Topological Tic Tac Toe

A nice source of generalizations for classic games played on a tabletop is porting them over to other surfaces. Today we'll look at Tic Tac Toe on more exotic surfaces than the standard $3 \times 3$ board.

Setup

This activity can be done with pencil and paper, but it's easier to visualize if one has a representation of the surface. For surfaces like planes, cylinders, and Möbius strips, a transparency sheet works great. A white piece of paper works well for planes and cylinders, but if you try this with a Möbius strip you have to be careful to interpret an X or O as bleeding through the paper, regardless which face of the page it was written on. (Without this interpretation, the game actually takes place again on a cylinder!) The local curvatures required to form a torus prohibit a transparency sheet from being manipulated into one in a nice way, but inner tubes can be gotten pretty cheaply.

Transparency sheets (3 for $12)
Intex inner tubes (3 for $12)

Additionally, you'll want a permanent marker for marking tic tac toe grid lines, dry erase markers for game play, and tape for making cylinders and Möbius strips from transparency sheets. (If you get printable transparency sheets, you can also print the grids out before cutting and taping.)

Here is a $3'' \times 9''$ rectangular grid I made using a sharpie and a cut up sheet protector, since those were what I had on-hand:

You can see that it's a pretty good dimension for making a cylinder, but that you might want an even more skewed set of dimensions to make the Möbius strip:

We'll use the standard rule set. One player will play as X and the other as O, alternating placing letters in the grid squares of an $n \times n$ grid. The first player to get $n$ of their letter in consecutive squares of a row, column, or diagonal wins. If the grid is filled with no $n$-in-a-rows, the game is a draw.

Challenges

If unfamiliar with classic Tic Tac Toe or needing a refresher, a few games of three-in-a-row on a $3 \times 3$ grid or four-in-a-row on a $4 \times 4$ grid can be a nice entry.

For these other games, students should think about what good strategies look like and whether the first or second player has the advantage:

Three-in-a-row on a $3 \times 3$ cylinder
Four-in-a-row on a $4 \times 4$ cylinder
Three-in-a-row on a $3 \times 3$ Möbius strip
Four-in-a-row on a $4 \times 4$ Möbius strip
Three-in-a-row on a $3 \times 3$ torus
Four-in-a-row on a $4 \times 4$ torus

Notes

In the following discussion, we'll assume X always plays first.

It might not be immediately obvious, but on the torus all grid squares are equivalent. By rotating the torus around its hole or "rotating" the grid square inside the hole to the outside and back, we can massage the torus to move a grid square to take the place of any other. This means there is no best first move, since they are all equivalent.

If the torus is gridded into a $3 \times 3$ board, any two squares form a line with a unique third square. This fact gives X an easy winning strategy:
- First, X plays anywhere.
- After O plays, X then plays on any square except the one on a line with the previously placed X and O.
- Since there are now two X's on a line that O has not played on, O must use their turn to block X. Once these two O's are in play, X must use their next turn to block O.
- Since there are three X's on the board and only one pair has been blocked from forming three-in-a-row, there are two other X-pair lines that must be blocked. Once O chooses to block one, X completes the other for the win.
Interestingly, the $n$-in-a-rows possible on the $n \times n$ cylinder are precisely the same ones as on the $n \times n$ torus, so any strategy that holds on one holds on the other.
On the $3 \times 3$ Möbius strip, the top row and bottom row are actually a single length-$6$ mega-row, and all squares in it are equivalent as a starting move. To win, X should play in this mega-row.
- If O does not play next to this X in the mega-row, then X can play an X next to their own X in the mega-row in a way that leaves XX with a gap on each side. O must block on one side, and X wins on the other.
- If O responds by placing an O in the mega-row right next to the X, then X places an X on the other side of the initial X. There are now two squares that can finish X's three-in-a-row. When O blocks one, X wins in the other.

Extra Challenges

Another couple interesting surfaces to play on are the projective plane and Klein bottle. These cannot be recreated in 3D space without self-intersection, however, so it is easiest to play with them as squares with identified sides. For example, below is a representation of a cylinder:

You can think about it as instructions for creating a cylinder from a sheet of paper by taping the blue edges together. For the Möbius strip, there must be a twist added, which has the net effect of changing the orientation of one of the blue edges before pasting. This is noted by reversing the direction of one of the blue arrows:

The torus looks like below (tape the blue edges to get a cylinder, then tape what have become red circles to get a donut):

The square with boundary pastings below represents a projective plane:

The square with boundary pastings below represents a Klein bottle:

You can use these rectangles to play by noting where the start of a three-in-a-row leaves through an edge and look for the twin oriented arrow to see where it would have to re-enter.
Since the net effect of turning the Tic Tac Toe board into a surface like a cylinder or Möbius strip is increasing the number of ways to get $n$-in-a-row, it in some sense makes it easier for the game to not end in a tie. In particular, on the $3 \times 3$ boards, the first player has winning strategies as opposed to the usual tie expected in standard Tic Tac Toe. To spice things up, you could play a misère version of the game: the loser is the first player to make $n$-in-a-row. How does this change gameplay on a standard $3 \times 3$ board?