2025:

1) is a square: 45²

2) is the product of two squares: 9² x 5²

3) is the sum of 3-squares: 40²+ 20²+5²

4) is the sum of cubes of all the single digits: 1³+2³+3³+4³+5³+6³+7³+8³+9³

5) sum of the single digits squared: (1+2+3+4+5+6+7+8+9)²

Some more (thanks to chatgpt-o1)

6) sum of the first 45 odd numbers: 1+3+5+...+89

7) is a Harshad number: https://en.m.wikipedia.org/wiki/Harshad_number

6 is kind of cheating. It's a restatement of 45^2.

3^2 is the sum of the first three odd numbers. 4^2 is the sum of the first four odd numbers. 5^2 is the sum of the first five odd numbers.

Edit: sorry, don't mean to be a pill.

I don't consider it cheating, I bet most of these rules have an internal relation.

They do indeed have an internal relation - they all add up to 2025.

Obviously all the formula will be equivalent to each other. They are, by construction, all restatements of each other.

I guess that means that every number is equivalent to a formula? Is there some sort of metric of how many formula produce the same number?

You’d have to at least exclude subtraction and division (and zero) to not have infinitely many formulas for every number.

I would say that a rule is "cheating" iff it is implied by another rule for any arbitrary N.

I think that it is a nice observation. Some people complain that explaining the formation of a rainbow scientifically makes it lose its "aweness" but I think it even deepens it.

Actually, property 5) trivially implies 1) but also 2), as `(1+2+...+n)² = n²(n+1)²/4` and either n or n+1 must be divisible by 2 hence one of the squares divisible by 4 hence it is a product of squares. But also property 4) as `(1+2+...+n)² = 1³+2³+...+n³` (easy to show by induction).

4 and 5 too

How so? I'm too dumb to see it.

The sum of the first n cubes is always the square of the sum of numbers from 1 to n. For example 1³+2³+3³+4³=(1+2+3+4)².

You can prove it by induction; just expand (n(n+1)/2)² – (n(n-1)/2)², the result is n³.

89 isn't 9^2, 81 is.

Huh? 89 is the 45th odd number.

(just reading wikipedia here, I didn't know about Harshad numbers)

There is no such thing as a Harshad number, there is a _Harshad number in a given base_. All integers between zero and n are n-harshad numbers.

Which is a pity, because apparenty it means the `joy-giver`. I think human kind could use a joy giver year

8) the sum of 2024 + 1 also

Oh I like these two.

How do people find these kinds of things out without idly brute forcing things?

Also, (20 + 25)^2 = 2,025! Happy New Year :)

Python:

    [x**2 for x in range(32,100) if x**2 // 100 + x**2 % 100 == x]
    [2025, 3025, 9801]

This decomposition is especially fun!

Great to know that someone else too keeps track of squares.

At the ages of perfect squares is when we all cross or achieve significant milestones in our lives as children, students, (young)adults, spouses, parents, grandparents, senior citizens of society and so on.

This year being a perfect square, I wish that it will be as much or more special as it was for everyone at those ages.

My youngest is fascinated by squares at the moment. Luckily for him, he is 4 years old, his older brother is 9, while I just turned 36. He will be delighted when I tell him that we are entering 45 squared!

Also, if you add your ages together… 7^2

If you multiply your ages… 36^2

This is the most Hacker News comment I’ve seen today. Well played.

Up there with Putnam and Dropbox.

Thank you for the Putnam; I did not know about it. For anyone else that did not understand this reference; https://news.ycombinator.com/item?id=35015. Legendary.

That's an epic thread. Thank you very much for sharing it :)

And now, looking back 17 years later, I'd say he succeeded. It's the tarsnap founder.

Indeed! And it was so much fun when dhouston popped up in the thread :-)

And "Less space than a Nomad"

Putnam? Of math competition fame?

Yep, specifically this comment: https://news.ycombinator.com/item?id=35083

Totally nothing bad happened in the decade following the last perfect square year in 1936. :')

Well things have already been a tad rough around this square, so if we follow the trend, the next square might turn bad even sooner. So maybe around, I dunno, 2101?

Unless something equivalent happened in 1849, 1764, 1691... I think we're OK :)

1225: ten years earlier, Magna Carta starting to limit monarchs and the seed of individual freedom

1681: eight years later was glorious revolution with a bill of rights, marking individual freedoms

1764: ten years later, beginning of American Revolution and being free of monarchs

1849: ten years-ish later, start of US civil war; was the time of an attempt by the British to end slavery around the world

1936: ten years later, colonial empires were being dismantled, UN established to attempt global cooperation, US in the ascendancy with a seed of ties being established more by economics than military force, great economic upswing lifting people out of poverty (60% in poverty then, 10% now) while the global population blossoms

2035: Majority of the global population in middle class or better, triumph of individuals over technocrats, bureaucrats, and corporatists :)

I love this! Haha, I was hoping someone would do that. :)

2025 = 515 (palindromic in base 20)

* https://www.numbersaplenty.com/2025

And

    (20+25)^(20/(2*5))

as well.

US President number 45 returns, kind of seems like squaring applies.

Donald = Donald E. Knuth? ;-)