The students started with 3x3 grids and found that the maxagon had 7 sides. When they used 3x4 grids, they found that the maxagon had 11 sides. Jack and Huon suggested a “rule†which would say how many sides the maxagon had, and Brendan used the rule to predict that a maxagon on a 3x5 grid would have 13 sides. Just at that moment, Cate, Ruby and Kristina finished their work on the 3x5 grid and showed us their maxagon and it did indeed have 13 sides. Everybody was very excited to see that it fitted the rule.
After some exploration, the students decided that the quickest way of making maxagons was by using a repeat unit as many times as possible (sometimes reflecting or rotating it) to fill up the grid.
Khubaib and Brendan drew an example of a very large maxagon, and explained how the number of sides grows by adding a repeat unit. They said :
This shape is always added to the next shape. It has four sides, but since it replaces the red side, the number of new sides becomes 3.
Fergus noticed that the pattern for maxagons in a grid which was 3 dots high could be described in two ways. The number of sides on the maxagon was the same as if you multiplied the number of dots which were on the sides of the grid and then subtracted 2. It was also the same as if you multiplied the number of dots on the sides of the next smallest grid and then added 1.
He said :
The rule is if you do a 3xn with dots, [then] to make as many sides possible it's the 3xn you are doing but minus 2, or the previous 3xn but plus 1.
And he gave an example :
3 x 3 = 9 [and] minus 2 is 7 [which is] as many as you can do.
Or 3 x 2 = 6 [and] plus 1 = 7 which once again is as many as you can do.
Cate said that the number of sides of the maxagon was (3 x n) - 2 and drew them up to a 3x7 grid.
This reminded Alex of a debate in our town over whether or not to build a light-rail system. He explained that constructing a maxagon is like building a train of carriages, with a fixed beginning and ending, but some number of repeating units in between. He drew a picture to illustrate.