Prejudiced Polygons

by Spencer Bowen

This week’s activity features a neighborhood populated by Squares and Triangles that are slightly “shapist” (but only slightly) and don’t seem to coexist too well.  We’ll be exploring “Parable of the Polygons” by Vi Hart and Nicky Case and giving some tips to get students started with this activity.

The Activity

Triangles and Squares live together in neighborhoods.  However, the Polygons all believe two things:

“I am unhappy if fewer than 1/3 of my immediate neighbors are like me.”

“I am unhappy if I have no immediate neighbors.”

Prejudiced Polygons 2.0 (4)
Image from Dr. Anne Ho

In Example 1, the middle Square has 6 neighbors. This Square is upset because only 1/6 of their immediate neighbors are similar, and 1/6 < 1/3.


In Example 2, the bottom-right Square is upset because they have no immediate neighbors.

The goal of the activity is place all your Square’s and Triangles so every shape is happy. 

Lesson Implementation

The challenge starts off easy enough, with Squares and Triangles on 3×3 and 4×4 grids.  This serve as an introduction to the rules and allowing some fraction work by making sure everyone is happy.

Once you get to placing 12 Squares and 7 Triangles on the 5×5, you may want to discuss strategy.

A Math Monday student's first attempt at solving the problem.

A common first solution to the problem is completely segregating the shapes, entirely sidestepping every Polygon’s biases.  Similarly, Polygon’s can be grouped so that even though some Triangles and Squares are neighbors, the Triangles and Squares form pockets that are entirely homogenous.  This phenomenon can be described to students as “an imaginary wall separating the Polygons.”

The next goal of this activity is to see if it’s possible to make the Polygon’s neighborhood more heterogenous while making sure everyone’s happy.  A possible way to word this question is:  “Can we place our Polygons so as many Triangles as possible are neighbors with Squares and vice versa?”  This is a much harder challenge and requires strategic thinking.

Watch students attempting to solve the problem during Math Monday Live.

The Parable of the polygons

The questions so far have tasked students with satisfying every Polygon’s biases without much discussion about whether this is a good or bad thing.  Should the shapes be trying to segregate?  Furthermore, students have total control of the neighborhood and can move any of the shapes around at will; and that’s not how real neighborhoods work.  

The true “Parable of the Polygons” and ties to real world social phenomenon come once students have no control over the Polygons at all!  On their activity page, Vi Hart and Nicky Case have an option for Polygons to move on their own.  In these simulations, unhappy Polygons will randomly move around the neighborhood.  This will go on forever unless the neighborhood reaches a point where every Polygon is happy.  Simulation also keep track of how many Polygons become entirely segregated from the other type of shape after every random move.

These are good shapes, nice shapes. And yet, though every individual only has a slight bias, the entire shape society cracks and splits.

These simulations show how rather minor biases can lead to large societal problems.  Students can control the simulation and play with the shapes biases. The problems had shapes moving if less than 1/3 of their neighbors were the same type of shape as them, but what if they wanted only 10% same shape neighbors? What about if they wanted 90% same shape neighbors?

For more of look into the real-world implications of societal biases on community segregation, mathematician Dr. Anne Ho and sociologist Dr. Jaime McCauley wrote an article about this activity last year. You can view it on our blog here.

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