This is an introduction to some ideas of combinatorial group theory. Writing a presentation for a group with generators and relations is like creating a new language with an alphabet and rules for determining equivalent words. This activity has close connections to tiling problems (in fact there is a very elegant description of tiling problems via combinatorial group theory), to braid theory and knot theory, and to geometric constrictions via Cayley graphs. Circle participants will need some explanation before they will be able to understand what is happening in this activity, but anyone who takes the time to understand it will be rewarded with beautiful mathematics and interesting problems. The AAH = HAA rule can be presented as a game. Players take turns trying to move all of the As to the left.
This activity should be paired with the Tile Activity.
This activity was included in the University of New England Julia Robinson Mathematics Festival. It was also demonstrated at the 2013 Circle on the Road workshop.
The attached template of n-gons may be printed on old file folders, so participants can construct the models for problems 7 and 8.