Symmetry
Basic

Lecture Notes

Symmetry is widely utilized to solve many competition math problems, and unsurprisingly, it is also an effective method for tackling certain brain teasers. Consider the following example, where each player has an infinite number of ways to place coins. To approach this problem, we can start with the simplest case: a table just large enough to hold one coin. In this scenario, it's clear that the first player is the winner, leading us to consider that the first player may have a winning strategy.

Given that there is an infinite number of ways for the second player to place his coin in each turn, a winning strategy must provide a definitive response to each unpredictable placement. This is where the principle of symmetry comes into play. By applying symmetry, we can often find a structured approach to respond to the vast array of choices the second player might make, potentially outlining a strategy that ensures victory for the first player.

Examples

(4710) $\textbf{Coins on a Table}$ Joe invites you to play a game with him by placing quarters on a rectangular shaped table. Each person places one coin in turn. Coins cannot overlap. The person who cannot find enough space to place the next coin loses the game. Do you want to play first or let Joe play first? |