Questions tagged [combinatorial-game-theory]
Two-player turn-based perfect-information games, surreal numbers, impartial games and Sprague-Grundy theory, partizan games
198
questions
2
votes
0answers
58 views
Winning moves in Hex
The game "Hex" is a simple game which apparently has been invented at least twice (Piet Hein and John Nash). The game consists of an n by n grid of hexagons, with two opposite sides marked ...
18
votes
3answers
509 views
Tic-tac-toe with one mark type
Parameters $a,b,c$ are given such that $c\leq\max(a,b)$. In an $a\times b$ board, two players take turns putting a mark on an empty square. Whoever gets $c$ consecutive marks horizontally, vertically, ...
1
vote
0answers
75 views
Winning criterion for a combinatorial game
Given $n$,
let $\mathcal{R}$ be a set of pairs $(\rho,A)$
where $A\subseteq n, \rho\in 2^A$.
Consider the following game between A and B.
At each round $t$, A enumerates an $m\in n$ (that has not been ...
3
votes
0answers
66 views
What values are representable by Hackenbush stalks?
It is known that every number can be represented by some red-blue Hackenbush stalk (see here, for instance). What values can be represented by red-blue-green Hackenbush stalks? In addition, what games ...
1
vote
0answers
90 views
Nim variant with minimum number of objects?
I'm wondering where I can find in the literature (if it exists) a discussion of a Nim variant where we impose the additional condition on Nim that we can remove only up to $c$ objects before the game ...
0
votes
0answers
38 views
The quadratic Markov game
Recently, I've asked about the linear Markov game: A "Markov game". It was solved by @JosephGordon (and I followed with a simplification). This time I'll ask about the quadratic Markov game. ...
3
votes
2answers
167 views
A “Markov game”
I call games similar to the one I describe below to be Markov games. I am selecting just that one or rather a 1-parameter series of games. The open challenge is to find out which of the players $\ 0\ $...
9
votes
1answer
241 views
An averaging game on finite multisets of integers
The following procedure is a variant of one suggested by
Patrek Ragnarsson (age 10). Let $M$ be a finite multiset of
integers. A move consists of choosing two elements
$a\neq b$ of $M$ of the same ...
5
votes
0answers
165 views
Quantum surreal numbers
Toward Quantum Combinatorial Games presents the definition of a "quantum game", allowing a superposition of moves rather than a single classical move. This leaves me wondering: Since surreal ...
6
votes
1answer
79 views
pursuit-evasion based on Schroeder's upper bound for graphs of genus $g$
I am following Schroeder's work on pursuit-evasion games on graphs (often called "cops and robbers"). In his 2001 publication ("The copnumber of a graph is bounded by $\lfloor 3/2 {\ \...
27
votes
1answer
803 views
Players alternate moving a $\{\swarrow,\uparrow,\rightarrow\}$ piece on a chessboard
Edit $4.$ $-$ Proposing to reopen the question (the related competition should be over by now).
Edit $3.$ $-$ I have just found out that the linked competition (see the "Edit $1$.") is still ...
12
votes
0answers
221 views
Does the Angel have to be really smart?
My question is about the computational complexity of the Angel's strategy in the Angels and Devils game, tl;dr does the Angel have a polynomial time strategy.
I'm a big Conway fan, so as you can ...
7
votes
0answers
169 views
How does Conway's proposed compromise for constructing the real numbers in ONAG actually work?
I have also asked this question on Math Stack Exchange (link).
In On Numbers and Games, after discussing the inclusion of the real numbers in the surreal numbers, No, Conway discusses the merits of ...
15
votes
1answer
575 views
In theory, how would Oneiric numbers be defined?
Background I am not a professional mathematician. I am researching Surreal numbers & games for fun (I think they are truly beautiful). If this question is not appropriate here, I beg forgiveness &...
51
votes
1answer
2k views
In the two-person Killing the Hydra game, what is the winning strategy?
My question is which player has a winning strategy in the
two-player version of the Killing the Hydra game?
In their amazing paper,
Kirby, Laurie; Paris, Jeff, Accessible independence results for ...
8
votes
1answer
284 views
Free category with product and coproduct
Is there a known description of the free category with both product and coproduct ?
That is given a small category $C$, I want to consider a category $U C$ which has product and coproduct, a functor $...
24
votes
1answer
2k views
Are Conway's combinatorial games the “monster model” of any familiar theory?
This is related to this question about a "mother of all" groups, and so seemed like it'd fit in better at MO than MSE.
If I understand the answer to that question correctly, the surreal numbers have ...
4
votes
0answers
59 views
Infinite positions in 3D chomp
I've recently come back to investigating ordinal chomp. See A winning move for the first player in $3 \times 3 \times \omega$ Ordinal Chomp for a definition. I made a new discovery, that the position \...
4
votes
0answers
159 views
Nimber $2^{2^k} - 1$ is a multiplicative generator of $[2^{2^k}]$?
Let $t = 2^{2^k}$, and consider the field $[t]$ of nimbers below $t$. For $k \leq 6$ one can check that $t - 1$ (in the usual arithmetic sense) is a multiplicative generator of $[t] \backslash \{0\}$. ...
19
votes
2answers
3k views
What is the winning strategy in this pebble game?
Consider the following two-player pebble game. We have finitely
many stones on a finite linear track of squares. We take turns, and
the allowed moves are:
move any one stone one square to the left, ...
1
vote
0answers
31 views
Suggestions for two-choice game played in ladder graph
I was just working on counting all the possible Nash Equilibrium solutions for a two-choice game played on a ladder graph (I got my results and all that for a generic number of players).
And I was ...
2
votes
1answer
210 views
Is following function a metric on the set of isomorphism classes of graphs with countably many vertices?
Suppose $\Gamma_1(V_1, E_1)$ and $\Gamma_2(V_2, E_2)$ are simple graphs with countably many vertices. And suppose $A_1$ and $A_2$ are initially empty sets. Suppose two players play the following game: ...
5
votes
0answers
181 views
Is Domineering on any finite approximation of the Sierpinski Carpet always a second-player win?
The game of Domineering can be played on any board consisting of some subset of $\mathbb{Z} \times \mathbb{Z}$.
In particular, consider the boards $K_n$ generated by iterating the following inductive ...
1
vote
1answer
69 views
Perturbation of the value of a general-sum game at a equilibirium
Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes ...
28
votes
2answers
881 views
Solution to simple mathematical game
Consider the following game (that I made up). Two players each attempt to name a target number. The first player begins by naming 1. On each subsequent turn, a player can name any larger number that ...
28
votes
0answers
775 views
Is this representation of Go (game) irreducible?
This post is freely inspired by the basic rules of Go (game), usually played on a $19 \times 19$ grid graph.
Consider the $\mathbb{Z}^2$ grid. We can assign to each vertex a state "black" ($b$), "...
4
votes
3answers
212 views
Best strategy for a combinatorial game
Consider the following scenario. We have 20 balls and 100 boxes. We put all 20 balls into the boxes, and each box can contain at most one ball.
Now suppose we are given 5 chances to pick 20 out of ...
1
vote
0answers
76 views
Strategy of Responder in Rényi Ulam Liar Games
I tried posting this in Math Stack Exchange but got no responses, so I figured I could try my luck here. My main concern is that I can't figure out how to get started on my "research" (bear with me, I'...
1
vote
0answers
82 views
Is there a well-posed definition of game on a graph? Or a well defined category of games on graphs?
All I ever found about this were natural language rules à la Asimov's three laws of robotics. The questions are straightforward questions:
1) Is there a well-posed mathematical definition of game on ...
2
votes
0answers
110 views
Game on groups (generalization of spinning switches puzzle)
Alice and Bob are playing a game as follows:
Initially
There're two subgroups $A,B$ of Sym(n) known to both Alice and Bob
There're $n$ slots $S_1, \cdots, S_n$ and $n$ boxes $B_1, \cdots, B_n$. ...
1
vote
0answers
24 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} \...
2
votes
0answers
199 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 ...
3
votes
2answers
142 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 $...
2
votes
0answers
96 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, ...
6
votes
1answer
142 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 ...
2
votes
1answer
241 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 ...
1
vote
0answers
105 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' <...
5
votes
1answer
189 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 ...
6
votes
0answers
97 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 ...
7
votes
1answer
308 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 ...
1
vote
2answers
243 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 \...
31
votes
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 ...
1
vote
1answer
73 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.
27
votes
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 one ...
20
votes
1answer
667 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). ...
10
votes
3answers
903 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 ...
34
votes
2answers
3k 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 ...
19
votes
3answers
863 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 ...
37
votes
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 $5$ ...
46
votes
3answers
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 ...
