# Mathematics and Tic-tac-toe: Why Are There So Many Draws?

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Tic-Tac-Toe, also known as noughts and crosses, is one of the oldest and most universally recognized games. While the game seems simple on the surface and easy understandable even for a kid, the underlying mathematics paints a complex and intriguing picture. One of the most intriguing aspects of Tic-Tac-Toe is the likelihood of the game ending in a draw, especially when played by two experienced players. But why is this?

## The Power of the First Move

The player who goes first has an inherent mathematical advantage. However, with perfect play, ‘O’ can always force a draw. Two players well-versed in the intricacies of Tic-Tac-Toe will often find their games ending in a tie.

## Symmetry in Tic-Tac-Toe

The Tic-Tac-Toe grid has several axes of symmetry. Many moves are mere rotations or reflections of others, which decreases the number of unique situations. With this symmetry understanding, it’s easier for players to counter potentially losing moves.

## The Cat’s Game Phenomenon

With a more significant number of games ending in a draw, the “Cat’s Game” phenomenon becomes even more prevalent among seasoned players. This term describes scenarios where neither player can force a win, resulting in a draw. This phenomenon is so recognized that many players invest time understanding strategies to avoid losses, inevitably ending up in draws more often. If you are interested you can delve deeper into the Cat’s Game in Tic-Tac-Toe here.

## Total Positions

There are 9 squares, each of which can be in one of three states: ‘X’, ‘O’, or empty. Thus, the total number of possible positions is $$3^9$$, because each of the 9 squares has 3 possible states.

$T = 3^9 = 19,683$

However, this includes positions that are physically impossible (for instance, where ‘X’ has 5 more marks on the board than ‘O’).

## Probability of ‘X’ Winning

To determine the winning probability for ‘X’, we need to know how many of these positions represent a win for ‘X’. Let’s denote this number as $$W_X$$. Then, the probability of ‘X’ winning is:

$P(\text{Win X}) = \frac{W_X}{T}$

## Probability of ‘O’ Winning

Similarly, if $$W_O$$ is the number of positions that represent a win for ‘O’, then the probability of ‘O’ winning is:

$P(\text{Win O}) = \frac{W_O}{T}$

## Probability of a Draw

If $$D$$ is the number of positions that don’t represent a win for either player, then:

$P(\text{Draw}) = \frac{D}{T}$

With optimal play, the probabilities (based on previous computations) are approximately:

• P(Win for X) ≈ 43.5%
• P(Win for O) = 5% (since ‘O’ can never win if both players play perfectly)
• P(Draw) ≈ 51.5%

Based on the rough estimate, in most cases, the result of the game turns out to be a draw.

## Refining Our Calculation

Now, let’s refine our calculation for $$T$$:

In reality, the number of positions will be less than 19,683 due to:

• The game ending as soon as one of the players wins.
• Positions being impossible where ‘X’ or ‘O’ have 5 or more marks exceeding the other.

Unfortunately, without a full exhaustive analysis of each of the 19,683 possible positions (which is typically done by a computer), it’s not feasible to precisely determine $$W_X, W_O,$$ and $$D$$.

## Conclusion

Tic-Tac-Toe, while seemingly straightforward, has a depth that many overlook. The mathematics behind it reveals a game where draws are the most likely outcome, especially when both players understand its intricacies. Like many things, Tic-Tac-Toe reminds us that sometimes, the most profound truths lie just beneath the surface.