deformed real numbers" introduced in our previous work and overview their main properties. We will also introduce $q$-deformed Conway-Coxeter friezes.
We introduce stuttering look and say sequences and describe their chemical structure in the spirit of Conway's work on audioactive decay. We show the growth rate of a stuttering look and say sequence is an algebraic integer of degree 415.
We consider a specific graph dynamical system inspired by the famous Conway's Game of Life in this work. We study the properties of the dynamical system on different graphs and introduce a new efficient heuristic for graph isomorphism testing. We use the evolution of our system to extract features from a graph in a deterministic way and observe that the extracted features are unique and the distance induced by that features satisfy triangle inequality for all connected graphs with up to ten vertices.
We define an all-small ruleset, Bipass, within the framework of normal-play combinatorial games. A game is played on finite strips of black and white stones. Stones of different colors are swapped provided they do not bypass one of their own kind. We find a surjective function from the strips to integer atomic weights (Berlekamp, Conway and Guy 1982) that measures the number of units in all-small games. This result provides explicit winning strategies for many games, and in cases where it does not, it gives narrow bounds for the canonical form game values. We prove that the game value *2 does …
The $\mathscr{P}$-position sets of some combinatorial games have special combinatorial structures. For example, the $\mathscr{P}$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, $\mathcal{D}_{\text{sh}}$. There were, however, few known games related to Steiner systems like the hexad game. For a given Steiner system, we construct a game whose $\mathscr{P}$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $\mathcal{D}_{\text{s…
Sprouts is a two-player pencil-and-paper game invented by John Conway and Michael Paterson in 1967. In the game, the players take turns in joining dots by curves according to simple rules, until one player cannot make a move. The game of Sprouts is very popular and simple-looking, so it may come as a surprise that there are essentially no AI Sprouts players available. This lack of computer opponents is caused by the fact that the game hides a surprisingly high combinatorial complexity and implementing it involves fascinating programming challenges. We overcome all the implementation barriers a…
Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205（里昂第一大学）；Instituto de Matemáticas, UNAM Juriquilla（墨西哥国立自治大学数学研究所）；National University of Singapore（新加坡国立大学）
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Physical and Mathematical School №146 (Perm)（第146号物理数学学校（彼尔姆））；Independent University of Moscow（莫斯科独立大学）
