Weapons of Maths Consumption

My name is Brendan Florio, and for this post I will be talking about a particularly fun activity for demonstration purposes, which has some hidden mathematics.

Mathematics can be a tough sell to high school students. It doesn’t have the colours, flames and explosions of chemistry nor the funky mechanical toys of physics or engineering. At university open days or other similar demonstrations aimed at high school students and younger, we mathematicians need to work harder to find activities that are both engaging and fun. In this post I will detail an activity which I found to be very entertaining and deceptively mathematical: a topological twist on the old Dots and Boxes game, which I first encountered at an open day of the University of Western Australia.

The standard rules of Dots and Boxes are as follows:

• A rectangular grid of dots is created with an arbitrary size. In the following examples, we will use 4×4 dots,
• Turn by turn, players must draw a vertical or horizontal line between two adjacent dots,
• If any player completes a square of four lines, they claim that square by writing their initial inside. They must then draw an additional line which may also complete a square prompting further compulsory additional lines,
• After all possible lines have been drawn, the number of squares claimed by each player is counted. The player with the greatest number of squares wins.

In the example below, the Red player draws a line first, then it is the Blue player’s turn.

Blue wins this game, with 6 squares.

As an interesting variant, we can use the same grid but make it so that their are no boundaries; the edges are “wrapped around” to each other. That is, any line drawn on the upper edge must be replicated on the lower edge, and any line drawn on the left edge must be replicated on the right edge. This is similar to how Pac-Man could navigate by leaving the screen on the left-hand side to appear on the right. To avoid confusion, it is useful to label the edge lines so that it is easy to see where the replicated line should be drawn, as I have done in the example below. If a line is drawn on the edge labelled “A”, then another line must be drawn on the other edge labelled “A”.

The interesting thing for a grid with these rules is that the game would be equivalent to playing Dots and Boxes on the surface of torus. Below is an example of a game. On the left we have the usual 4×4 grid, but with “wrapped” edges. On the right we have the same game being played on the equivalent torus surface.

Red wins with 6 squares.

Another fun surface to play on is the Möbius strip. On our 4×4 grid, the top and bottom edges remain as boundaries while the side boundaries are connected to each other with a twist: The upper edge on the right-hand side is connected to the lower edge on the left-hand side. Below is an example game played on the 4×4 Möbius grid, alongside the same game being played on the equivalent Möbius strip.

Red wins with all 9 squares!

We can even play Dots and Boxes on the mind-bending Klein bottle. This is similar to the torus in that the upper and lower boundaries are connected, but similar to the Möbius strip in that the left and right boundaries are connected but twisted. To achieve this topologically, the Klein bottle must turn itself inside-out! Below is an example game.

Red wins with 6 squares.

When introducing this game to students, you can simply play the Dots and Boxes game on the grid of your choice. Then, if you have pictures of a Torus, Möbius strip or Klein bottle available, you can explain that the grid you were playing on is equivalent to one of these crazy surfaces. This approach means that initially, mathematics is avoided entirely; they are simply introduced to a simple game with simple rules. Then, once they are comfortable with the grid, we can show that the grid can be abstracted to these surfaces by the concept of topological equivalence (perhaps the terminology can be skipped when demonstrating).

There are a few ways this principle can be extended. First, there are other surfaces not mentioned will will also be appropriate for this game. Secondly, there are many other games that can be played on these fun surfaces. Would anyone like to play Tic-Tac-Toe on a torus?