## Mondrian Art Puzzles

Piet Mondrian was a Dutch painter who is now considered one of the great artists of the 20th century [1]. Among his other works, some of Mondrian’s art had a unique, geometric style that (no surprise) attracted the eyes and minds of mathematicians. His art looked a little something like this:

From the clashing of two worlds, math and art, the “Mondrian art puzzle” was born: a fun, creative, and colorful math activity built for almost any age!

## The Rules of the Game

The basic rules of Mondrian art puzzles are simple:

- Start with an nxn grid – you pick the size.
- Now, split the grid up into (at least two) any-sized squares and rectangles, as long as you don’t repeat the same sized square or the same sized rectangle twice.
- Note: you can alternate colors however you choose – the colors don’t matter except for aesthetic purposes.

That’s it! Pretty simple, right? But you might be wondering: “What’s the point?” That’s where it gets a little more tricky. The goal is to find the **lowest possible score** of whatever n-sized grid you choose. To “score” your Mondrian art, take the area of the largest rectangle or square and subtract from that the area of the smallest.

Here’s a simple example:

To score this 3×3 Mondrian art, we take the largest area (the blue rectangle) and subtract from that the smallest area (the yellow square). This gives us 6 – 1 = 5.

Cool! But what if we broke this up into a different set of rectangles and squares – could we get a lower score?

Yes, actually, we can! In fact, this second 3×3 square is the only other possible combination of squares and rectangles for a 3×3 Mondrian art puzzle, and it yields the score 4 – 2 = 2.

## Getting Students Started

These puzzles get exponentially more complicated as we increase the size of the board, so starting the kiddos out with a 3×3 or 4×4 and working up slowly to at most an 8×8 board (unless skill level warrants larger ones) will help keep them engaged and interested.

**Giving students a limited amount of tries** (say, 3) for each nxn board and **disallowing erasing** can also keep faster students from racing ahead. This way, they have to think a little harder about their strategy before they start drawing.

Once students are set with their answers, asking them, “How did you find your answer?” or “Did you use any specific strategies to make the score lower?” or “Do you think you could find a lower score?” can help spark interesting discussion.

If the challenge isn’t enough for students, you can also add the extra layer of coloring. Along with finding the lowest possible score, ask the students to find the least possible number of colors that allows them to color their art so that no two same-colored squares/rectangles touch on an edge or vertex.

## Strategy

The actual algorithm for solving Mondrian art puzzles is much too complicated for our purposes, and better explained by Hannes Bassen in his article, “Further Insight into the Mondrian Art Problem,” if you’re curious.

In Math Pickle’s video on “Mondrian Art Puzzles,” Gordon Hamilton demonstrates some of this complexity by noting how the sequence of lowest scores for each nxn Mondrian art puzzle has no discernible pattern, at least to the average person. Here are examples of some of those solutions:

n = 4

min score: 4

n = 5

min score: 4

n = 6

min score: 5

n = 7

min score: 5

n = 8

min score: 6

n = 9

min score: 6

n = 10

min score: 8

n = 11

min score: 6

Notice how the lowest score seems to have an upward trend with n, but then goes back down to 6 at n=11. Although the minimum score does generally increase as n increases, it doesn’t do so strictly, which makes it more difficult for us to predict.

Note: the above pictures aren’t the *only* ways you can set up the board to get these scores!

For example, a 6×6 board can achieve the lowest score of 5 in both of these graphs (and probably more):

If you’re feeling adventurous, see how many different lowest-scoring layouts you can find for a single nxn board!

But I digress – back to strategy. If the algorithm for finding solutions is too complicated, then how can we strategize? Here are a couple of tips that might marginally help:

- In order to find the smallest possible score, we need the biggest area to be as small as possible and the smallest area to be as big as possible – that means we should aim to
**make the range of areas as small as possible**. This may have already been obvious to you, but it helped me find lower scores by visualizing boards with all similar-sized rectangles and squares. - Note that while we can’t have the same sized rectangle twice or the same sized square twice on on board, we
*can*have a rectangle and a square with the same area.**Maximizing the pairs of rectangles and square with the same area**can help shorten the range and get you a lower score!