[globnomulous]

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.

[fifilura]

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

[roenxi]

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.

[mr_toad]

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?

[layer8]

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

[kdmccormick]

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

[freehorse]

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).

[bonzini]

4 and 5 too

[globnomulous]

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

[bonzini]

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.

[xerox13ster]

89 isn't 9^2, 81 is.

[mkl]

Huh? 89 is the 45th odd number.