All Questions
Tagged with combinatorial-game-theory nt.number-theory
14 questions
93
votes
3
answers
6k
views
A little number theoretic game
I came up with this little two player game:
The players take turns naming a positive integer. When one player says the number $n$, the other player can only reply in two different ways: They can ...
- 861
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
17
votes
3
answers
2k
views
Traversing the infinite square grid
Starting somewhere on an infinite square grid, is it possible to visit every square exactly once, if at move $n$, one must jump $a_n$ steps in one of the directions north,south,east or west, and mark ...
- 171
13
votes
0
answers
221
views
A game based on the Euclidean algorithm
The following game is based on a somewhat "stupid" version of the Euclidean algorithm (where we allow only subtractions).
Positions are given by finite non-empty multisets (repeated elements ...
- 17.9k
12
votes
1
answer
392
views
Euclid's algorithm as a combinatorial game
Consider the following two player game based on the Euclidean algorithm: Positions are given by $(a,b)$ in $\mathbb N^2\setminus\{(0,0)\}$ (where $\mathbb N=\{0,1,2,\ldots\}$) defining a greatest ...
- 17.9k
9
votes
1
answer
351
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 $a_1=1$. ...
- 3,717
8
votes
1
answer
16k
views
Analysis of Misere Nim?
My friend likes to impress people by playing 3-5-7 which has three piles of counters of sizes 3, 5 and 7. You can remove any number of coins from a single pile, the last player to move loses.
ooo
...
- 22.8k
6
votes
1
answer
196
views
A combinatorial game with seemingly curious arithmetic properties
We consider the following combinatorial game (with two players
alternatively playing optimally). Posititions are given by heaps containing $b\geq 0$ black and $w\geq 0$ white stones and are
encoded by ...
- 17.9k
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 ...
- 3,144
3
votes
1
answer
462
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, ...
- 22.8k
3
votes
0
answers
175
views
Does a generalized Queen split the upper P-positions of Wythoff Nim into two new beams of P-positions?
Wythoff Nim is an impartial game where 2 players take turns in reducing the heights of two finite heaps of tokens. Two types of moves are allowed
(I) Remove any number of tokens from precisely one ...
- 71
2
votes
3
answers
462
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 ...
- 13k
1
vote
1
answer
147
views
Name for an easy combinatorial game
What is the name of the following combinatorial game:
Two players, moving in turn.
Positions: $0,1,2,\ldots$.
Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$
if $n>0$.
No move for $0$...
- 17.9k
1
vote
0
answers
256
views
The maximum number of moves in a game of Nim [closed]
I was assigned a fun, but also quite hard problem for my computer science class - I have to write a java program that computes the maximum number of turns in an optimal game of Nim.
In case you are ...
- 11