Welcome to 2025! A lovely year, filled with excellent things. Obviously, we’re not talking about the state of the planet – that’s pretty terrible, all things considered. But the math? That is delectable.

The spatial design of 2025

One of the most straightforward observations about the upcoming year is that 2025 is a perfect square. Specifically, it can be expressed as 45 multiplied by 45, indicating that if we were to create a large square with each side measuring 45 units, the overall area would amount to 2025 square units. 

That’s not all, though: because it’s an odd square, it’s also a centered octagonal number – which, much like with square numbers, is precisely what it sounds like: it means we can draw a perfect octagon using exactly 2025 pieces.

Let's delve into something even more intricate: 2025 is classified as an enneadecagonal number (for those who aren't familiar with geometric terminology, that's a shape with 19 sides). However, there's a catch – it happens to be a negative enneadecagonal number, specifically the -15th one, which makes it quite impossible to visualize. 

We are confident in its accuracy because every enneadecagonal number can be expressed using the following formula:

N_m = m(17m - 15)/2

and plopping m = -15 into this recipe gives us 2025.

The titles of 2025

Along with square, octagonal, and enneadecagonal, 2025 has a few pretty names. It’s a powerful number: an integer m such that if p|m, then p²|m. The reason for that is fairly simple – it’s 45², which is equal to (3²)²×5² – or in other words, every prime factor of it turns up at least twice.

It’s classified as a refactorable number, or tau number, indicating that it can be divided by the total count of its divisors. For instance, consider the number 18: it has six divisors, which are 1, 2, 3, 6, 9, and 18, and it can be evenly divided by six. In the same vein, the number 2025 possesses 15 divisors, with one of them being 15. To enumerate them all, the complete list of divisors for 2025 includes 1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, and 2025.

Theorems from the year 2025

Let’s get onto the good stuff, shall we? We’ve already seen that 2025 is a perfect square, but dig a little deeper and we see some even prettier patterns. Forty-five, the number’s square root, is also a triangular number, and that means we can write it as a sum of consecutive numbers. Like this:

45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9.

This indicates that 

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)², 

which is nice for sure, but that’s not all. Thanks to Nicomachus, an ancient Greek follower of Pythagoras who lived between around 60 CE and 120 CE, we know that numbers that can be written like this – the squares of triangular numbers – also have another interesting property: they can be rewritten as the sum of the cubes of those same numbers. In other words, because 

2025 = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)²,

we are also aware that 

2025 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³.

Isn't that awesome?! There are several methods to demonstrate this – one of the most elegant is through this proof that doesn't require any words:

An alternative approach involves utilizing the characteristics of square and cube numbers directly. In reality, it is this method that earns old Nicomachus his recognition, rather than the mere observation of the theorem. His namesake result takes a step back and, in a technical sense, states the following:

∀n∈N>0 : n³ = (n² − n+1) + (n² − n+3) + … + (n² + n−1).

That might look… well, like it’s written in another language, and it sort of is, but really it just means that any number n cubed can be written as the sum of n consecutive odd numbers beginning at (n² − n+1). Like this: 

1 等于 1

8 can be expressed as the sum of 3 and 5.

27 can be expressed as the sum of 7, 9, and 11.

64 can be expressed as the sum of the numbers 13, 15, 17, and 19.

125 = 21 + 23 + 25 + 27 + 29

and so forth.

Now, written out like that, you can probably see a nice pattern already, right? If you sum up the first k cubed numbers, you’re going to get

1³ + 2³ + 3³ + … + k³ = 1 + 3 + 5 + 7 + 9 + 11 + … + (k² − k+1) + (k² − k+3) + … + (k² + k−1).

Now, let's examine square numbers. They exhibit a similar pattern and can be expressed in the following manner:

1 等于 1

4 可以表示为 1 加 3。

9 = 1 + 2 + 6

16 can be expressed as the sum of the first four odd numbers: 1 + 3 + 5 + 7.

25 = 1 + 3 + 5 + 7 + 9

and so on – that is, the square of n is equal to the sum of the first n odd numbers. 

But guess what? That sum we found before is precisely that – it’s the sum of the first (k² + k)/2 odd numbers! In other words,

1 + 3 + 5 + 7 + 9 + 11 + … + (k² − k+1) + (k² − k+3) + … + (k² + k−1) = ((k² + k)/2)².

So there’s just one thing left to prove, and that’s that (k² + k)/2 is equal to the sum of the first k natural numbers. Luckily, that’s pretty easy – it’s the definition of a triangular number (or, if you prefer, you can do it visually: 

However you prove it, though, the math doesn’t lie: the sum of (n cubes) equals the (sum of n) squared. And now is as good a time as any to rave about this nice little result, since 2025 proves it perfectly. Happy New Year!