Find a coloring of these vertices so that no equilateral triangle can be found that has all three corners the same color. Here we failed because the highlighted triangle has three red corners.

It also fails because this highlighted triangle has all its corners yellow.

Your students first task is to try to solve this even larger triangle by coloring each vertex one of three colors. Try not go forward in this presentation till you give it a try.

Here is one solution.

Here is another solution. I like this one better because it has a kind of rotational symmetry.

Students who complete that first task can get the much more difficult task of solving this larger triangle. Again – try this before proceeding.

Here is an example of me failing. The next slide has my solution.

This took me a long time to find, but I’m sure it is not unique. There may be hundreds or thousands of answers.

The inspiration for this puzzle came directly from Bill Garach. In 2009 he started to ask questions like: Is it possible to color this 17×17 square with four colors so that every little rectangle doesn’t have all four corners the same color.

As you can see any rectangle oriented up-down does not have all four vertices the same color.

There are other interesting questions here. What happens if you are allowed to rotate the rectangle? What happens if the rectangle must be a square?

Bill Gasarch is actually interested in puzzles as “simple” as this attempt at 2-coloring this 5×5 square. Can it be done so that no rectangle has all four vertices the same. This example fails because all four corners are the same.

Three Color Equilateral Triangle

(Inspired by Bill Gasarch, 2009)

Equilateral triangle hunting is a light hearted puzzle that becomes perplexingly difficult as the size of triangle is increased. Download pdf puzzle-sheets here.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)