All Questions
Tagged with combinatorial-game-theory infinite-games
22 questions
12
votes
0
answers
495
views
Connection properties of a single stone on an infinite Hex board
This includes a series of questions.
One of the most typical examples is shown as the picture below.
An half-infinite Hex board with an one row of black stones. Black stones are separated by one ...
- 661
4
votes
1
answer
432
views
"Infinity": A card game based on prime factorization and a question
I have been developing a card game called "Infinity", which involves a unique play mechanic based on card interactions. In this game, each card displays a set of symbols, and players match ...
- 1,887
1
vote
0
answers
132
views
Are gaps and loopy games interchangeable in the Surreal Numbers?
The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and ...
- 39.9k
67
votes
5
answers
10k
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Decidability of chess on an infinite board
The recent question Do there exist chess positions that require exponentially many moves to reach? of Tim Chow reminds me of a problem I have been interested in. Is chess with finitely many men on an ...
- 236k
128
votes
13
answers
24k
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Checkmate in $\omega$ moves?
Is there a chess position with a finite number of pieces on the infinite chess board $\mathbb{Z}^2$ such that White to move has a forced win, but Black can stave off mate for at least $n$ moves for ...
- 5,492
24
votes
2
answers
1k
views
What is the complexity of the winning condition in infinite Hex? In particular, is infinite Hex a Borel game?
Consider the game of infinite Hex, where two players Red and Blue alternately place their stones on the infinite hex grid, each aiming to create a winning configuration.
Red wins after infinite play, ...
- 740
41
votes
3
answers
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 $5$ ...
- 236k
28
votes
7
answers
6k
views
Why is game theory formulated in terms of equilibrium instead of winning strategies?
Game theory, on the outset, seems to invite the questions,
"what can I do to win" or "how do I beat my opponent?"
So many people who are not familiar with game theory look to game ...
- 887
9
votes
3
answers
1k
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 ...
- 32.1k
25
votes
4
answers
2k
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The Chocolatier's game: can the Glutton win with a restricted form of strategy?
I have a question about the Chocolatier's game, which I had
introduced in my recent answer to a question of Richard
Stanley.
To recap the game quickly, the Chocolatier offers up at each stage
a finite ...
- 12.8k
4
votes
0
answers
165
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 \...
- 63.9k
28
votes
2
answers
1k
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 ...
- 1,616
1
vote
0
answers
96
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 ...
- 11
47
votes
3
answers
5k
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 ...
CommunityBot
- 1
9
votes
1
answer
460
views
Infinite-dimensional hex
Suppose $n$ players take turns selecting vertices of the grid $[k]^n = \left\{0, 1, 2, \ldots, k-1\right\}^n$. Each player is assigned a pair of opposite faces of the grid, and wins the game if they ...
- 129
7
votes
2
answers
671
views
Determinacy of (infinite, possibly loopy) combinatorial games
I am looking for references and hopefully enlightening proofs of the following statement(s) concerning the determinacy of not-necessarily-well-founded (i.e., possibly infinite, possibly loopy) ...
- 129
0
votes
1
answer
495
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Infinite board games: sentences about
As a unified approach if we have an ( read any) infinite board game described as $\mathcal{G}$ using a particular axiom set A..
can a sentence be devised in A which automatically answers the basic ...
- 129
2
votes
2
answers
320
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 \...
- 129
13
votes
1
answer
3k
views
The infinite X in Conway's game of life
In Conway's game of life, take the initial position to be two infinite diagonal lines of live cells, with a single cell in common. Does this thing converge to a stable configuration? I.e., is the ...
- 129
33
votes
1
answer
3k
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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 ...
- 236k
9
votes
2
answers
691
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\...
- 236k
19
votes
3
answers
1k
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 ...
- 236k