134
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 ...
- 236k
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 ...
- 236k
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 ...
- 236k
42
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 ...
- 236k
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. ...
- 236k
32
votes
Accepted
The 1-step vanishing polyplets on Conway's game of life
Here are two more vanishing 12-plets similar to yours:
$$
\substack{
\displaystyle{◻◻◼◻◻◻} \cr
\displaystyle{◻◻◻◼◻◻} \cr
\displaystyle{◻◻◼◼◼◼} \cr
\displaystyle{◼◼◼◼◻◻} \cr
\displaystyle{◻◻◼◻◻◻} \cr
\...
- 1,289
32
votes
Accepted
What is the winning strategy in this pebble game?
The positions which are a win for the second player are those with:
an even number of pebbles in odd-numbered squares, and
an even number of pebbles in even-numbered squares.
Indeed, from a position ...
- 32.4k
32
votes
Why is game theory formulated in terms of equilibrium instead of winning strategies?
There's a few issues that need to be distinguished here. First, one can distinguish the question of how you find the winning strategy from the question of how you define what the winning strategy even ...
- 148k
31
votes
Accepted
Are Conway's combinatorial games the "monster model" of any familiar theory?
In On a conjecture of Conway (Illinois J. Math. 46 (2002), no. 2, 497–506), Jacob Lurie
proved Conway's conjecture that the class $G$ of games together with Conway's addition defined thereon is (up to ...
- 6,453
26
votes
In the two-person Killing the Hydra game, what is the winning strategy?
We can think of this as a game of "omega-nim;" to more precise since the game you are describing is impartial, operating under the normal play convention, and finite we have that the Sprague-Grundy ...
- 841
25
votes
Does knight behave like a king in his infinite odyssey?
(Moved from a comment.)
Questions 2, 3, and 4 are answered negatively by [W], which decomposes the plane into concentric annuli of width two.
OP observes that this construction is still spiral (...
- 576
21
votes
Checkmate in $\omega$ moves?
Dropping the assumption of finitely many pieces as in this answer, we construct for any countable ordinal $\alpha$ a position having mate in $\beta > \alpha$, so $\gamma = \omega_1$ in the context ...
- 531
21
votes
The 1-step vanishing polyplets on Conway's game of life
Yes, there are others, such as the alternative $n=9$ example
$$\substack{
\displaystyle{◻◼◻◻◻} \cr
\displaystyle{◻◻◼◻◼} \cr
\displaystyle{◻◼◼◼◻} \cr
\displaystyle{◼◻◼◻◻} \cr
\displaystyle{◻◻◻◼◻}
}$$
...
- 79.9k
21
votes
Accepted
A hat puzzle question—how to prove the standard solution is optimal?
Suppose we have some mixed strategy for this hat puzzle. For each of the $8$ possible hat assignments and each of the $3$ people involved, we can ask about the probability that the person guesses ...
- 3,068
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 ...
- 82.6k
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 ...
- 236k
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 ...
- 32.4k
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
- 151
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-...
- 1,823
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.
...
- 255
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 ...
- 236k
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 ...
- 231
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 ...
- 5,958
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 ...
- 10.6k
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 ...
- 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 ...
- 82.6k
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-...
- 590
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.
- 50.8k
13
votes
A hat puzzle question—how to prove the standard solution is optimal?
Suppose you and your two friends have names A, B, C.
For each case, assign a 3-letter string for what colours the people get. For example, TGG means A gets tan while B and C get gray.
Suppose when A ...
- 328
12
votes
Checkmate in $\omega$ moves?
Introducing a position of game value ω² with finitely many pieces
In the following position with finitely many pieces, White has mate-in-ω² (Black to move): (For convenience, I put the position into a ...
- 271
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