Four Tac Toe Rules

Four Tac Toe (or four dimensional Tic Tac Toe) is a variation of a childhood classic. Typically played on a 3x3 grid Four Tac Toe is instead played on nine 3x3 grids arranged in a 3x3 pattern forming the metaphorical four dimensional 3x3x3x3 board. The objective is to complete as many 3-cell lines as possible.

Competitors take turns placing either an X or an O in one cell of one of the grids, with the only restriction that no player can claim the center square of the middle board.

Players score points by completing lines of three cells across any of the four dimensions, across any of the nine boards. Once all cells have been filled the player with most points wins.

Scoring in four dimensions

In four tac toe there are 232 possible lines that can be completed for points. Each line is worth one point, and players can score multiple points in a single turn by completing multiple lines at once.

A line is any three cells that form a straight line. Lines can run horizontally, vertically, diagonally, or any combinations of those and may pass through multiple boards when following that same direction.

as a general rule of thumb, a valid line can be formed by starting at a given cell and following the same movement across cells or boards one step at a time in any number of dimensions - a mantra like "One cell left, one board up" constitutes a line; as does "one cell up, one cell right, one board right" can help when figuring out what is a line and what is not.

Examples:

Example 1: Four different lines are filled in. The red circle line always goes one step to the right. The blue square line goes one cell up or down. The yellow star line goes one cell down, then one board down. The green triangle line goes one cell up, one board to the right.

Example 2: All lines running through the cell at the bottom left corner, each with a different color and symbol. Can you find the mantra for all of them?

When you hover over a cell the game will show you all lines running through that cell, each with a different color and symbol to help you keep track of them. If you hover over an empty cell or a cell you control only lines that you could potentially complete by claiming that cell will be highlighted, while if you hover over a cell your opponent controls, only lines that they could potentially complete involving that cell are highlighted. All dead lines (lines neither player could complete or are already completed) are filtered out

The center cell

The center cell at the middle of the middle board is off limits to both players, and cannot be claimed. This is simply for balancing reasons - the center cell had 40 different lines running through it encompassing every other cell. As a result, the player who claimed that center would always win by a wide margin.

Hyper-geometry

This section is a small aside going a bit into the geometry involved to build up an intuition for what is going on, but at this point you ought to have a good enough understanding of the game to start playing.

Building an intuition for how the boards and lines work and interact can feel extremely complicated, but in reality the game is just a 4D extension of tic tac toe, and the same principles that govern the 2D version apply here as well. It might take a bit of practice to get used to it but it doesn't take very many games until you start to get a feel for how the lines work and how to spot them.

However, for the more geometrically inclined it is surprisingly easy and elegant to build up a mental image on how the board works, and we start with the 1D line

1D - A line

In one dimension the game is just a line of three cells. Not very exciting and makes for a dreadful game, but it is the basis for everything else.

2D - A grid

In two dimensions we have the classic 3x3 grid, with three rows, three columns, and two diagonals for a total of 8 lines. However, you can see how a 2D grid is just three 1D lines stacked on top of each other to form a grid. Each of the three 1D lines still has their one line running through it, but now you also have the option to draw a line vertically or diagonally - giving us 8 total valid lines on a tic tac toe board

3D - A cube

Going a step further, if you take three 3D grids and stack them on top of each other you get a cube! A cube is a 3D extension of our 3x3 grid. You can still reasonably see where the 49 lines are: Each 2D board making up the cube still has their 8 lines, but now you can also have lines running through the cube itself - straight down, corner to corner, cutting diagonally from one edge to another and so on and so forth. You could very easily figure out how to take a cube and play tic tac toe on it: first to color in three mini-cubes in a row wins!

Each of the examples below show the same cube with the same straight lines coloured, with one of the examples just three 2D grids for reference to see how we might represent the cube as 3x3 grids.

Take a moment to examine it, because we're about to take a rather drastic leap to get to our destination: The fourth dimension.

4D - A hypercube

This is where things start to get a bit tricky, because we - limited three dimensional creatures as we are - cannot visualize the fourth dimension nor the hypercube we're about to construct. However, let's try!

If we take three 3D cubes and stack them on top of each other, we get a hypercube. A hypercube is a 4D extension of the idea we've been working with so far. While we can't visualize it, we can still understand how the lines work by applying the same principles as before - each 3D cube making up the hypercube has its usual 49 lines, but now we can also have lines running through the hypercube itself, cutting across multiple cubes

However, despite it not being easy to visualize the pattern is still the same: We start with a line, and then stack three lines together to form a grid. We take three copies of the grid to form a cube, and then we take three cubes and stack them together to form a hypercube. The same principles apply at every step of the way, and the same patterns of movement that formed lines in the previous dimensions still form lines in the next one, we just have to apply them across more dimensions.

And while drawing a hypercube on screen wouldn't make for a very good game, we can of course just cheat a bit by flattening the whole thing out to reach our nine grids. As long as we obey the established rules of what cells connect where it will all come out to the same result.

And so, we've arrived back to two dimensions. Just like how we could represent a cube with three Tic tac toe boards our 4d playing field is back to being represented as nine boards, arranged in a three by three grid. If we wanted we could extend the idea further into 5D, or 6D, or any D we want. However, nine boards is already plenty to contend with.... for now.

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