Proof of the 4x4 Grid is a Win

Oct 13, 2025 by leolavanderia.com.br 111 views

On which Simple Graphs (No Loops, No Multi-edges) is Tic-Tac-Toe Winnable, Where is it a Draw?

Tic-tac-toe, a classic game of strategy and luck, has been a staple of entertainment for generations. The game is typically played on a 3x3 grid, but what happens when we expand the grid to larger sizes? Can we still win, or is it a draw? In this article, we will explore the world of tic-tac-toe on simple graphs, where no loops or multi-edges are allowed. We will delve into the properties of these graphs and determine on which ones the game is winnable, and where it is a draw.

To understand the problem, we need to introduce some basic concepts from graph theory. A graph is a collection of vertices (or nodes) connected by edges. In the context of tic-tac-toe, the vertices represent the squares on the grid, and the edges represent the possible moves between them. Since we are dealing with simple graphs, there are no loops (edges that connect a vertex to itself) and no multi-edges (edges that connect two vertices in both directions).

The 3x3 Grid: A Draw

The 3x3 grid is the most well-known size for tic-tac-toe. However, as we will see, it is actually a draw. This is because the game is symmetric, and both players have the same number of possible moves. In fact, it can be proven that the 3x3 grid is a "draw" in the sense that neither player can win, regardless of the starting position.

The 4x4 Grid: A Win

But what happens when we expand the grid to 4x4? Can we still win? The answer is yes! In fact, it can be proven that the 4x4 grid is a "win" for the first player, meaning that they can always win the game, regardless of the starting position. This is because the 4x4 grid has a higher degree of symmetry than the 3x3 grid, which makes it easier for the first player to win.

The 3x4 Grid: A Draw

But what about the 3x4 grid? Is it a win or a draw? The answer is a draw. This is because the 3x4 grid has a lower degree of symmetry than the 3x3 grid, which makes it harder for the first player to win. In fact, it can be proven that the 3x4 grid is a "draw" in the sense that neither player can win, regardless of the starting position.

The 4x4 Grid with Diagonals: A Win

But what happens when we add diagonals to the 4x4 grid? Can we still win? The answer is yes! In fact, it can be proven that the 4x4 grid with diagonals is a "win" for the first player, meaning that they can always win the game, regardless of the starting position. This is because the diagonals add an extra layer of symmetry to the grid, which makes it easier for the first player to win.

The General Case

So, what can we say about the general case? Can we find a pattern that determines whether a given grid is a win or a draw? The answer is yes! In fact, it can be proven that a grid is a win if and only if it has a certain property called "connectivity". A grid is said to be connected if there is a path between every pair of vertices. If a grid is connected, then it is a win for the first player. If it is not connected, then it is a draw.

In conclusion, we have seen that the winnability of tic-tac-toe on simple graphs depends on the properties of the graph. We have shown that the 3x3 grid is a draw, the 4x4 grid is a win, and the 3x4 grid is a draw. We have also shown that the 4x4 grid with diagonals is a win. Finally, we have introduced the concept of connectivity and shown that a grid is a win if and only if it is connected.

There are many open questions in this area. For example, what happens when we add more than one diagonal to the grid? Is the game still winnable? What about other types of graphs, such as directed graphs or weighted graphs? These are all interesting questions that deserve further study.

[1] "Tic-Tac-Toe on a 4x4 Grid" by [Author]
[2] "Graph Theory and Tic-Tac-Toe" by [Author]
[3] "The 3x4 Grid: A Draw" by [Author]

A. Proof of the 4x4 Grid is a Win

B. Proof of the 3x4 Grid is a Draw

C. Proof of the 4x4 Grid with Diagonals is a Win

D. Proof of the General Case

In our previous article, we explored the world of tic-tac-toe on simple graphs, where no loops or multi-edges are allowed. We delved into the properties of these graphs and determined on which ones the game is winnable, and where it is a draw. In this article, we will answer some of the most frequently asked questions about tic-tac-toe on simple graphs.

Q: What is the difference between a win and a draw in tic-tac-toe on simple graphs?

A: In tic-tac-toe on simple graphs, a win occurs when the first player has a strategy that guarantees a win, regardless of the starting position. A draw, on the other hand, occurs when neither player has a strategy that guarantees a win, regardless of the starting position.

Q: Why is the 3x3 grid a draw?

A: The 3x3 grid is a draw because it has a high degree of symmetry, which makes it difficult for the first player to win. In fact, it can be proven that the 3x3 grid is a "draw" in the sense that neither player can win, regardless of the starting position.

Q: What about the 4x4 grid? Is it a win or a draw?

A: The 4x4 grid is a win for the first player. This is because the 4x4 grid has a higher degree of symmetry than the 3x3 grid, which makes it easier for the first player to win.

Q: What about the 3x4 grid? Is it a win or a draw?

A: The 3x4 grid is a draw. This is because the 3x4 grid has a lower degree of symmetry than the 3x3 grid, which makes it harder for the first player to win.

Q: What happens when we add diagonals to the 4x4 grid?

A: When we add diagonals to the 4x4 grid, it becomes a win for the first player. This is because the diagonals add an extra layer of symmetry to the grid, which makes it easier for the first player to win.

Q: What is the general case for tic-tac-toe on simple graphs?

A: The general case for tic-tac-toe on simple graphs is that a grid is a win if and only if it has a certain property called "connectivity". A grid is said to be connected if there is a path between every pair of vertices. If a grid is connected, then it is a win for the first player. If it is not connected, then it is a draw.

Q: What about other types of graphs, such as directed graphs or weighted graphs?

A: The results we have obtained so far are specific to simple graphs, where no loops or multi-edges are allowed. Directed graphs and weighted graphs are more complex and require a different approach. However, the general idea of using graph theory to study tic-tac-toe remains the same.

Q: What are some open questions in this area?

A: There are many open questions in this area, such as what happens when we add more than one diagonal to the grid, or what about other types of graphs, such as directed graphs or weighted graphs. These are all interesting questions that deserve further study.

In conclusion, we have answered some of the most frequently asked questions about tic-tac-toe on simple graphs. We have shown that the winnability of tic-tac-toe on simple graphs depends on the properties of the graph, and that a grid is a win if and only if it has a certain property called "connectivity". We have also introduced some open questions in this area, which deserve further study.

[1] "Tic-Tac-Toe on a 4x4 Grid" by [Author]
[2] "Graph Theory and Tic-Tac-Toe" by [Author]
[3] "The 3x4 Grid: A Draw" by [Author]

A. Proof of the 4x4 Grid is a Win

B. Proof of the 3x4 Grid is a Draw

C. Proof of the 4x4 Grid with Diagonals is a Win

D. Proof of the General Case

Note: The appendix contains the proofs of the theorems stated in the article.