Have you ever heard of Piet Mondrian? He was a Dutch artist famous for his abstract paintings made up of rectangles. Inspired by his work, mathematicians came up with a fun puzzle that challenges us to cover a square canvas with unique rectangles. Let’s dive into this puzzle and see how it works!
The goal is to cover a square canvas completely with rectangles that don’t overlap. Each rectangle must be unique, meaning if you use a 1×4 rectangle, you can’t use a 4×1 rectangle elsewhere. However, a 2×2 rectangle would be fine since it’s different.
Let’s start with a 4×4 canvas. If you try to split it directly in half, you’ll get two identical 2×4 rectangles, which isn’t allowed. Instead, you could divide it into a 3×4 rectangle and a 1×4 rectangle. This works, but there’s more to the puzzle!
Now, take the area of the largest rectangle and subtract the area of the smallest rectangle. This difference is your score, and the goal is to get the lowest score possible. In this case, the largest area is 12 (3×4) and the smallest is 4 (1×4), giving us a score of 8. We can do better!
Let’s keep the 1×4 rectangle and split the 3×4 into a 3×3 and a 3×1. Now, the largest area is 9 (3×3) and the smallest is 3 (3×1), resulting in a score of 6. That’s an improvement!
What happens when we try a bigger canvas, like 8×8? Start by dividing it roughly in two, giving you a 5×8 rectangle (area 40) and a 3×8 rectangle (area 24), which results in a score of 16. Not great!
Try splitting the 5×8 into a 5×5 and a 5×3. This gives a score of 10, which is better but still not ideal. The trick is to keep the areas of the rectangles close together to minimize the score.
The total area of the 8×8 canvas is 64. We need rectangles whose areas add up to 64. Let’s list possible rectangles and their areas. We want a range of areas that spans 9 or less to get a low score.
Some rectangles won’t fit, like 1×13 or 2×9. If you use a rectangle with an odd area like 5, 9, or 15, you’ll need another odd-area rectangle to make an even sum.
After some trial and error, using rectangles in the 8 to 14 range (excluding 3×3) fits perfectly, resulting in a score of 6. This is the best score you can achieve!
There’s no magic formula for this puzzle—it’s a mix of logic and creativity. Even expert mathematicians aren’t sure if they’ve found the lowest scores for larger grids. So, why not give it a try? How would you divide a 4×4, 10×10, or 32×32 canvas? Experiment and see what scores you can achieve!
Grab some graph paper and draw a 4×4 grid. Your challenge is to fill this grid with unique rectangles, just like the Mondrian Squares puzzle. Remember, each rectangle must be different in size. Once you’ve filled the grid, calculate your score by subtracting the area of the smallest rectangle from the largest. Try to get the lowest score possible!
Using colored paper or cardboard, cut out rectangles of different sizes. Arrange them on a table to cover an 8×8 square area. Experiment with different configurations to minimize the score, just like in the puzzle. Discuss with your classmates which configurations work best and why.
Inspired by Piet Mondrian, create an abstract art piece using rectangles. Use colored pencils or markers to fill in the rectangles on a blank canvas. As you work, think about how the sizes and colors of the rectangles affect the overall composition. Share your artwork with the class and explain your design choices.
Visit an online puzzle website that features Mondrian Squares or similar rectangle puzzles. Challenge yourself to solve puzzles of increasing difficulty. Keep track of your scores and see if you can improve over time. Discuss strategies with your classmates to find the most efficient ways to solve the puzzles.
Form small groups and compete to solve a Mondrian Squares puzzle on a larger canvas, such as 10×10. Each group should try to achieve the lowest score possible. Afterward, present your solutions to the class and explain your strategies. Celebrate the group with the best score!
Here’s a sanitized version of the provided YouTube transcript:
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Dutch artist Piet Mondrian’s abstract rectangular paintings inspired mathematicians to create a two-fold challenge. First, we must completely cover a square canvas with non-overlapping rectangles. All rectangles must be unique; for example, if we use a 1×4 rectangle, we can’t use a 4×1 in another spot, but a 2×2 rectangle would be acceptable.
Let’s try this with a canvas measuring 4×4. We can’t chop it directly in half, as that would give us identical rectangles of 2×4. However, the next closest option—3×4 and 1×4—works. That was easy, but we’re not done yet. Now, take the area of the largest rectangle and subtract the area of the smallest. The result is our score, and the goal is to achieve the lowest score possible. Here, the largest area is 12 and the smallest is 4, giving us a score of 8. Since we didn’t aim for a low score that time, we can probably do better.
Let’s keep our 1×4 while breaking the 3×4 into a 3×3 and a 3×1. Now our score is 9 minus 3, or 6. Still not optimal, but better. With such a small canvas, there are only a few options. But let’s see what happens when the canvas gets bigger. Try out an 8×8; what’s the lowest score you can get?
To get our bearings, we can start as before by dividing the canvas roughly in two. That gives us a 5×8 rectangle with an area of 40 and a 3×8 with an area of 24, resulting in a score of 16. That’s not great. Dividing that 5×8 into a 5×5 and a 5×3 leaves us with a score of 10. Better, but still not ideal.
We could keep dividing the biggest rectangle, but that would lead to increasingly tiny rectangles, which would increase the range between the largest and smallest. What we really want is for all our rectangles to fall within a small range of area values. Since the total area of the canvas is 64, the areas need to add up to that.
Let’s make a list of possible rectangles and areas. To improve on our previous score, we can try to pick a range of values spanning 9 or less that add up to 64. You’ll notice that some values are left out because rectangles like 1×13 or 2×9 won’t fit on the canvas. You might also realize that if you use one of the rectangles with an odd area like 5, 9, or 15, you need to use another odd-value rectangle to get an even sum.
With that in mind, let’s see what works. Starting with an area of 20 or more puts us over the limit too quickly. However, we can reach 64 using rectangles in the 14-18 range, leaving out 15. Unfortunately, there’s no way to make them fit. Using the 2×7 leaves a gap that can only be filled by a rectangle with a width of 1.
Going lower, the next range that works is 8 to 14, leaving out the 3×3 square. This time, the pieces fit, resulting in a score of 6. Can we do even better? No. We can achieve the same score by replacing the 2×7 and 1×8 with a 3×3, 1×7, and 1×6. However, if we go any lower down the list, the numbers become so small that we’d need a wider range of sizes to cover the canvas, which would increase the score.
There’s no trick or formula here—just a bit of intuition. It’s more art than science. For larger grids, expert mathematicians aren’t sure whether they’ve found the lowest possible scores. So how would you divide a 4×4, 10×10, or 32×32 canvas? Give it a try and share your results in the comments.
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This version maintains the original content’s essence while ensuring clarity and readability.
Canvas – A surface on which artists create paintings or drawings. – The artist used a large canvas to paint a geometric design featuring various shapes.
Rectangles – Four-sided shapes with opposite sides that are equal and all angles that are right angles. – In math class, we learned how to calculate the perimeter of rectangles by adding the lengths of all four sides.
Area – The amount of space inside the boundary of a flat object, such as a rectangle or circle. – To find the area of a rectangle, multiply its length by its width.
Score – A number that represents the result of a test or an evaluation. – In the art competition, each painting was given a score based on creativity and technique.
Unique – Being the only one of its kind; unlike anything else. – The artist’s unique style made her paintings stand out in the gallery.
Overlap – To extend over and cover a part of something else. – In the Venn diagram, the overlap between the two circles represents the elements they have in common.
Divide – To separate into parts or groups. – In geometry, we learned how to divide a circle into equal sections using a protractor.
Fit – To be the right size or shape for something. – The puzzle pieces fit together perfectly to form a complete picture.
Logic – A systematic way of thinking that helps solve problems and make decisions. – Using logic, we can solve complex math problems by breaking them down into simpler steps.
Creativity – The use of imagination or original ideas to create something. – Creativity is important in both art and math, as it helps us find new ways to express ideas and solve problems.
