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