Welcome to 2025 ! A lovely twelvemonth , filled with excellent thing . plain , we ’re not blab out about the country of the planet – that’spretty dire , all thing considered . But the math?Thatis delectable .

The geometry of 2025

countenance ’s start with arguably the simple fact about this new twelvemonth : 2025 is a perfect square . It ’s adequate to 45 × 45 , mean that if we draw a heavy quondam square with side distance 45 units , the full surface area would be 2025 unit squared .

That ’s not all , though : because it ’s an unpaired square , it ’s also a centered octagonal number – which , much like with square act , is precisely what it sounds like : it mean we can draw a perfectoctagonusing exactly 2025 pieces .

We can get even more complex : 2025 is an enneadecagonal number ( for the few among you who are n’t professional shape - name - knowers , that is of course a 19 - sided shape ) . Unfortunately for us , it ’s a negatively charged enneadecagonal number – the -15th – so it ’s a piece unacceptable to draw .

grid of 45 by 45 squares

Go on, count them if you don’t believe us.Image credit: ©IFLScience

We know it ’s veracious , though , because all enneadecagonal numbers are given by the following formula :

Nm = m(17m- 15)/2

and ploppingm= -15 into this recipe gives us 2025 .

A visual proof of the theorem that the sum of consecutive cubes is equal to the square of the sum of those same consecutive numbers.

Neat, huh?Image credit: Cmglee,CC BY-SA 3.0, viaWikimedia Commons

The names of 2025

Along with solid , octagonal , and enneadecagonal , 2025 has a few pretty names . It ’s a knock-down routine : an integermsuch that ifp|m , thenp2|m . The reason for that is fairly simple – it ’s 452 , which is adequate to ( 32)2×52 – or in other intelligence , every prime factor of it become up at least twice .

It ’s also a refactorable identification number , or tau turn , which means it ’s divisible by the number of divisors it has . To take a simple-minded example , intend of 18 : it has six divisor – 1 , 2 , 3 , 6 , 9 , and 18 – and it ’s divisible by six . Similarly , 2025 has 15 factors , and one of them is indeed 15 – the intact listing , for the record , being 1 , 3 , 5 , 9 , 15 , 25 , 27 , 45 , 75 , 81 , 135 , 225 , 405 , 675 , and 2025 .

The theorems of 2025

Let ’s get onto the good stuff , shall we ? We ’ve already see that 2025 is a perfect second power , but compass a little deeper and we see some even prettier patterns . Forty - five , the number ’s straight ascendent , is also a triangular number , andthatmeans we can indite it as a sum of consecutive numbers . Like this :

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

That means that

four rectangles subdivided into squares, ranging from left to right in size: 2 by 1, 3 by 2, 4 by 3, and 5 by 4. In each, half of the squares are shaded in right angled triangular pattern to illustrate triangular numbers

The triangular number – the shaded section – is half of thenx (n+1) rectangle.Image credit: ©IFLScience

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

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

we also bonk that

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

Ai n’t that cool ? ! There ’s a few ways to turn out this – one of the nicest is this proof without words :

Another path is to use the properties of the satisfying and regular hexahedron numbers themselves – in fact , this is where erstwhile Nicomachus actually gets the quotation , rather than noticing the theorem itself . His eponymous upshot is a pace back , and technically say this :

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

That might depend … well , like it ’s write in another language , and it sort of is , but really it just mean that any numberncubed can be publish as the sum ofnconsecutive singular phone number beginning at ( n2−n+1 ) . Like this :

1 = 1

8 = 3 + 5

27 = 7 + 9 + 11

64 = 13 + 15 + 17 + 19

125 = 21 + 23 + 25 + 27 + 29

and so on .

Now , save out like that , you’re able to in all probability see a nice shape already , ripe ? If you sum up up the firstkcubed number , you ’re going to get

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

But now let ’s take a expression at the square numbers . They have a interchangeable sort of practice going on as well – they can be indite like this :

4 = 1 + 3

9 = 1 + 3 + 5

16 = 1 + 3 + 5 + 7

25 = 1 + 3 + 5 + 7 + 9

and so on – that is , the straight ofnis equal to the sum of the firstnodd numbers .

But guess what ? That gist we see before isprecisely that – it ’s the sum of the first ( k2+k)/2 odd figure ! In other wrangle ,

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 rise , and that ’s that ( k2+k)/2 is equal to the sum of the firstknatural identification number . fortuitously , that ’s pretty easy – it ’s the definition of a triangular number ( or , if you prefer , you may do it visually :

However you prove it , though , themathdoesn’t Trygve Halvden Lie : the total of ( ncubes ) equals the ( sum ofn ) squared . And now is as good a time as any to gush about this nice little result , since 2025 proves it perfectly . Happy New Year !