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How many squares fit in a rectangle?

By David, 10 March 2022 News

A rectangle of coloured squares diminishing in size

Continue this pattern and you can fit infinity squares inside a rectangle!
Image: CSIRO/David Shaw

It might seem like a silly question to ask. After all, it depends on how big the rectangle is, and also the size of the squares you’re packing it with. With a big rectangle and small squares, you could fit a million, a billion or even more. But what if you have infinity squares? Is there a way to fit them into a rectangle that doesn’t go on forever?

Shrinking squares

To tackle this question, we need a few quick rules. First, the squares and rectangle must have sensible side lengths. No zero length sides, or infinite ones, and definitely no negative sides. And when you’re packing the squares, there’s no overlapping squares or going outside the edges of the rectangle.

If you want to fit infinity squares into one rectangle, then your squares can’t all be the same size. No matter how small they are, if you have infinity of them, the total area will be infinity. And you can’t get around it by saying your squares have zero length sides – that’s against the rules!

However, if your squares keep getting smaller, there is a way to keep the area small. For example, you could have a square with ½ length sides, and one with ⅓ length sides, one with ¼ length sides and so on.

If you lined up all these squares, one next to the other, they would stretch forever. But if you calculate the area of each and add them up, you don’t get infinity. In fact, the total area is π2 ÷ 6 – 1, or about 0.645. The area of all those squares added up is less than 1!

Approaching an answer

It turns out that you can fit all those squares into a rectangle. Back in 1968, mathematicians Aram Meir and Leo Moser managed to fit all these fractional squares into a single square that was less than 1 wide.

Ever since, people have been trying to fit these squares into smaller and smaller rectangles. The best result so far is from Marc Paulhus. In his rectangle, only an area of about 0.000,000,000,8 is not covered by squares.

Meanwhile, Australian mathematician Terry Tao has a different approach. He’s been looking at squares that are just a smidge larger than the fractions mentioned above. And he’s found a way to pack them into a rectangle without any gaps at all!

So, is it possible to pack our original squares into a rectangle with no gaps? We still don’t know. But with 2 different approaches getting closer and closer to an answer, we might find out soon!

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