Risky Business -- board game math lessons

[I'm going to be doing a math games retrospective for the next few weeks mixing new posts with reprints of some old favorites. The following first appeared in 2012.]




"Let's play Twister; let's play Risk."

No, seriously, let's play Risk, or at least talk about playing Risk. In case you're not familiar with the game, here's the Wikipedia summary

Risk is a turn-based game for two to six players. The standard version is played on a board depicting a political map of the Earth, divided into forty-two territories, which are grouped into six continents. The primary object of the game is "world domination," or "to occupy every territory on the board and in so doing, eliminate all other players."[1] Players control armies with which they attempt to capture territories from other players, with results determined by dice rolls.

(invented by the director of the Red Balloon -- who knew?)

Risk can be a good taking off point for a number of lessons and assignments like:

1. Strategic thinking -- break the class into groups, have them write up rules and recommendations for the game then test these ideas in a tournament

2. (not math but why should that stop you?) -- what modern countries and provinces correspond to the regions on the board? What are they like? You might add naming these countries as a condition for conquering a region.

3. Probability -- what are the chances of taking a country with j attackers and k defenders? What's the expected strength of j if there is a conquest, k if the attack is repulsed?

But the main lesson I'd like to suggest is an introduction to graphs. We have other graph based games and puzzles in our tool chest such as doublets and the six degrees game but Risk is probably the most familiar to the kids. More importantly, it requires actually working with graphs as part of a larger problem.

After the students are acquainted with the game, explain the basic terms of graph theory then show them something like this subgraph of North America

Ask the students to do the following (After checking to make sure I got this right) :

1. Fill in the other countries

2. Explain what's special about nodes like Alaska and Greenland

3. Draw subgraphs of each continent

4. Find the shortest path between various pairs of countries
  4b Find the shortest path between various pairs of countries when certain territories (particularly Russia) are blocked
  4c (advanced) With randomly placed armies, see which path would be easiest to conquer

5. Play a game on a node and edge board

6. Make up new node and edge boards. Give the territories real or mythical names. Try playing a few games on them.