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Questions tagged [combinatorial-game-theory]

Two-player turn-based perfect-information games, surreal numbers, impartial games and Sprague-Grundy theory, partizan games

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18 views

Bound for the additive period length of certain Sprague-Grundy functions

Let $\left( Y_x \right)_{x=0}^\infty $ be a sequence of finite subsets of $\mathbb{Z}$, and let $G : \mathbb{N}_0 \to \mathbb{N}_0$ be a greedy permutation, defined by $$ G(x) = \operatorname{mex} \...
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0answers
40 views

Combining a Nim-variation and Wyrthoff's game. How to find a winning strategy? [closed]

Wythoff's game is a variation of the classical Nim - There are two heaps and the players take turns either taking any amount from one heap, or the same amount of both heaps. The winner is the one ...
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0answers
128 views

How often do random games of go end in illegal moves?

Suppose that moves are generated from two players in accordance with three rules: each move is chosen uniformly at random among places on the board ($19 \times 19$, $9 \times 9$, or $k \times k$ with ...
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2answers
124 views

Satisfier-Falsifier games

In a Maker-Breaker game, there is a finite set of elements $X$, and a family $F$ of subsets of $X$ called the "winning sets". Two players, Maker and Breaker, take turns picking untaken elements from $...
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0answers
86 views

A combinatorial number game

Alice and Bob play the following (base 10) number game. A target N is fixed, N being a positive integer. Alice then writes the number 1 on the blackboard. Bob responds with the number 2. Thereafter, ...
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1answer
107 views

What is the minimum worst-case length of an element removal game?

A game is played as follows. There is a set $X = \{1, \ldots, n\}$. Player 1 is trying to find a "locally minimal subset" $M \subseteq X$ - that is, player 2 has said that $M$ is good, and also that ...
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1answer
178 views

Combinatorial games with infinite paths, and generalized Sprague-Grundy theory

Generalized Sprague-Grundy theory has been used to analyze finite impartial loopy games with normal play. There is a nice short account by Mark S. in this answer. It was introduced by Cedric Smith in ...
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0answers
90 views

Is there only one meaningful definition of product of games?

Work in the context of combinatorial games as introduced by Conway. For surreals, the definition of the product is forced by the requirement that surreals should form an ordered field. Say, if $s' <...
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1answer
176 views

A set-family game

Two players, Green and Red, play a zero-sum game. It is parametrized by two integers $n\geq 0, k\geq 0$, and a finite family $F$ of sets of size $n$ (each set may appear multiple times in $F$). Each ...
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0answers
82 views

Length of optimal play in Hex as a function of size

Consider Hex on an $n \times n$ board without a swap rule, so that the first player wins. Assume the first player tries to minimize the length of the game, and the second player tries to maximize the ...
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0answers
84 views

Decidability of mate-in-n for infinite chess with huygens piece

Consider a game like chess on an infinite board, where we have the usual chess piece types and an additional piece which moves a prime number of square horizontally or vertically. If we assume a ...
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0answers
240 views

surreal form star is a xor cipher [closed]

i have been doing a lot of surreal analysis lately & it dawn'd on me that star in the surreal numbers has the same properties of a simple xor cipher. star operations are given as: ...
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1answer
265 views

A Bitwise Xor Problem

Consider a sequence $a_i$ defined by $$ \begin{align*} a_1&=p,\\ a_2&=q,\\ a_i&=a_{i-1} \oplus a_{i-2}+1, \end{align*}$$ where $\oplus$ is the bitwise xor operation. How can we give an ...
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2answers
211 views

How to describe the common boundaries between regions in a infinite Sudoku?

This relates to the answer to a question "Who wins two player sudoku?" and this awesome blog. A Sudoku can be $N \times N$ where $\sqrt{N}$ is a natural number because $N \times N / \sqrt{N} \times \...
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1answer
2k views

Is there a position in infinite Go for which the life of a particular stone has transfinite game value?

As follow up to Checkmate in $\omega$ moves?, we can ask the same question about go. Is there a position on a $\mathbb Z \times \mathbb Z$ goban such that either black can kill a white group, but ...
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1answer
65 views

Effective way to find Nash equilibrium

Is there any good algorithm for finding Nash equilibrium point, for one and repeated game theory? Thansk a lot for giving me some guidance.
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4answers
3k views

Alice and Bob playing on a circle

I want to solve this problem: Let there be $n \ge 2$ points around a circle. Alice and Bob play a game on the circle. They take moves in turn with Alice beginning. At each move: Alice takes ...
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1answer
441 views

Who wins the Rubik's cube game?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier). Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...
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3answers
704 views

The Sudoku game: Solver-Spoiler variation

Consider the Sudoku Solver-Spoiler game, a natural variation of the Sudoku game recently appearing in the question Who wins two-player Sudoku? posted by user PyRulez. In that game, the players attempt ...
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2answers
2k views

Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...
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3answers
512 views

The arithmetic progression game and its variations: can you find optimal play?

Consider the arithmetic progression game, a two-player game of perfect information, in which the players take turns playing natural numbers, or finite sets of natural numbers, all distinct, and the ...
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3answers
4k views

A game on integers

$A$ and $B$ take turns to pick integers: $A$ picks one integer and then $B$ picks $k > 1$ integers ($k$ being fixed). A player cannot pick a number that his opponent has picked. If $A$ has $5$ ...
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2answers
4k views

Does knight behave like a king in his infinite odyssey?

The Knight's Tour is a well-known mathematical chess problem. There is an extensive amount of research concerning this question in two/higher dimensional finite boards. Here, I would like to tackle ...
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0answers
128 views

Games in non-standard models

Has anyone studied Combinatorial game theory in non-standard models? In particular, we can work in either non-standard models of set theory, or we can work in non-standard models of arithmetic, where ...
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1answer
2k views

How many Tic Tac Toe games are possible? [closed]

Consider the average game of Tic Tac Toe or Noughts and Crosses. The game is played on a 3 by 3 two dimentional board. The game is played by two people and each person is allowed to only add one type ...
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2answers
217 views

For which number of pairs is it an advantage to start in memory

Players A and B play memory starting with $n$ pairs of cards. We assume that they can remember all cards which have been turned. At his turn a player will first recall if two cards already turned ...
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4answers
2k views

The 1-step vanishing polyplets on Conway's game of life

A $n$-polyplet is a collection of $n$ cells on a grid which are orthogonally or diagonally connected. The number of $n$-polyplets is given by the OEIS sequence A030222: $1, 2, 5, 22, 94, 524, 3031, \...
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1answer
777 views

Vanishing line on Conway's game of life

If the initial state of Conway's game of life is a line of $n \in [0,100]$ alive cells, then it vanishes completely after some steps iff $n \in \{0,1,2,6,14,15,18,19,23,24 \}$. See below for $n=24$. ...
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5answers
870 views

When is a game tree the game tree of a board game?

This question arises from what I find interesting in the recently asked question What is a chess piece mathematically? My answer to that question was that mathematically, game pieces are in general ...
9
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1answer
225 views

Ordered Nim game

Consider the following variant of Nim: There are two players and $n$ piles of stones, with sizes $a_1,\dots,a_n$, such that $a_i\leq a_j$ for any $i<j$. A move consists of removing a positive ...
4
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1answer
246 views

What ordinal corresponds to the T(3)?

Let's play a game. You start with the ordinal $\alpha$ and I start with the empty sequence. Each turn, you decrease your ordinal, and I add a tree (where each node can have one of three labels), ...
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0answers
156 views

Two-player independent set game

Let $G = (V, E)$ be a finite graph, and $S \subseteq V$ initially be an empty set. Alice and Bob play a game, making moves in turns starting with Alice. A move consists of choosing a vertex $v \in V \...
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2answers
198 views

Study of Hex on the Torus

Hex is usually played on a parallelogram shaped board. What if you play it on a Torus? One thing I notice is that the idea of connecting opposite sides doesn't make much sense anymore, since a torus ...
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0answers
202 views

Mistake in ONAG?

In the second edition of the book "On Numbers and Games" by Conway there is a theorem 88 (p. 194) on comparison of sums of ${\uparrow} x$ games. It contains a weird statement: ... (If $X$ is a sum ...
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0answers
119 views

Modern advances in combinatorial game theory

I'm going to take part in teaching a course in combinatorial game theory in the best of ONAG's spirit. I was wondering if there are interesting post-ONAG results that are worth mentioning in (a later ...
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2answers
455 views

What surreal numbers are representable by Red-Blue Hackenbush games?

Every game of Red-Blue Hackenbush represents a surreal number. Is the converse true? Assuming that it is false, what can be said about the class of surreal numbers that are representable by such ...
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1answer
329 views

Are Surreal Numbers the same as Trans-series?

I recently found the paper of Berarducci + Mantova [1, 2] saying that surreal numbers are equivalent to trans-series. These are very different objects: trans-series are used in physics to correct, ...
5
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1answer
381 views

Can a game be an option of itself?

My question is, can a game contain itself as an option? and can it be a surreal number? For example $A=\{A|\}$ or $B=\{C|B\}$ where $C$ is a surreal number. from the point of view of games, it is ...
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0answers
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Is there a better way to understand this particle?

I've been reading through Winning Ways and was working through some examples of my own related to cooling and particles, and I managed to stump myself. If we let ...
9
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1answer
414 views

Is every ordinal the nimber of a ring?

This question is about the game of Noetherian rings, see MO/93276. Here I will include the zero ring in order to get better formulas. The nimber of a Noetherian ring is an ordinal number. It is ...
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1answer
290 views

A game of singletons

Alice and Bob play the following zero-sum game, parametrized by two integers $m$ and $k$: Alice picks $m$ sets, each of which has $k$ items. Bob colors some items in green. Bob's score is the number ...
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3answers
381 views

A faster way to spoil an injection?

Ultimately this is about how primes jump. I will abstract the situation somewhat as there may be related applications which do not spring to my mind. I want to find small spoilers to Hall's Marriage ...
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0answers
82 views

Difficulty of 3-color forest Hackenbush

"Forest Hackenbush" (for lack of a better name) is the particular case of the game of Hackenbush where the initial position (and therefore all subsequent positions) is a (finite) forest (:= disjoint ...
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1answer
177 views

Maximum $2$-D bootstrap percolation time for $n$ points on an $n\times n$ grid

I hesitate to ask this question here, but since it remained unanswered after a bounty on MSE, I ask it here with some reservation. Is the maximum bootstrap percolation time for $n$ points on an $n\...
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0answers
247 views

A game played on binary matrices ($2$-dimension coin-turning game)

Let $r\geq 1$ be a natural number. I am interested in the following (two-player, impartial, perfect-information) game: The state of the game is an $n\times n$ matrix with coefficients in $\mathbb{F}...
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1answer
314 views

A Combinatorial Game with Integer Sequences

Two players, Alice and Bob, take turns constructing a sequence $a_1,a_2,a_3,\dots$, of distinct positive integers, none greater than a given parameter $K$. Alice plays first and makes $a_1=1$. ...
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1answer
178 views

A game: building a bounded degree graph

Given positive integers $n$ and $d\leqslant n-1$. Two players build a graph, starting with $n$ vertices and no edges. On each turn, a player joins two yet not joined vertices by an edge. It is ...
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2answers
537 views

Choosing subsets of $\mathbb R$ of cardinality $\frak c$, who wins?

Consider the following infinite game: two players, I and II, are alternating and choosing a descending sequence of subsets of $\mathbb R$ of cardinality $\frak c$, so I chooses a set $A_1\subseteq\...
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0answers
129 views

Combinatorial game similar to Sprouts

Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy? Basically this game is "Sprouts without midpoints". One starts with $n$ points in the ...
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0answers
136 views

Local Connection Game

A local connection game is given by a set of vertices and graph G where connection is built by adding edges. If the cost to user a(user at node a) is given by $$C(u)=\alpha n_u + \beta \sum_v(dist(u,...