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12:10 AM
@flawr: Updated the syntax spec!
Q: Simplify a square root

xnorGiven a positive integer n, simplify the square root √n into the form a√b by extracting all square factors. The outputted a,b should be positive integers with n = a^2 * b with b as small as possible. You may output a and b in either order in any reasonable format. You may not omit outputs of 1 ...

12:26 AM
Q: Ninja Assassins - which ninja stays alive?

HacktivatorNinja Assassins - which ninja stays alive some basic information given an int N as the number of ninjas, return the number of the winning ninja. At the end only 1 ninja survives,the question is which one? what was his number?. Each ninja kills the ninja who stands after him. Afterwards, he pas...

12:37 AM
I recently saw in a bookstore a JS book that said "how to code safely in the world's most dangerous language"
if that isn't a turn off I don't know what is
12:49 AM
I'm cringing right now. Watching Bruce almighty
1:13 AM
@CᴏɴᴏʀO'Bʀɪᴇɴ Better than Evan Almighty.
@PhiNotPi true
TBS, I thought Bruce Almighty was very educational about religion and spirituality, and about man's relationship to God. /s
TBS = to be sarcastic
Interesting domain name for sale, 3.cymru
@PhiNotPi I actually consider Bruce Almighty to be more of a Christian movie than most movies that are explicitly (labeled as) Christian.
Just finished it. nice movie
1:29 AM
@El'endiaStarman TBH, it's been a long time since I've seen either of them.
@PhiNotPi I thought you were making a joke
By cringing, I meant, plot-wise.
@CᴏɴᴏʀO'Bʀɪᴇɴ Evan Almighty is a very real movie.
@PhiNotPi I just looked at it
Another interesting domain for sale: s.et I'm not even sure how it's for sale, since according to all sources I can find second-level .et aren't available.
@PhiNotPi how is that a domain for sale has a logo?
1:34 AM
@LeakyNun People like to make logos to promote their domain name. Cool logo = more interest, apparently.
I see
lots of logos there
that could be a potential pop-con: generate distinct 4-letter logos :P
CMC: Write a CMC that is a duplicate of another challenge
1:44 AM
CMC: Write a CMC that is a duplicate of another challenge
CMC: Write a CMC that is a duplicate of another challenge
Okay that's enough. :P
CMC: Write a CMC that is a duplicate of another challenge
12 secs ago, by El'endia Starman
Okay that's enough. :P
VTC as too broad, duplicate, unclear what you're asking
1:45 AM
*smacks Leaky Nun with a salmon*
hey, does this remind anyone of that xkcd what if, "how many possible tweets are there?"
@CᴏɴᴏʀO'Bʀɪᴇɴ 1114112^140
@LeakyNun That's not what Randall calculated.
Link to it: What-If: Twitter
I see
He's trying to calculate English, which is about impossible.
1:47 AM
I think his estimate is about as close as you can get, though.
is there a nice bit hack to convert all non-leading zeroes to ones?
i.e. 10100 -> 11111
> Based on the rates of correct guesses—and rigorous mathematical analysis—Shannon determined that the information content of typical written English was around 1.0 to 1.2 bits per letter. This means that a good compression algorithm should be able to compress ASCII English text—which is eight bits per letter—to about 1/8th of its original size. Indeed, if you use a good file compressor on a .txt ebook, that’s about what you’ll find.
@xnor in which language?
@LeakyNun have you tried doing PE with J? It's wonderful
@CᴏɴᴏʀO'Bʀɪᴇɴ and what is PE?
@LeakyNun I think he's just generally talking about bits
@LeakyNun project euler, sorry
1:48 AM
@LeakyNun python, or anything with C-style bit arithmetic
@CᴏɴᴏʀO'Bʀɪᴇɴ not really
@xnor it involves loops
@xnor basically, round up to the next power of two minus 1?
for(int i=1; 2^i < n; i++){
    n |= n>>i;
@LeakyNun basically, yes
1:51 AM
@LeakyNun it was exactly this conversion I was trying to avoid because it's lengthy in Python
@xnor i*=2 instead of i++ for faster performance
is there a nice criterion for what functions can be computed in bit arithmetic expressions?
maybe something of bounded circuit depth?
@xnor every
can you make a function that makes children
@CᴏɴᴏʀO'Bʀɪᴇɴ If there were, you know someone would stupidly put it inside an infinite loop, and boom, extreme overpopulation!
1:53 AM
@LeakyNun how? I'm assuming only +, -, and, or, flip, and shifts
isn't that china
@CᴏɴᴏʀO'Bʀɪᴇɴ People are functions that make children.
@xnor isn't every operation bitwise in its purest form?
@Geobits I guess the question now is when/if each of us are going to make one of those functions
1:54 AM
@LeakyNun i imagine they can involve conditionals or loops, whereas I'm looking for pure arithmetic
@CᴏɴᴏʀO'Bʀɪᴇɴ Already did mine. Someone else's turn.
@Geobits I'll wait at least a decade before taking my turn
Please do :P
@xnor I see.
A: Sandbox for Proposed Challenges

Cᴏɴᴏʀ O'BʀɪᴇɴLanguage succession answer-chaining string Given two words (two strings of lowercase-only letters separated by a space) as input to a program PX written in language X, output two programs in languages Y and Z such that program PY outputs the first word and program PZ outputs the second word. Yo...

^ any more feedback? upvote if you think it looks okay
2:06 AM
@xnor shit
I just helped you golf your algorithm
@LeakyNun oh man. how do you do it?
lambda n:n&n*2|1Leaky Nun 35 secs ago
precedence abuse at its best
that returns 1 for 5?
oh, right
lambda n:int(bin(n)[3:],2)+1Leaky Nun 23 secs ago
you need to start with a 2*, which gives the same length
2:12 AM
oh, right
and it fails for 1 because int refuses empty strings, even though they are clearly 0
my current solution came from that :)
2:43 AM
@xnor I was wondering if you saw this comment.
@xnor Finite number of operations?
@Sp3000 constant number of operations
@Dennis oh yes, the trick of spreading the +n over all the loops is quite nice
i was playing around trying to see if I could do better
looks like not
i'll edit in your improvement
@Sp3000 yes
@DrGreenEggsandIronMan I'll add it as soon as my internet starts to behave. My SSH connections keep dying...
Is it just me, or did they remove the blue from the PPCG's color scheme?
2:57 AM
@xnor Got nothing :( not even sure if there's a good one for next power of two or leftmost bit
@LegionMammal978 where?
@Sp3000 yes, i suspect there's some invariant they all satisfy that stops you from doing anything too interesting
I still see blue everywhere, unless you're talking about something specific.
@Sp3000 I have a fuzzy idea of a candidate: you can parallelize two operations by putting them in far-off place values separated by sufficiently large gaps of zeroes
Problem then is whether you can, say, combine the two separate results
3:02 AM
@Sp3000 i'm suggesting this as a limitation, that you can't get the next power of two because it would have to do it separately for each "island", which doesn't combine right into result for the combination
Hm k...
May 26 at 11:04, by Leaky Nun
Chat mini-challenge: implement the set theoretic definition of the natural numbers (starting from 0).

You will be given a non-negative integer, and you should return the set-theoretic notation of it.

0 is defined as {} (the empty set), and n+1 = n ∪ {n}.

Therefore, 1 = 0 ∪ {0} = {} ∪ {{}} = {{}}.

2 = 1 ∪ {1} = {{}} ∪ {{{}}} = {{},{{}}}
Should I post this in main?
@LeakyNun it's a dupe, i remember doing it
I'm fairly sure that's a dupe of something
Q: Set Theoretic Arithmetic (+ and *)

LiamSet Theoretic Arithmetic Premise There have been a couple challenges already that involve multiplying without the multiplication operator ( here and here ) and this challenge is in the same vein (most similar to the second link). This challenge, unlike those previous, will use a set theoretic...

3:04 AM
Q: Length of String Using Set Theory

jing3142From Wikipedia Set-theoretic definition of natural numbers The set N of natural numbers is defined as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The first few numbers defined this way are 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{},{{...

Q: Length of String Using Set Theory

jing3142From Wikipedia Set-theoretic definition of natural numbers The set N of natural numbers is defined as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The first few numbers defined this way are 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{},{{...

CMC: Partition function
Reference: oeis.org/A000041
@CᴏɴᴏʀO'Bʀɪᴇɴ C_BU
Off by one error :P
@Sp3000 Oops, C_B+U
@CᴏɴᴏʀO'Bʀɪᴇɴ Jelly: E
3:22 AM
@Dennis Thanks! I really appreciate it!
@Dr.ZOMBOOS welcome back
@DrGreenEggsandIronMan Btw, your interpreter currently requires you to run it from its directory.
3:23 AM
@Dr.ZOMBOOS join our site! ppcg.SE
joined it!
Oh haha. V has the same problem
@Dr.ZOMBOOS welcome
Next time I put a language on TIO, I'll test it from another directory.
3:28 AM
Not a problem for TIO (I have to write a wrapper anyway), but it would be nicer in general.
CMC: Giving an array of positive integers and a number n, split the array into different bins so that the sum of each bin is as evenly distributed as possible
In general, what do you think the solution to that is? Is defining a system variable overkill?
@LeakyNun Isn't that the knapsack problem? Or subset sum problem?
Define: "evenly distributed as possible"
@Sp3000 the variance is smallest
Actually: CMC: variance.
3:47 AM
What do you mean? Example plz
given an array of positive integers, compute its variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean, and it informally measures how far a set of (random) numbers are spread out from their mean. The variance has a central role in statistics. It is used in descriptive statistics, statistical inference, hypothesis testing, goodness of fit, Monte Carlo sampling, amongst many others. This makes it a central quantity in numerous fields such as physics, biology, chemistry, economics, and finance. The variance is the square of the standard deviation, the second central moment...
example: [1,2,3,4,5,6] -> 35/12 (or floating point)
4:02 AM
@Quill Hello (even though I don't remember joining the room at that time)
@LeakyNun kdb+, 3 bytes: var
@Dennis alright
Jelly, 8 bytes: S÷L¶_DzÇ
quick question. how can I prove that (a+b) mod n = (a mod n) + (b mod n)?
@CᴏɴᴏʀO'Bʀɪᴇɴ 0 = (3+5) mod 8 =/= 3 mod 8 + 5 mod 8 = 3 + 5 = 8
4:11 AM
You're missing a mod n on the right side.
@zʏᴀʙiɴ101 maybe we should do what CR did and create another room for off-topic chatter and leave this room for code golfing.
@CᴏɴᴏʀO'Bʀɪᴇɴ Let a = q1 n + r1, b = q2 n + r2, then a+b = (q1+q2)n + (r1+r2)
something along this line
nicd idea. thnks
,410 ,742 <-- this is the number of voters who voted for UK to leave EU
4:23 AM
@zʏᴀʙiɴ101 glad you enjoyed it

 Code Golfer's Corner

Discussion specific to golfing code. For general PPCG chatter ...
Please no. It turned out bad
oh ok
There's no non-sitebusiness talk in CR chat anymore without someone saying "let's move it"
or the conversation being like five messages
what's wrong with moving it?
4:32 AM
@LeakyNun I don't have my TI-84 with me, but it'd be like 9 bytes in TI-Basic
@Quill which 9?
@Upgoat then the chat can't hold conversations without being moved
@LeakyNun I wrote similar code a while ago, and I remember it being particularly small
@Quill btw, I want to get v0.3 of Cheddar out but the tests aren't working (seems to be either babel-node or babel-istanbul), is there a way to make istanbul just return 100% so I can merge them?
uh, no. you can't fake test coverage
5:03 AM
I'm not trying to fake it. I'm just trying to manually set it
I'm not trying to do "x', I'm trying to do the letter after "w".
2 hours later…
7:01 AM
A: Sandbox for Proposed Challenges

DerpfacePythonStacks and Stacks and Stacks... Write a program that, with the input as n, finds the first n-gonal and n-gonal pyramid number that is NOT 1. n is guaranteed to be larger than or equal to 3. Examples: n = 3: 10 n = 4: 4900 n = 2: The output can be nothing, False, or anything that you want, ju...

@Upgoat I'm interested, what's the config for the gource command?
it's on the readme.md for gource's wikipedia page
there's not much config needed, unless you wanna speed it up
I mean, how did the gource command looks.
I'm confused ;-;
I'm about this
8:15 AM
@Quill do you like rubber ducks as pictures on the screen? :3
depends whether those rubber ducks have the Code Review logo on them
@Quill This one, sadly, doesn't.
In the meantime, if you still do like rubber ducks not from Code Review as pictures on the screen, there's a giveaway of one.
in Sandbox, 13 hours ago, by zʏᴀʙiɴ101
+-+-+-+-+       0 - NotPunchedYet
|█|█|█|3|       1 - ProgramFOX
+-+-+-+-+       2 - DanPantry
|4|5|6|█|       3 - NotPunchedYet
+-+-+-+-+       4 - NotPunchedYet
|8|█|A|█|       5 - NotPunchedYet
+-+-+-+-+       6 - NotPunchedYet
|█|█|█|█|       7 - FOX9000
+-+-+-+-+       8 - NotPunchedYet
9 - FOX9000     A - NotPunchedYet
B - ConorOBrien C - DanPantry
D - Mast        E - ProgramFOX
F - ConorOBrien
I checked--both Gource and Logstalgia work!
8:47 AM
Q: Brainfuck cat - encode a file as a brainfuck program

Yeow_MengChallenge Implement a program that satisfies the man page below: Name bfcat - even more useless than useless use of cat. Synopsis bfcat <FILE> Description Given a FILE, bfcat writes to standard output a Brainfuck source file. The outputted source file satisfies the following conditions: ...

9:29 AM
@El'endiaStarman If you already have function chaining, you need to make currying a thing + pointless notation a thing! =)
@flawr Do you like rubber ducks as pictures on the screen? =)
I like spaceshuttles way better. I have a relative who is a rubber duck collector.
And he's not on SE? :(
9:37 AM
I don't think so
In the meantime, there's a giveaway of a rubber duck as a picture on the screen.
in Sandbox, 14 hours ago, by zʏᴀʙiɴ101
+-+-+-+-+       0 - NotPunchedYet
|█|█|█|3|       1 - ProgramFOX
+-+-+-+-+       2 - DanPantry
|4|5|6|█|       3 - NotPunchedYet
+-+-+-+-+       4 - NotPunchedYet
|8|█|A|█|       5 - NotPunchedYet
+-+-+-+-+       6 - NotPunchedYet
|█|█|█|█|       7 - FOX9000
+-+-+-+-+       8 - NotPunchedYet
9 - FOX9000     A - NotPunchedYet
B - ConorOBrien C - DanPantry
D - Mast        E - ProgramFOX
F - ConorOBrien
What is this?
in Sandbox, 17 hours ago, by zʏᴀʙiɴ101
I have a rubber duck. I'm going to give it away to the person who guesses the right hexadecimal digit.
Apply in the Sandbox.
(Note: the duck will be provided as a picture on the screen.)
10:06 AM
oh, then I can save it anyways
I'm learning with the new SEDE tutorial. Here's my first tutorial query: data.stackexchange.com/stackoverflow/query/504298/…
This query casually touches WHERE and parameters.
10:22 AM
@zʏᴀʙiɴ101 I don't get it. It seems all digits have already been guessed?
@flawr Nope.
When the punchcard is fully punched, the result will be announced.
A concise edition of my first tutorial query: data.stackexchange.com/stackoverflow/query/504681/…
So, how the query works:
SELECT ==> Top (Cost: 0%) ==> Clustered Index Scan / [PostsWithDeleted].[UIX_PostsWithDe... (Cost: 421994%)
That is the execution plan.
Also, WTF? 421994%? o_O
10:52 AM
The Raspberry Pi magazine is called MagPI.....
@Quill That's a reference to Alex A.!
It's a bad joke from RPi
11:27 AM
Your answer is invalid, as the highest snippet length, 2, is larger than 1. You have to delete every snippet longer than 1. — zyabin101 8 secs ago
Or have a person with rollback delete these snippets.
@Quill do you have rollback? >_>
I don't have rollback privs on Code Golf
my rep is too low
On the same question, I have updated the UGL answer! :D
Is anyone else affected by this? It affects PPCG for me, but I'm not sure if it's site specific or user specific
@cc @LeakyNun
What is 0^i?
11:42 AM
@orlp Surely 0, although I confused myself with powers of i the other day so I will wait for someone who knows what they are talking about...
@orlp Python 3 says ZeroDivisionError
It points out that you can't raise 0 to a negative power (understandable as that would be dividing by zero) but also you can't raise zero to a complex power, which I don't know the reason for
@trichoplax This has always been like this. The host parameter prioritizes the site-specific rooms in the list (i.e. they'll appear first).
Thank you
@mınxomaτ Would you like to post an answer with that, or shall I answer quoting you?
Do it, I'm lazy :P
12:11 PM
I still can't find an intuitive reason for why all complex numbers would give an undefined result (apart from those with imaginary part zero). I might ask on math.SE
(just the corrected version)
@trichoplax The question probably is more on how you define a^b in the first place
So maybe I need to ask if there is a consistent way of extending the definition?
I have no answer yet, but usally a^b is defined via e^(b*log(a))
where e^x = exp(x) = sum x^k / k!
and log(x) is just the inverse power series of exp(x) (inverse wrt composition)
on the other hand if b is rational, you do not have to use exp and log
@zʏᴀʙiɴ101 nice
@orlp a^b is really exp(b*ln(a)), where exp is exponential and ln is natural logarithm
in this case, ln(0) is undefined
if i'm correct, lim(z->0) ln(z) = lim(r->0) ln(r)+i*theta = -infinity (it means no matter which direction you come from, ln(0) approaches -infinity)
12:27 PM
@LeakyNun It does not have to be that way.
@flawr it is defined to be that way
There is actually an interesting read (not exactly what we are looking for) on 0^0 : en.wikipedia.org/wiki/…
at least that is how we extend it to the real numbers and the complex numbers
@LeakyNun Always depends on the context
Different approaches have different definitions
@flawr how does your link contradict me?
12:28 PM
Luckily for practical purposes they agree, but when it comes to pathological cases, they might indeed differ
@flawr I hadn't seen that before. It is beautiful. I like that they give attribution in the description to the math.SE post too.
(I have no idea why Wolfram's trying to take a 2-sided limit there)
@LeakyNun Look at the different justifications on 0^0
@flawr 0^0 is undefined, but you can assign values to it so that it makes sense in other contexts
just like how you assigned 0! to be 1
because it is convenient
12:31 PM
@LeakyNun Well that is just true if you assume the definition you just used
But different fields might consider different values as convenient
@Sp3000 I don't know what (x->i) even means. from which direction are you approaching i? usually we only approach a limit from the left or the right, but you're making it complex
my point is that there might be no "true" convention, but it is always a matter of perspective and a matter of what you defined
0^0 is often defined
@flawr 0^0 is very convenient: 255.wf/FormulaCompiler
@LeakyNun I don't think a two-sided limit is sufficient for the complex plane, hence why I'm confused by Wolfram's result attempt
12:34 PM
@Sp3000 so you tried to break it
what is going on?
It's usually just taken to be 1 by convention.
@mınxomaτ I don't really get it, what is it supposed to do?
@EricTressler Not always
@flawr Read the proof.
12:38 PM
@EricTressler by which convention?
@flawr That's what USUALLY means
@EricTressler As aslways, it heavily depends on the field=)
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: b n = b × ⋯ × b ⏟ n ...
and @LeakyNun conventions don't have names. It's the convention that 0^0 is taken to be 1? I'm a mathematician
@EricTressler I thought it's the convention to leave 0^0 as undefined...
@EricTressler What field are you working in?
12:42 PM
Combinatorics. @LeakyNun If you're doing integrals or probability, and 0^0 shows up, it's going to be in the context of a parametrized function
and you will almost certainly be trying to integrate across that point
@EricTressler I see
@mınxomaτ What proof?
@flawr Click the button where it says "Read the proof"...
Anyway, there are some impassioned arguments for just taking it to be 1. It doesn't matter to me, because I mostly deal with discrete things
The main one is to make the binomial theorem work in all cases
@mınxomaτ A ton of equations, but no text/explanation?
12:45 PM
Try expanding (x + 0)^n by the binomial theorem
@flawr Try actually reading.
Chapter 2, Claim 1 is the basis for formulas constructed by the Formula Compiler.
> This follows from the fact 0^0 = 1 while 0^n = 0 for any n > 0. Readers
who would prefer not to use this fact (which might feel rather like cheating) are
invited to replace the definition of δ using any of a number of formulas listed
for sequence A000007 at the OEIS [3].
@LeakyNun The only term that doesn't die is (x choose 0)x^n0^0 = x^n * 0^0
@mınxomaτ Oh I got the wrong link, my browser seems to have trouble downloading that pdf
@LeakyNun So unless you want to put the condition on the binomial theorem that neither a nor b in (a + b)^n is 0, it's more convenient to take it to be 1
@EricTressler alright.
12:55 PM
@mınxomaτ I cannot open that pdf, do you have a mirror?
That is the mirror :D
Try wget-ing it.
Well where can i find the original?
That's no longer available.
@mınxomaτ It seems that this is a linux thing, I'm on windows=/
But it definitely works. So ... get a better browser.
12:58 PM
@flawr I don't know/remember what those are. Explain, please? :)
@mınxomaτ Thank you
Ok, given a BASIC program, thsi compiler writes a mathematical function for it. However I do not really see the point, those functions it does return seem way too complicated?
Is it some kind of obfuscation?
> involving only basic arithmetic and summation notation
now I understand=)
That does indeed make a lot of sense=)
(that sentence should perhaps be included in the main website=)
@flawr It is!
It's the first sentence on the whole site :D
I just browsed your github site
"code size enthusiast" -> "code-golf nerd"
1:25 PM
Not sure if this has been posted yet, but anyone feel like some XSS golf? escape.alf.nu
Solved level zero in 20 chars!
I believe 12 is possible (no cheating by looking at the scoreboards btw, cos people accidentally put their XSS as their username)
I realized about the function eval(), which can alert(1) in 20 chars.
However, level one... well, nope.
They sanitise double quotes o_o
I got 13 chars
@aditsu For level zero?
1:31 PM
Welp, I got... 20 chars.
The "+ disarm is not so golfy.
ah, 12
thanks to a horrible js feature :)
1:34 PM
zyabin101@avista MINGW32 ~/Desktop/ws1a00/ws1a00
$ diff /dev/null README.md
> This is the index repo for ws1a00.
I didn't expect that diff format.
I expected something like that:
@@@ (whatever means 0a1,3 in the git-like diff format) @@@
+This is the index repo for ws1a00.
Hmm 14 for level 1
Hmm I can't solve it, and I can't ask for hints for level one
regex golf was fun
The --git option for diff probably won't work, as it might involve some weird mechanics that check for execution outside of git, parameters other than a/... and b/... and other things.
not sure I'm interested in XSS golf. it has a lot more garbage at the bottom of the screen (not as clean as regex golf)
1:41 PM
yep, 14 :)
I didn't check the diffutils code, so I don't exactly know.
The "garbage" is the messy scoreboard, unfortunately :/ it's also spoilery, so it's a bit annoying
... The -c option seems to work. :)
@Sp3000 Yeah, I saw that. It's the same domain as regex golf, so I assume it's by the same person? It's a weird design decision for a puzzle
That was my assumption too, so yeah it's a bit weird...
Personally I find the puzzles interesting enough and I just ignore the scoreboard anyway, so I'm not too bothered
1:46 PM
Q: Teach GabeN the number 3 using code Golf!

Matthew RohGabeN is the CEO of Valve. He wants to make Half life 3, But he can't. He just can't count to 3.Now You will attempt to teach him the number 3, by doing Code Golf! What you should do: Input a list of integers between -10000 to 10000. if you can make 3 using all the numbers in the list and th...

2:06 PM
A: Sandbox for Proposed Challenges

Matthew RohGabeN is the CEO of Valve. He wants to make Half life 3, But he can't. He just can't count to 3.Now You will attempt to teach him the number 3, by doing Code Golf! What you should do: Input a list of integers between -10000 to 10000. if you can make 3 using all the numbers in the list and th...

2:28 PM
lulz ^
@aditsu I thought it was going to be the subset sum problem in disguise, but it's much less interesting
oh I didn't read it
2:43 PM
@flawr I like that the numbers kinda stay in the same place relative to each other, compared to the original notation

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