2025 is a perfect square year: What makes that so interesting to mathematicians?

Photo courtesy of the University of Waterloo.

Mathematicians aren’t just ringing in 2025, they’re also celebrating the start of a perfect square year.

“A perfect square is what you get when you take a counting number – 1, 2, 3, 4, etc. – and multiply it by itself,” explained Craig S. Kaplan, a professor in the school of computer science at the University of Waterloo. “The first few perfect squares are 1 [1x1=1], and then 2x2=4, and 3x3=9, 4x4=16, and so on.”

When you multiply 45 by 45, you get 2,025.

“It is a perfect square year and they are arguably interesting because they are relatively rare,” Kaplan added.

The last perfect square year would have been 1936, as 44x44=1,936, and we’ll have to wait until 2116 for the next one.

Kaplan’s work primarily focuses on the interaction between mathematics and art. He also made it onto TIME’s Best Inventions of 2023 list for helping discover an “einstein” – a mathematical problem that was considered impossible for more than 60 years.

Craig Kaplan holds "the hat." (Spencer Turcotte/CTV Kitchener)

As for 2025, the number has other special qualities too.

“We know that it’s 45x45, but what’s cool about the number 45 is that 45 is equal to 1+2+3+4+5+6+7+8+9,” Kaplan explained. “If you add up all the numbers from 1 to 9, you get 45. 45 is sometimes called a triangular number, or a pyramid number.”

He said if you drew 1 circle, following by 2 underneath and 3 below that, all the way to 9 circles, the number of circles would total 45.

“When is the next time we’ll get a number like that? Well, the next number you have to add, after adding up the numbers from 1 to 9, would be 10. That would give you 55, and 55x 55=3025.”

The last one of those before 2025 would have been 1296.

But that’s not all.

“Any number that is the square of a triangular number is also a sum of cubes. Take 1 to the power of 3, which is 1x1x1=1. 2 to the power of 3, or 2x2x2=8. 3 to the power of 3, or 3x3x3=27,” Kaplan explained. “If you add up a bunch of consecutive cubes like that, you get the square of the corresponding triangular number. So, 1 to the power of 3, plus 2 to the power of 3, plus 3 to the power of 3… up to 9 to the power of 3, also adds up to 2025.”

He then did a bit more math.

“Take the numbers 1 to 9. Let’s divide them up into two groups. The first group I’m going to put 5, 6 and 9. What does that add up to? 5+6=11, plus 9 is 20. And then all of the other ones, 1, 2, 3, 4, 7, 8. Those add up to 25. You end up with the result… 2025 [which is] is 20+25, all squared.”

Kaplan admits its all a bit arbitrary.

“But it gives mathematicians something interesting to talk about because every year is a different number with its own weird set of properties.”

There’s another year that has also piqued his interest.

“One that we should really be looking out for is the year 2048, which is a power of 2,” Kaplan explained. “Those come along very rarely. The next one after that is 4096.”