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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.

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?

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.

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 is \(X\) while you are \(O\).

The initial gameplay is as shown below:

Assuming that your opponent will play optimally from now on, where must you place your next mark in order to obtain a winning position?

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