133 votes

What is a chess piece mathematically?

In terms of mathematical analysis and combinatorial game theory, the essence of any game is captured by its game tree, the tree whose nodes represent the current game state, and to make a move in the ...
Joel David Hamkins's user avatar
65 votes
Accepted

A game on integers

I claim that Player A has a winning strategy in your game, and furthermore, it is a winning strategy for her simply to play the smallest available number. Let me consider the game along with several ...
Joel David Hamkins's user avatar
58 votes
Accepted

Does knight behave like a king in his infinite odyssey?

Consider the following open knight's tour on a $5\times 5$ board, starting at position $1$ and then touring the $5\times 5$ board in the indicated move order. The final position is $25$, from which ...
Joel David Hamkins's user avatar
39 votes
Accepted

Who wins two player sudoku?

Update. I made a blog post about Infinite Sudoku and the Sudoku game, following up on ideas in this post and the comments below. I claim that the second player wins the even-sized empty Sudoku ...
Joel David Hamkins's user avatar
38 votes
Accepted

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

This is a really great question! Previous attempts to make sense of infinite Go have sometimes had problems because it wasn't clear how to define the winner of a game of Go after transfinite play. ...
Joel David Hamkins's user avatar
32 votes
Accepted

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

Here are two more vanishing 12-plets similar to yours: $$ \substack{ \displaystyle{◻◻◼◻◻◻} \cr \displaystyle{◻◻◻◼◻◻} \cr \displaystyle{◻◻◼◼◼◼} \cr \displaystyle{◼◼◼◼◻◻} \cr \displaystyle{◻◻◼◻◻◻} \cr \...
Ilmari Karonen's user avatar
32 votes
Accepted

What is the winning strategy in this pebble game?

The positions which are a win for the second player are those with: an even number of pebbles in odd-numbered squares, and an even number of pebbles in even-numbered squares. Indeed, from a position ...
Gro-Tsen's user avatar
  • 29.8k
32 votes

Why is game theory formulated in terms of equilibrium instead of winning strategies?

There's a few issues that need to be distinguished here. First, one can distinguish the question of how you find the winning strategy from the question of how you define what the winning strategy even ...
Will Sawin's user avatar
  • 135k
31 votes
Accepted

Are Conway's combinatorial games the "monster model" of any familiar theory?

In On a conjecture of Conway (Illinois J. Math. 46 (2002), no. 2, 497–506), Jacob Lurie proved Conway's conjecture that the class $G$ of games together with Conway's addition defined thereon is (up to ...
Philip Ehrlich's user avatar
30 votes
Accepted

Can one make high-level proofs about chess positions?

Let me prove, for example, that the following 7-piece position is a draw. 7-piece positions are about the borderline of what's doable by brute force: they were tabulated around 2010. Black draws as ...
Kostya_I's user avatar
  • 8,642
26 votes

In the two-person Killing the Hydra game, what is the winning strategy?

We can think of this as a game of "omega-nim;" to more precise since the game you are describing is impartial, operating under the normal play convention, and finite we have that the Sprague-Grundy ...
Pedro Juan Soto's user avatar
25 votes

Does knight behave like a king in his infinite odyssey?

(Moved from a comment.) Questions 2, 3, and 4 are answered negatively by [W], which decomposes the plane into concentric annuli of width two. OP observes that this construction is still spiral (...
Eric Towers's user avatar
21 votes

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

Yes, there are others, such as the alternative $n=9$ example $$\substack{ \displaystyle{◻◼◻◻◻} \cr \displaystyle{◻◻◼◻◼} \cr \displaystyle{◻◼◼◼◻} \cr \displaystyle{◼◻◼◻◻} \cr \displaystyle{◻◻◻◼◻} }$$ ...
Noam D. Elkies's user avatar
20 votes

Why is game theory formulated in terms of equilibrium instead of winning strategies?

Let me address the criticism that the Nash equilibrium is of questionable real-life significance. I'll begin by openly admitting something that theorists often are reluctant to admit: One big reason ...
Timothy Chow's user avatar
  • 78.1k
19 votes

Checkmate in $\omega$ moves?

Dropping the assumption of finitely many pieces as in this answer, we construct for any countable ordinal $\alpha$ a position having mate in $\beta > \alpha$, so $\gamma = \omega_1$ in the context ...
Matthew Bolan's user avatar
16 votes

Alice and Bob playing on a circle

For even $n$, I claim that nobody has a winning strategy, and therefore both players have drawing strategies. To see this, observe first that by the fundamental theorem of finite games, we know that ...
Joel David Hamkins's user avatar
16 votes

A little number theoretic game

(This is not an answer, but an extensive comment and numerical simulation about Grundy values.) I believe there is some level of confusion because there are actually two very similar games under ...
Gro-Tsen's user avatar
  • 29.8k
15 votes

Can one make high-level proofs about chess positions?

It's not chess, but you might like the book "Mathematical Go: Chilling gets the Last Point" by Berlekamp and Wolfe, about mathematical analysis of Go endgames. IIRC they used transfinite (or was it ...
none's user avatar
  • 151
15 votes

The Chocolatier's game: can the Glutton win with a restricted form of strategy?

Yes. See Theorem 1.2 in K. Ciesielski and R. Laver, A game of D. Gale in which one of the players has limited memory, Period. Math. Hungar. 21 (1990), no. 2, 153–158
dragonsmilk's user avatar
15 votes

A little number theoretic game

Edit: I added code to compute the Nim values for the first $N$ positions of this game after the original post, as requested by @Timothy-Chow. Unfortunately my results don't match those given by @Peter-...
I. J. Kennedy's user avatar
14 votes

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

Since it looks like no one else has tried programmatic search I thought I'd give it a try. I wrote the following Haskell program which generates finds vanishing polyplets. ...
Atsma Nayim's user avatar
13 votes
Accepted

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

In ZFC, the player aiming for the empty set has a winning strategy in the game played on any infinite set, including the reals. Using the axiom of choice, we can well-order the set and thereby pretend ...
Joel David Hamkins's user avatar
13 votes

What is a chess piece mathematically?

Approaching this from the perspective of a computer programmer rather than a mathematician, my instinct is to try to isolate those properties of a chess piece that are unique to that piece, separating ...
Michael Kay's user avatar
13 votes

Alice and Bob playing on a circle

Here is an argument that Alice wins for odd $n$. Label the points $1$ to $n.$ Lemma: If at any point Alice has marked $k$ points in a row and no other points, Bob has marked the two points ...
dhy's user avatar
  • 5,888
13 votes
Accepted

Free category with product and coproduct

The general problem of giving a categorical construction of the free category with finite coproducts and products (or "free sum–product category") seems to still be open, though there are ...
varkor's user avatar
  • 8,675
13 votes

In theory, how would Oneiric numbers be defined?

I emailed John Conway about this very thing over ten years ago. His response (paraphrasing) was along the lines of RP’s comment; if you treat 1/up as a formal entity nothing breaks, but there wasn’t ...
Jeffrey's user avatar
  • 231
13 votes
Accepted

Tic-tac-toe with one mark type

The case $a=1$ and $c=3$ is known as Treblecross. It is an octal game with code .007 and there is some computational data available on Achim Flammenkamp's webpage, but as far as I know, the game has ...
Timothy Chow's user avatar
  • 78.1k
13 votes
Accepted

The Chocolatier's game: can the Glutton win with a restricted form of strategy?

(Not an answer; promoted from a comment on another answer) If we modify the game so that the glutton can remember (only) the last chocolate they ate, they have a winning strategy as follows: Well-...
Milo Brandt's user avatar
13 votes
Accepted

Who wins this two player game of making squares?

This just the game of Dots and Boxes. There is a huge literature on this game. In particular, Berlekamp's book referenced in the above link shows how difficult this game is.
Richard Stanley's user avatar
12 votes
Accepted

What does "game theory" cover and how should it be called?

If we include the larger research community -- economics, computer science, social sciences, business schools, operations research, etc -- I think there really is a partition between combinatorial ...
usul's user avatar
  • 4,429

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