Mondrian Art Puzzles

People who gaze upon a piece of Mondrian art should ask how he decided which colors to paint where.  The disgruntled person may even ask what the piece of artwork might have looked like if Mondrian had taken the time to color it all in…

In these puzzles you’ll be Mondrian’s nasty mathematical boss.  Instead of allowing Mondrian to randomly draw rectangles and colors - you lay out precise requirements...

Before we start... here is a real section of a piece of Mondrian art. You should google him to see more, because the art work that we produce is only going to be a fun imitation of the real thing.

Mondrian must cover the canvass with rectangles.

Here he has been successful, but it is not a very good solution. Why?

Mondrian's score is equal to the area of the largest rectangle minus the area of the smallest rectangle. This score must be made as small as possible. Here it is 42 - 7 = 35 which is much too large.

This is a better solution. Here Mondrian's score is 30 - 7 = 23 which is still too large, but much better than before.

This is still better a solution. Here Mondrian's score is 25 - 7 = 18.

This might look like the best solution yet, but it is not. It violates the second rule:

No two rectangles can have the same dimensions. Here we have a 3x4 and a 4x3 rectangle. That's a disaster.

PS. The 2x6 rectangle is different from the 3x4 rectangle, so those two can co-habit the same solution.

When you are introducing the Mondrian Art Puzzles, do not start with the rules. Instead start with asking one student after another to determine the dimension of one of the tiling rectangles.

This technique is true in all MathPickle puzzles. The first thing to do is focus on engagement. You do this by having students participate in a ridiculous attempt at solving a problem for which they have not even been told the rules.

The second advantage of this technique is it gives the class practice failing.

This is the first puzzle-sheet. All students should start with the 4x4. Students may not use erasers. They have three attempts to try to find the lowest possible score.

Danger: Many teachers allow fast students to streak ahead. This can be demotivating to slow, methodical thinkers. To avoid this in your classroom, allow slower students - after they complete the 4x4 - to jump up to a larger square than anyone else is working on.

The last rule is that the solutions must be colored using the fewest number of colors possible so that no two rectangles of the same color touch along an edge or vertex.

This aesthetic puzzle-within-a-puzzle increases engagement levels for those students who appreciate beauty. If any of these students wants to redo their messy sheet - give them a new sheet to let them create something of beauty.

 Reward answers that are most beautiful as well as having the lowest score.

There are additional puzzle-sheets in this pdf.

Mondrian Art Puzzles

(MathPickle, 2015)

Mondrian Art Puzzles are the most successful MathPickle puzzle yet. To see Brady Haran’s Numberphile video, more pictures of Mondrian in the classroom, and optimal solutions click here.

As well as practicing multiplication, students in higher grades should tackle generalized results using algebra.




Anastasia–a grade four student in Buckalew Elementary in The Woodlands, Texas discovered this solution for the 7×7 canvas. The best I found scored “6” so she beat me by one 😉

Thanks to Anastasia’s teacher, Kelly Reynolds, for sharing!

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 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, etc.

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)