11
votes
1answer
606 views
+100

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go . . . ] Here's my description of the game: There's a $4\times 4$ grid with some random, numbered cards on. The numbers are either one, two, or multiples of three. ...
2
votes
2answers
82 views

Winning a restricted game of Nim?

Given the following piles, find the Grundy number of the initial position and make the first move in a winning strategy given that no more than two sticks may be removed from a pile at any time. Pile ...
1
vote
1answer
36 views

Determining Grundy Numbers for an inverted takeaway game

Given the following game, I need to determine a winning strategy and find the set of positions in the kernel. I figure the best way to do so would be with Grundy numbers. Rules: The game consists ...
1
vote
1answer
62 views

Game Of Strings

There are two strings A and B. Initially, some strings A’ and B’ are written on the sheet of paper. A’ is always a substring of A and B’ is always a substring of B. A move consists of appending a ...
3
votes
0answers
46 views

News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
2
votes
1answer
107 views

Play with pairs of numbers

Two players are playing a game. The game is played on a sequence of positive integer pairs. The players make their moves alternatively. During his move the player chooses a pair and decreases the ...
5
votes
1answer
91 views

How to simply show that there are “78 'strict ordinal' 2x2 game matrices”

In "Theory of Moves", Steven J. Brams analyses two-player games with two strategies per player, where each player can totally rank his payoffs, although payoffs need not be comparable among players. ...
1
vote
2answers
99 views

Problem regarding filling squares inside a $n\times n$ grid.

Assuming a $n\times n$ square grid, what is the most number of squares that can be filled in such that there are no completed rows, columns, or diagonals? Is there a formula to calculate this? ...
6
votes
1answer
194 views

Man, Woman, Dog, seeking stable relationship.

There is a classic problem in combinatorics dealing with a stable pairing between a set of men and a set of women as spouses. (Gale-Shapely algorithm) ...
1
vote
1answer
72 views

Game of Stones - Count the ways

We are given a number of piles of stones. and we can remove two stones , where both stones come from different piles. We do this until all the piles are finished or only one pile is left as we cannot ...
1
vote
2answers
48 views

Game with two players and 120 points in total

Assume the following game: The game has two players $P_{1}$ and $P_{2}$ and 15 rounds in which they play against each other. Each round gives an amount of points equal to its number, i.e. the ...
11
votes
1answer
221 views

Determining the number of valid TicTacToe board states in terms of board dimension

I am attempting to find a closed form equation in terms of $n$, for the number of valid Tic-Tac-Toe board states (ignoring symmetry), where the board has dimension $n \times n ,\; 0 \lt n,\;n \in \Bbb ...
1
vote
1answer
58 views

What is the relative strength of each of the players in this game?

This is a real life problem. A group of people meet once a week to play a game between two teams. Each round 2 people are randomly appointed captains. Each captain takes turns picking people to be on ...
4
votes
1answer
80 views

Shapley value: an alternative representation

It is my belief that the more common representation of the Shapley value is given by $$ \phi_i(v)=\sum_{S\subseteq N-i} \frac{|S|!(|N|-|S|-1)!}{|N|!}(v(S\cup\{i\})-v(S)) $$ where $v \in ...
2
votes
1answer
181 views

Von Neumann's minimax theroem and Carathéodory's theorem

In J.F. Mertens(1986)(Paywall), there's a neat proof of a version of Von Neumann's minimax theroem. But I can't understand how Carathéodory's theorem is invoked. Suppose, in a two-person zero sum ...
4
votes
0answers
73 views

Linear Independence Game

Suppose you have a set $X$ of vectors in $\mathbb{F}_2^n$, with $|X| \ge n+1$, and consider the following game. On their turn, each player (2 player game) chooses from $X$ one vector and sets it aside ...
0
votes
1answer
97 views

How to calculate Team Strength for future prediction?

You are given with $4$ players name, namely Player $A$, Player $B$, Player $C$ and Player $D$. These players are grouped into two teams with two players each. A Game is played between the two team.For ...
1
vote
1answer
58 views

Probability of winning of teacher

Teacher is playing a game with his students. He is having $k$ red balls. Each of his student is either having a red or black ball. $M$ students have red balls and $N$ students have black balls. Now ...
3
votes
2answers
52 views

Obtaining certain pairs of numbers using three “machines”

Each of three machines can read a card on which is written a pair of whole numbers $(m,n)$ and print a new card. Machine $\text{A}$ reads $(m,n)$ and prints $(m-n,n)$. Machine $\text{B}$ ...
0
votes
0answers
105 views

Finding all number combination which XOR results to 0

Let's say I have a fixed list of numbers: $2, 3, 1, 2$ and I can reduce every number from $n$ to $0$, for instance: $1,3,1,2$ or $0,3,0,1$ etc. I am looking for all combinations of this sort, where ...
1
vote
2answers
113 views

Some Questions About Chess

I have to questions about the chess game: please help me to understand it. 1- How can a computer program know if this move or that move is better? It calculates all possbile continuation and examine? ...
1
vote
0answers
105 views

Card game-ordering a deck [duplicate]

Possible Duplicate: Game Theory Matching a Deck of Cards Suppose we take a blank deck of $52$ cards, write the number $1$ on the first card, $2$ on the second card, and so on until we write ...
2
votes
2answers
121 views

Poker, number of three of a kind, multiple formulaes

I wanted to calculate some poker hands, for a three of a kind I infered, 1) every card rank can form a 'three of a kind' and there are 13 card ranks, 2) there are $\binom{4}{3}$ ways to choose three ...
30
votes
1answer
1k views

Is War necessarily finite?

War is an cardgame played by children and drunk college students which involves no strategic choices on either side. The outcome is determined by the dealing of the cards. These are the rules. A ...
3
votes
1answer
284 views

Calculating the Shapley value in a weighted voting game.

Given a special case of WVG (Weighted Voting Game) of $a$ 1s and $b$ 2s and a quota q, $ [q:1,1,1,1..1,2,2,..2] $. I need help with calculating the Shapley value of a player with a weight of $2$ and a ...
8
votes
3answers
133 views

Seemingly similar but different probability games

Burger King is currently running its "family food" game in which each piece can be modeled as a scratch off game where exactly one of three slots is a winner and you may only scratch one slot as your ...
4
votes
3answers
138 views

Consider a card game-parity

Consider a card game where the deck consists of 63 distinct cards. The deck is created in the following manner: each card consists of some number of symbols, where no two symbols are the same. There ...
1
vote
0answers
100 views

Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
5
votes
1answer
253 views

Olympic Badminton, or How to Design a Tournament

Hearing the recent news about disqualified Badminton players in the ongoing 2012 London Olympics got me wondering about how best to design tournaments to avoid situations where players are ...
1
vote
2answers
524 views

Is there a winning strategy for Scrabble?

I am sure many of us are addicted to the popular Facebook app: Words with Friends, which is basically an online version of Scrabble. In Playing Games with Algorithms:Algorithmic Combinatorial Game ...
1
vote
0answers
130 views

Is Bingo Games Solved?

So we have numbers 5 rows by 5 columns. Different players have the same table. The numbers on each cell of the tables are different. Different players choose which numbers will be marked. We select a ...
0
votes
1answer
77 views

Solution for assigning independent tasks to independent individuals

I have $n$ tasks that I wish to delegate to $m$ independent individuals, where $m$ is a factor or divisor of $n$. Each of the tasks $T_{1} ... T_{n}$ is independent. From the following two extremes, ...
2
votes
1answer
337 views

A checkerboard problem

If $mn$ squares out of a $2m\times n$ white checkerboard are colored black, and a move consists of interchanging the color on any two squares who share a side, how many moves at maximum can it take to ...
18
votes
1answer
1k views

Number of moves to solve a flood-it/sock-dye game

[ Question based on the sock dye game ] [ Update: It appears that this game is better known as "Flood it" and is NP-hard. Also, "the number of moves required to flood the whole board is $\Omega(n)$ ...
12
votes
3answers
545 views

Best Strategy for a die game

You are allowed to roll a die up to six times. Anytime you stop, you get the dollar amount of the face value of your last roll. Question: What is the best strategy? According to my calculation, for ...