Articles of combinatorial game theory

Game of replacing number with divisors

In a game , there are N numbers and 2 player(A and B) . If A and B pick a number and replace it with one of it’s divisors other than itself alternatively, how would I conclude who would make the last move? (Notice that eventually when the list gets replaced by all 1’s , […]

A Tic-Tac-Toe Variant

Let’s imagine a game like tic-tac-toe, but you have to have 3 of the same sign in the same line or colon… So the diagonals don’t count! Then I have to prove that the second player to play can always be sure not to lose… But how do I do that? I am searching a […]

Game: Group and Multi-Dimensional Chessboard

Let $G$ be a group and $S\subseteq G$. Consider a $d$-dimensional chessboard of size $n_1\times n_2\times \ldots \times n_d$, where $n_1,n_2,\ldots,n_d\in\mathbb{N}$. Each unit hypercube of the chessboard contains an element of $G$, which is called the label of the hypercube. A move consists of three parts: (1) selecting an element $s\in S$, (2) choosing a […]

Counting all possible board positions in Quoridor

I’m trying to figure out how many possible board positions there are for the game Quoridor. I think sorting out the legal positions from the illegal positions will be difficult, so to start I’m trying to count the total number of board positions ignoring legality. The board is 9×9. There are 81 spaces and two […]

Ring structure on subsets of the natural numbers

Let $$\mathcal{N}=\{\{k_1,\ldots,k_s\}:\ s>0,\ \mbox{and the}\ k_i\ \mbox{are non-negative and pairwise different integers}\}\cup\{\emptyset\}.$$ Note that there is a bijection with the naturals, $$ \begin{array}{rccc} B:&\mathcal{N}&\longrightarrow &\mathbb{N}\\ &\emptyset& \longmapsto & 0\\ &\{k_1,\ldots,k_s\}& \longmapsto & 2^{k_1}+\cdots+2^{k_s} \end{array}. $$ For $K,L\in\mathcal{N}$ define $$K\oplus L=(K\cup L)\setminus(K\cap L),$$ $$K\otimes L=\bigoplus_{k\in K,\ l\in L}\{k+l\}.$$ The definition of the product doesn’t make sense […]

Are there non-zero combinatorial games of odd order?

Are there any combinatorial games with odd order (under the usual addition of combinatorial games), apart from $0$? In Are there combinatorial games of finite order different from $1$ or $2$? I asked about games of finite order greater than $2$, and was given a really nice example of a game of order $4$ (and […]

Why does the strategy-stealing argument for tic-tac-toe work?

On the Wikipedia page for strategy-stealing arguments, there is an example of such an argument applied to tic-tac-toe: A strategy-stealing argument for tic-tac-toe goes like this: suppose that the second player has a guaranteed winning strategy, which we will call S. We can convert S into a winning strategy for the first player. The first […]

Splitting Stacks Nim

A game is played with two players and an initial stack of $n$ pennies ($n\geq 3$). The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. When a player makes a move that causes all the stacks to be of height $1$ or $2$ at […]

Counting all possible legal board states in Quoridor

Ignoring pawns there are 1,375,968,129,062,134,174,771 possible ways to place 0 to 20 walls on the Quoridor board, as answered here. Ignoring walls there are 81 * 81 = 6410 ways to place the two pawns on the board. Multiplying these numbers gives us all possible legal and illegal board states. How do I separate out […]

A variation of Nim game

There are two players X and Y . They write N integers on paper ( A_1 , A_2 , A_3 , …. A_N ). They have also p integers (b_1 , b_2 , b_3 , …. b_p ) . Now , Player X always takes turn first . He can choose any integer A_i from […]