Understanding Winning and Losing States

In any game-based problem, we can classify all possible states into two categories: winning states and losing states.

1. A winning state is a state from which the player can ensure a win by playing optimally.
2. A losing state is a state where the player will inevitably lose if the opponent plays optimally.

The classification of states into winning and losing is based on the concept of optimality. If a player can make a move that leads to a losing state for the opponent, then the current state is considered a winning state. Conversely, if all moves from the current state lead to winning states for the opponent, then the current state is a losing state.

The process of classifying states begins with the identification of obvious losing states. In most cases, these are states where no moves are possible. For instance, in a game where players take turns removing balls from a heap, a state with zero balls is a losing state because the player cannot make a move.

Once the losing states are identified, we can determine the winning states by finding states that can reach a losing state in one move. This process continues until all states are classified as either winning or losing.

In Competitive Programming, we often encounter problems where two players are playing a game optimally and we need to find the winner of the game. In order to solve such problems, we should have a clear understanding of Winning and Losing states of the game. In this article, we will go through the basics of Winning and Losing States and how to identify them using State Graph.