From: Eric Burton (brilanon@gmail.com)
Date: Sun Mar 11 2012 - 15:37:14 MDT
Just this prints something not draws it. I don't know if it's as good
as that thing the other day where you divide 1 by something and it
prints every real number up to 100 or something buffered by zeroes.
But this is pretty good. I think I found it on shrooms. I forget how.
Bear with me. I try some things in this
math one
Check out what this is. If you try to print powers of two this particular
wrong way, in order to represent them using fewer digits, it prints powers
of two instead, in certain bases, including ours. It prints powers of two
you can't represent with that rule because it becomes ambiguous with more
than two digits per number. You need a high and a low one. But I've tried to
extend it here too. Which took some time. Darn. Please continue. Brix'll be
shat.
if you mistakenly (ha, ha) represent the powers of two that have two digits
by subtracting the low one from the high one until the rule breaks at 128
which has three digits, ah, you see,
OK I just realized you can extend the rule and ...
look. If you say 16 by going 6-1 = 5, 32 by saying 3-2 = 1, and 64 by saying
6-4 = 2, for some crazy reason, and place them after the powers of 2 before
them, that way, you get:
1 2 4 8 5 1 2
See how it spells 512. One of the first powers of 2 you won't be able to
represent that way since the rule should break down at two digits but I'll
come back to that. Even representing them that way for the powers of two
that are only two digits long you get 1 and 2 again which aren't unique
though 5 is. And I just figured out, you can extend the rule to the
three-digit powers of two, by figuring out which two digits add up to a
number large enough to subtract the remaining digit from, if there isn't
already one digit long enough... the way I did it, I got,
128... 8 - (2 + 1) = 5
256... (2 + 5) - 6 = 1
512... 5 - (1 - 3) = 2
So 512 serves to print itself suddenly. Holy shit! Let's see if you can use
that rule on those numbers another way.
Trying 128 again:
If we add 2 to 8 to get 10 and subtract 1 we get 9. Good, there's another
way. I just happened to find one that did spell 512 again. Well let's try to
do it this way, making the biggest number we can from two digits and
subtracting the smallest one, and see what it spells.
128 : (8 + 2) - 1 = 9
256: (5 + 6) - 2 = 9
512: (5 + 2) - 1 = 6
Wow then it says 996. See that system isn't any good. That's what it's
trying to tell me. Look what happens if you do it in octal, where 8, 16,
and 32 are said 10, 20, and 40. If you try that rule you go 1 - 0, 2 - 0 and
4 - 0 and just get 1, 2, 4 again. It says,
124124
Then you can use the same rule on the higher powers of two, in octal, and
only one way, because all the other digits will be zeroes. So you extend it
and just get
124124124...
From,
1
2
4
10
20
40
100
200
400
...
No good at all. So I figured this out walking around just now, about what it
does in octal and guess what, in binary, every power of two also has only
one high digit, the rest zero, so you can use the rule in that too, where
you get,
111111111
Now maybe that's not saying much
So I got home and thought about other bases of course, what about really
weird numbers of fingers or whatever? In hexadecimal, a power-of-two base
like octal, you can say "128" as "80" but at 256 it's "100" which in this
weird system I made up goes "8", "1". It goes,
124812481248
Because unlike octal it has 8, 80, 800... ok. So,
Seriously what is going on. I got this javascript open that does bases 1
through 20. So let's try the powers of 2 and spell them this way and see
what -that- spells, in each. Hell I'll even do the ones I just did, over
again. If you've forgotten what we're doing,
We're representing powers of 2 by subtracting the lower digits from the
higher one, if we can. Because of the amazing effects on the human mind it
has. Like something is trying to tell us we're doing it wrong if we try to
represent them that way, the numbers. What would happen if you did this to
other numbers than powers of 2? Other series? Well hold. Let's try this first
I would do this in python but I'm too lazy.
Binary: 1 1 1 1 1 1 1 1.....
Trinary [way one]: 1 2 0 0 0 0 0 0 2 0 0 2 2 1
Trinary [way two]: 1 2 0 0 0 1 2 4 8 8 10 10 8 10
Sorry I did the rules on the digits of trinary powers of two but I showed
the decimal equivalents there. Interestingly way one is unchanged since it
didn't print any values over 2, weird. In trinary that's:
Trinary [way one]: 1 2 0 0 0 0 0 0 2 0 0 2 2 1
Trinary [way two]: 1 2 0 0 0 1 2 4 22 22 101 101 22 101
Notes
Trinary: This was weird. I always got zeroes. For instance 32 goes 1012 in
trinary so you go 2 - (1+1+0) and get 0. Then 64 is 2101 so you do the
-same thing- and get 0 again, that's just up to 64 though. If you do 128,
11202, you could either go (2+2)-(1+1), 4-2 and get 2, or you can apparently
go (1+2)-(1+2) and get another 0. So what I did, I figured if you spell 32
as 2-2=0 and 64 as 3-1=2 although they have the same digits and it'd be
arbitrary do so, well golly you can get a "2" out of 11202 too, and if you
add the 1's in 100111, trinary for 256, and subtract the zeroes, you get a
4... but you could also subtract the ones from each other and get another 0
that's what I did in way one.
So this is cool, 512 goes 20022 where even the first rule cannot make another
zero, it can only make a 2, unless you subtract the twos "by each other" which
look there's an odd number of them you can't, haha, maybe that is a joke about
that. -goes cross-eyes, throws up an alien-. So otherwise you add up the twos
and subtract the zeroes and get yes the next power of 2, grrr, and now I
really want to know if it's going to print 16 next or say something else so
let's continue. 1024=1101221. Both rules are ambiguous now I think, I mean,
one was just to get zeroes and one was just to get powers of two, so probably
with this many digits there are new rules we can try on these to get other
kinds of numbers. But ah, I think rule 1 might go (2+2)-(1+1+1+1) and get
another 0. Weird that on 512 it couldn't. Rule 2 though adds em and
subtracts the zero by now and we get oh my god, another 8. Look it doesn't
even try to say sixteen. 2048? 2210212. Rule 1 can go 5-5 for another 0.
OK... rule 2 goes 5+5, gets 10. Well. These rules are to avoid saying any
2-digit numbers. But what will that -do-?
God I'm still on base 3. 4096 = 12121201 in base 3. By rule 1 we can say
(1+2+1+2)-(1+2+1) and get another 2. By some other damn old rule we could
say ie (2+2+2)-(1+1+1+1) and still get 2, ok. Or (2+2+2+1)-(1+1+1) and get
4. Or (2+2+2+1+1+1)-1 and get 8, another power of two. OOPS
But you cannot say 0. So rule 1 gets another 2 instead, three zeroes after
the last one. So who cares? I'm really interested in what rule 2 will say
after that "10". 1+2+1+2+1+2+1-0 = 10. No it says 10 again. Ok.
8192 = 102020102 . Rule 1 can't make a zero the lowest number it can make is
another 2. Rule 2 adds em all up and we get ... an 8. 16384 = 211110211.
Rule 1 goes "5-4" and gets a 1. Rule 2 goes 2+1+1+1+1+2+1+1, another 10. 101
in trinary. Just realized I'm looking at these in decimal, "wrongly". I'll
make a copy of the trinary chart showing it in that too. OK
I've decided it doesn't spell anything in trinary. I'll try it in base 5
then give up for now
Base 5: 1 2 4 [4/2] [4/2] [4/2/0] [8/4/0] [4/2] [4/2/0] [8/4/0] [8/6/4/2/0]
Wow that was pretty cool. Look how the sums of the digits all different
ways, adding and subtracting them from each other, are always powers of 2
somehow. I mingled the digits every way I could think of up to 1024. 0 is
the only non power of two you see til you get that 6 which you get by adding
1 to 4 and 4 to get 9 and removing 3, weirdly. Oh shit I better represent it
in base 5 too. Though I don't think it does say anything, even in that. In
decimal though it says "512" (!) so it should say something in base 5 too.
Here it is again...
Base 5: 1 2 4 [4/2] [4/2] [4/2/0] [13/4/0] [4/2] [4/2/0] [13/4/0] [13/6/4/2/0]
Dunno. OK. So for base 5 we could just take the low ones and say
12422002000. That doesn't spell anything. The next one the lowest number you
can get is 2 not 0 so now we can see 20020002. Maybe it continues 00002
0000002. Then it's kind of funny that it starts with 22. Ah. WE DON'T KNOW.
Spelling it with the high ones it's
1 2 4 4 4 4 8 4 4 8 8
Eh
I've probably bored you all. The juicy bit is at the top, what this does in
decimal (especially), binary, octal and hex. Well. OK. I will try base 20,
though.
So 32 (in base 20) is C1, C - 1 is B. Uh
64 is 34 and becomes 4 - 3 = 1. B1
128 = 68, 8 - 6 = 2. B12
256 = CG... G - C = 4. B124
512 = 15C ... C - (5+1) = 6. B1246. Or if we say C+5, 17, -1, we get 16,
wow. Of course we could add them together to get 18 but why. At least it has
one and eight not a lousy six.
1024 = 2B4... B - (2+4) = 5. B12465. -or- (B+4) - 2 = 13; or (B+2) - 4 = 9,
who cares
2048 = 528... 5+8 - 2 = 11... uh-oh another weird 11. Can't (2+5)-8.
(8+2)-5=3. So you can say eleven or three. Or five or nine.
4096, A4G, ... well... In decimal they seem to go,
Base 20: 1 2 4 8 16 12 1 2 4 [18/16/6] [15/11/3] [15/13/5/9] [31/23/3]
Notice I spelled the "B" 12 up there and the next thing it prints is a "12",
1 and 2. Pretty cute. But I'm not totally sure. Well
Listen. I better go
flamoot
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