random math/logic puzzle

[youre_a_squib_now]

I've spent way too much time thinking about slosher and at this point I have more questions than answers. (How tall are inklings? What are the physical properties of ink? When am I going to learn fluid dynamics??) I think I might have to cut the math and stick to general observations. Either way, it'll still be a while.

so in the meantime, here's something random my brain thought of. See if you can figure it out.

I created a system that can represent any integer greater than or equal to 2 using only parentheses. The parentheses are always nested properly (meaning that the number of opening and closing parentheses are the same and that the set of parentheses to the left of a given parenthesis always contains at least as many opening parentheses as closing parentheses); incorrectly nested parentheses do not represent a number and are invalid. A single integer may correspond to more than one string of parentheses, but every valid string of parentheses corresponds to exactly one number.

Here are some examples.

10 = ()((()))
23 = ((())(()))
39 = (())(()(()))
73 = ((()())(()))
86 = (()(()()))()
127 = (((((())))))

The goal is to figure out how it works. Show that you have figured it out by converting (())(()((()()(())(()())))) to a regularly-formatted number.

idk how hard this is going to be so I might drop hints and/or more examples periodically, we'll see

 

[missingno]

For a second I thought this was von Neumann ordinals.

Spoiler

My first intuition is that this is a base two system so it perhaps translates to binary in some way. That seems to fit 10, 23, and 39 as they have the same number of parenthesis as base two digits, but then that breaks for 73, 86, and 127 - seven digits in base two, but only six sets of parens.

127 looks like a key here, 2^7-1 and it's just six open six close. If it was seven open seven close I think I'd be on the right track, but evidently I'm not. So if my initial assumption doesn't work, what other pattern must it fit...

 

[youre_a_squib_now]

This should give you something to think about.

Spoiler: another example

I said that it can represent any integer greater than or equal to 2, but depending on how you think about it, it's also possible to write 1. The way you write it is by writing nothing, which is part of why I didn't count it before, but here it is in the format of the previous examples:

1 =

 

[youre_a_squib_now]

Here are some examples.

10 = ()((()))
23 = ((())(()))
39 = (())(()(()))
73 = ((()())(()))
86 = (()(()()))()
127 = (((((())))))

btw, is there anything better to call these than "examples"? It feels like kind of a weird word to use but I can't think of anything else

 

[Aiko.Octo]

I love these kinds of puzzles and this is so fascinating especially since upon initial examination I haven't a clue, but I am avoiding thinking too hard into it because it WILL eat my entire day if I let myself lock into trying to figure it out, lol.

 

[Aiko.Octo]

btw, is there anything better to call these than "examples"? It feels like kind of a weird word to use but I can't think of anything else

I think examples describes it. If you want to give more hints, I'm curious what 9, 11, and 100 look like.

 

[youre_a_squib_now]

I think examples describes it. If you want to give more hints, I'm curious what 9, 11, and 100 look like.

I'll do 101 because it looks cool. It's not based around base 10, if you're wondering.
I'm giving you these, but I still think analyzing the example I gave before will be more helpful.

Spoiler: 9, 11, and 101

9 = (())(())
11 = (((())))
101 = (()(()(())))

Spoiler: all of them so far, in one place

1 =
9 = (())(())
10 = ()((()))
11 = (((())))
23 = ((())(()))
39 = (())(()(()))
73 = ((()())(()))
86 = (()(()()))()
101 = (()(()(())))
127 = (((((())))))

I think this should be enough examples for now. although I might still do more at some point

 

[youre_a_squib_now]

I think examples describes it. If you want to give more hints, I'm curious what 9, 11, and 100 look like.

ok fine here's 100 fr this time

Spoiler: 100

((()))()()((()))

 

[Bennet]

I am so ficking confused on this even tho I am good at math this is so Goddamn confusing

 

[youre_a_squib_now]

I guess it has been a while. Here's a pretty big hint for you guys.

Spoiler: the hint

I don't know if this is a good way to explain this, but the parentheses are commutative. Some numbers can be represented multiple ways with parentheses, but they are always just moved-around versions of each other.

For example, 10 can be written exactly 2 ways: ()((())) and ((()))().
23 can only be written 1 way, which is ((())(())), but 39 can be written 4 ways: (())(()(())), (())((())()), (()(()))(()), and ((())())(())

also, remember that you can work together if you want. You don't have to solve it individually