Three Color Triangle Puzzles

(MathPickle 2021)

This logic puzzle is inspired by the brilliant Mathologer video below. The tricks presented above are proved below. 

Playing with these puzzles is good for upper elementary, but younger kids can get a lot of value from the methodical application of the triangle-filling algorithm. The results are beautiful. Even Kindergarten kids can fill a 5-edge triangle given the bottom row.

If you want a bigger question: how many red triangles are possible in a completed 5-edge triangle? Of course you can get zero by filling the bottom row with all blues. Can you find a way to get 1, 2, 3, 4… 15? I don’t think all of these are possible.

You can get printable puzzle-sheets here. They include some blank triangles to create your own puzzles and explore.

Today’s scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.
Nikola Tesla

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)