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Combinatorial Games

Whether you’re a master of games or just playing around, learn how combinatorial ideas can be used to analyze and solve games such as Nim.

Concept Quizzes

Level 2

         

You are playing a game of Tic-tac-toe. You are playing as \(O\) while your opponent plays as \(X\). The game plays as shown:

Where should you put your next \(O\) in order to save yourself from losing?

Andy and Bob are playing a game on a \( 9 \times 9 \) checkerboard. They take turns to move a move: on his turn, Andy places an X in an empty square while Bob places an O. When the entire checkerboard is filled, Andy scores a point for each row or column that contains more X's than O's, while Bob scores a point for each row or column that contains more O's than X's. The winner of the game is the person with (strictly) more points.

Given that Andy makes the first move, who has the winning strategy?

Note: If they both scored 9 points, then it is considered a draw.

Daniel and Cody are playing a game on an infinite unit square grid. Daniel places unit square sticky notes with a letter \(D\), while Cody places unit square sticky notes with a letter \(C\). They take turns placing sticky notes on squares in the grid, with Daniel going first. Cody wants to have four of his sticky notes form the corners of a perfect square with sides parallel to the grid lines, while Daniel wants to prevent him from doing so.

Can Daniel succeed?

Assume that both players make optimal moves.

Alice and Carla are playing a game often learned in elementary school known as Say 16. The rules for the game are as follows:

  • Each player takes turns saying between 1 and 3 consecutive numbers, with the first player starting with the number 1. For example, Player 1 could say the numbers 1 and 2, then Player 2 can say "3, 4, 5", then Player 1 can say "6" and so on.

  • The goal of the game is to be the one to say "16".

Carla decides that she'll go first and that Alice will go second. Is there a way to tell which player is going to win before the game even starts?

Details and Assumptions:

  • Assume that each player plays "perfectly", meaning that if there was an optimal way of playing, both players would be playing the best that the game allows them to play.

You're playing tic tac toe with an opponent who plays optimally. You are \(O\) while your opponent is \(X\).

The game plays as shown:

Where must you place your next mark in order to obtain a winning position?

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