Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license.

This is a math education question that I've been thinking of when taking and teaching topology.

For a few years now I've had an idea for a board game that could help teach students topology. However, I've had trouble working out the specifics, and wondered if the community would be able to help. Of course, this may be closed for being off topic, but if so, I'll post a link where we could continue the conversation elsewhere for those interested.

Setup: Basically like battleship. The playing board would be squares of transparent paper (or chessboards, go boards, etc.).

1. Players secretly place markers (which are circles) indicating where gardens are (or mines, etc.) on their paper, hidden from the other player.
2. A card is then flipped over selecting a topology.
3. Players then send five agents to the other player's board; the agents are points. If the points are placed directly in the garden, the player gets his opponents garden. If not, each agent has the option of moving (in a topological path) or of setting off a bomb. Bombs explode to form an open set; any agents or gardens caught in the bomb perish (so you may have to sacrifice your agents). Players describe the shape of the bomb, and their opponent tells them if they've hit the hidden gardens.
4. At the end of the round, players gain one point for each garden they possess and lose two points for each agent lost.

Possible topologies include:

-Discrete topology: Bombs can take any shape, but agents cannot move.

-Indiscrete topology: Agents can move anywhere, but the only possible bomb is a total nuke.

-Finite complement topology: Agents can still move anywhere, but bombs can miss the agents.

-Dictionary order: Most interesting if agents aren't allowed to move through each other.

-Product topology: Each direction is one of discrete/indiscrete/finite complement/standard

-Metric topology: In metric topology, we require bombs to be formed of metric balls. Then we have the standard metric, the square metric, etc.

-Torus topology: Identify opposing edges (could do other surfaces, use orientation perhaps)

-Subspace topology: Players place an overlay on their boards marking out a subset (like topologist's sine curve, etc.) and then flip over another topology to combine with the overlay.

Now, I think this could be made more interesting. Possible variants could include that agents don't find gardens when placed in them until they do a "search" which means they can detect gardens in a compact connected set containing them (but the test only detects if there is at least one garden, so if the only compact set in the whole space is the space itself, the test is always positive).

I'm sure you all could think of many improvements and better rules. I think this could really help people learning topology for the first time. I would have loved to had it for my students this last semester. What ideas do you have to incorporate other parts of topology (like connectedness or algebraic topology) and how could scoring and set up be improved? In other words, how could this be made playable with real strategies? I don't want this to end up like Quidditch. Thanks, and happy Boxing Day!

Edit: I forgot to mention, I don't know whether it would be better to do analog (paper and pen) or discrete (pegs on a go board).

This isn't really a direct answer but some related information. I couldn't find it with a quick web search, but I'm fairly sure a topology board game already exists. I certainly remember being disappointed as an undergrad when my algebraic topology lecturer brought up topological tic-tac-toe, a game my friends and I had been playing for a couple of years in less-than-scintillating lectures and I was planning on making into a board game. On the other hand it did give us some new topologies to play with. For a discussion of topological tic-tac-toe see the three posts starting here. Readers interested in this thread may also enjoy Singularity chess, also available as an android app.

I'm not sure I like the subspace topology for this.

I think the torus bit is good, though; perhaps expand that to flipping over two cards, one for the space and one for the topology, where the space is given by an identification diagram, which would yield the cylinder, moebius strip, torus, klein bottle, sphere, and the real projective space on R2 (that I know of) as spaces to topologize, which gives a good variety of objects for your students to become familiar with.

I decided to post this as an answer instead of part of the post since it cluttered the main post and is the kind of answer that would be helpful:

5/25/13 Edit: I realized today that this game can be adapted as a fun reworking of the board game Clue. Instead of rolling a die, each player draws a topology card, then moves according to the topology card (in a continuous path). The doors are too restrictive, so players can move through walls. Besides movement, the game is the same.

-Discrete topology: Player cannot move.

-Indiscrete topology: Player can move anywhere.

-Dictionary order: Can freely move left to right, but not up and down.

-Product topology: Each direction is one of discrete/indiscrete/standard metric. Movement in each direction is as described in other sections. In standard metric, one moves five steps. Some overlap with dictionary order.

-Metric topology: We have the standard metric, the square metric, etc. Players can only move 5 steps at a time (in square metric, they can move in a diagonal for free).

-Torus topology: Players that walk off one side of the board come back on the other. All identifications of the square can be cards as well, with players moving 5 steps. It could also be fine having a sphere card where the entire boundary is identified (so moving onto the boundary, one can step off anywhere else in the boundary).