/sci/ - Science & Math

Okay, so what about, we find an approximation of <span class="math">r[/spoiler], if we're lucky we guess the exact form of <span class="math">r[/spoiler] from there, and then we start thinking about the proof?

("Too-lazy-to-think" approach, starting now)

>>4322281
Here's the first few terms of the sequence of indices <span class="math">i[/spoiler] of the sequence <span class="math">(s_i)_{i\geq 0}[/spoiler] from the problem's text such that <span class="math">s_i=2[/spoiler]:
0, 3, 7, 11, 14, 18, 22, 26, 29, 33, 37, 41, 44, 48, 52, 55, 59, 63, 67, 70, 74, 78, 82, 85, 89, 93, 97, 100, 104, 108, 111, 115, 119, 123, 126, 130, 134, 138, 141, 145, 149, 153, 156, 160, 164, 167, 171, 175, 179, 182, 186, 190, 194, 197, 201, 205, 208, 212, 216, 220, 223, 227, 231, 235, 238, 242, 246, 250, 253, 257, 261, 264, 268, 272, 276, 279, 283, 287, 291, 294, 298, 302, 306, 309, 313, 317, 320, 324, 328, 332, 335, 339, 343, 347, 350, 354, 358, 362, 365, 369, 373, 376, 380, 384, 388, 391, 395, 399, 403, 406, 410, 414, 417, 421, 425, 429, 432, 436, 440, 444, 447, 451, 455, 459, 462, 466, 470, 473, 477, 481, 485, 488, 492, 496, 500, 503, 507, 511, 515, 518, 522, 526, 529, 533, 537, 541, 544, 548, 552, 556, 559, 563, 567, 571, 574, 578, 582, 585, 589, 593, 597, 600, 604, 608, 612, 615, 619, 623, 626, 630, 634, 638, 641, 645, 649, 653, 656, 660, 664, 668, 671, 675, 679, 682, 686, 690, 694, 697, 701, 705, 709, 712, 716, 720, 724, 727, 731, 735, 738, 742, 746, 750, 753, 757, 761, 765, 768, 772, 776, 779, 783, 787, 791, 794, 798, 802, 806, 809, 813, 817, 821, 824, 828, 832, 835, 839, 843, 847, 850, 854, 858, 862, 865, 869, 873, 877, 880, 884, 888, 891, 895, 899, 903, 906, 910, 914, 918, 921, 925, 929, 933, 936, 940, 944, 947, 951, 955, 959, 962, 966, 970, 974, 977, 981, 985, 988, 992, 996, ...

>>4322318
From this, <span class="math">r\approx \frac{1}{3.731343}=0.2680000[/spoiler]. Interesting, isn't it?

>("Too-lazy-to-think" approach, starting now)
Fuck, now this doesn't work anymore. Coding is done, thinking is required. Help me, /sci/!

> if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence.

I'm still kinda blown away by that. I'm really not good at math in general. I usually don't even understand these stickies.

>>4322388
The formulation is a bit weird. Basically, you take the first sequence. It starts with a 2. That means the 2nd sequence starts with 2332 (the space between the 2s is "2"). Then there's a 3. That means that after 2332, there are 3 3s before the next 2. So the 2nd sequence is 23323332. Then again a 3, so the sequence is 233233323332. And the sequence defines itself like that. You read it and while reading it, you add stuff at the end.

so all I've really thought of so far is what is perhaps a better way to think about the sequence.
The fundamental building block is 23323332333
Now let a=233, b=2333.
the sequence goes abbabbbabbb...
Similarly, if we let c=abb, d=abbb, then the sequence starts with cddcdddcddd...

>>4322330
Okay, so that post was stupid. First, I'm not talking about <span class="math">r[/spoiler] but about its inverse. And second, I thought my algo had converged but it hadn't, and the precision was bad. Better precision on <span class="math">r[/spoiler] is <span class="math">3.732051[/spoiler], which doesn't look like anything special to me.

>>4322431
Oh and, the fact that it looked rational in the previous post was good because it would have meant the sequence is periodic, and proving that the property is true on a period would have been not only easy (by an algo) but enough to prove the whole result (by periodicity). So I'm back to square <span class="math">\approx 0[/spoiler].

>>4322431
Google knows 3.73205 as an approximation of the "sacred ratio" <span class="math">2+\sqrt{3}[/spoiler] (what the hell is the sacred ratio?). Now hoping that <span class="math">r=2+\sqrt{3}[/spoiler], even though that breaks my proof by periodicity, but because that sounds cute.

2 -> 233
3 -> 2333
and it's the sequence generated from 2 is this sequence.

So the portion of 2s is r, and the portion of 3s is 1-r, and the portion of 2s is also 1/(3r + 4(1-r)) = 1/(4-r) = r, so r^2 - 4 r + 1 = 0, and 2 +- sqrt(3) is the root of that.

Hey, TN5.

>>4322440
It's 2 - sqrt(3)

What if we're given an infinite sequence of 3's and 5's? Would r be <span class="math">3 + \sqrt{5}[/spoiler]?

>>4322720
Hi!
Hmm, <span class="math">r=2+\sqrt{3}[/spoiler] for sure. I don't have any idea about the perfect ratio though if that's what you're talking about.

>>4322720
I mean to say, the proportion of 2s must be 2 - sqrt(3), so the number would have to be 1/(2-sqrt(3)) = 2 + sqrt(3).

<span class="math">(\[Pi] (2 x-1)-I log(1-E^(-2 I \[Pi] x))+I log(1-E^(2 I \[Pi] x)))/(2 \[Pi])[/spoiler]

>>4322743
Oh okay. So now we have a real proof of what <span class="math">r[/spoiler] is, instead of a strong intuition. How do we go on from here?

So now the sequence is pretty well defined, and we have to prove that if n < m(2+sqrt(3)) < n + 1 that floor((n+1)(2+sqrt(3)) - floor(n(2 + sqrt(3)) = 2, or something very very similar.

>>4322773
I just checked, and I think that's what we need to show. Here's why.
If sequence element n is 2, then that means that sequence portion created by expanding the sequence instead of contracting it proves that there are only two 3s after the nth occurrence of 2. (Here I'm ignoring the first element. Each element in the sequence generates a 2, so the number of 2s before a sequence 233 is the same as the number of digits before the original 2.)

Therefore, the condition I gave.

>>4322793
Kinda cool. I didn't think it would be this quick. Nice job!

>>4322807
I haven't shown it, though. I don't know how.

So, I normally at least try these, and I usually do okay. Today though... fuck. Idk if it's actually that hard, maybe the phrasing is just confusing me.

>>4322425
Someone post di caprio saying we need to go one level deeper!

n < m(2+sqrt(3)) < n + 1
(sqrt(3) - 2) n < m (2+sqrt(3))(sqrt(3) - 2) < (n+1)(sqrt(3)-2)

so (sqrt(3) - 2) n < -m < (n+1)(sqrt(3)-2), so sqrt(3) n < 2 n - m < sqrt(3)(n+1) - 2
well, fuck, that's the opposite of what I was trying to prove.

>>4322821
Yeah but we're down to something algebraic that is equivalent to the statement we're trying to prove. That's basically the point at which I stop caring ^^'

>>4322942
TN5, you have to come up with something. I just proved the exact opposite of what I wanted, so chances are I did something horribly wrong. Any ideas?

Okay, okay.

So we're supposed to show that if <span class="math">n < m(2+\sqrt{3}) < n + 1[/spoiler], then <span class="math">\lfloor(n+1)(2+\sqrt{3})\rfloor - \lfloor n(2 + \sqrt{3})\rfloor = 2[/spoiler]. Actually that can't be right.
<span class="math">\lfloor(n+1)(2+\sqrt{3})\rfloor\geq (n+1)(2+\sqrt{3})-1[/spoiler]
<span class="math">\lfloor n(2 + \sqrt{3})\rfloor \leq n(2 + \sqrt{3})[/spoiler]
So the difference is at least <span class="math">(n+1)(2+\sqrt{3})-1 - n(2 + \sqrt{3})=1+\sqrt{3}>2[/spoiler].

>>4323020
Oh but the distance should be 3, not 2.

>>4322942
Oh, shit, the difference has to be 3, because there are 3 digits in "233"
My bad.

>>4323029
Thanks for catching my mistake.

Np. That's the kind of mistake that is easy to make: understanding how it works but going to fast on the simple parts because they are boring.

Anyway, I don't have any simple realistic approach here. We need to prove the equivalence between <span class="math">n < m(2+\sqrt{3}) < n + 1[/spoiler] and <span class="math">\lfloor(n+1)(2+\sqrt{3})\rfloor - \lfloor n(2 + \sqrt{3})\rfloor = 3[/spoiler]. The little gain I can think of is that proving either the direct or the reverse implication should be enough because we know how many <span class="math">n[/spoiler]'s we have that satisfy either of these equations, so I think any direction we prove is enough, if you see what I mean.

>>4323075
Oh, that's an interesting gain.

Well, how about this: in the inequality
<span class="math">n< m(2+\sqrt 3) < n+1[/spoiler] each inequality difference is less than 1, so when we multiply by something smaller than 1, we get the inequalities each being less than 1, so we know that the difference between (n+1) \sqrt 3 - 2 and 2n - m is less than 1, so <span class="math">\lfloor (n+1) \sqrt 3 \rfloor = 2n - m + 2, \lfloor n \sqrt 3 \rfloor = 2n - m - 1[/spoiler]


WHEEEEEEE!

>>4323075
If your gain is appropriate, then we've done it.

>>4323129
reported for ban evasion. reported for offtopic. enjoy your permaban

>>4323129
Occifer, problem?

>>4323139
Thanks, I liked yesterday's so much I thought I'd have another.

>>4322942
Multiplied an inequality by a negative number and didn't flip the inequality. FAIL

>>4323128
Okay, cool!
So we have <span class="math">n < m(2+\sqrt{3}) < n + 1 \Rightarrow \lfloor(n+1)(2+\sqrt{3})\rfloor - \lfloor n(2 + \sqrt{3})\rfloor = 3[/spoiler]. Let's prove that it's enough for the reverse implication. Actually that's easier than what I had in mind (assuming I'm correct). The distance between "2"s is either 3 or 4 (easy to prove by induction). We already have a lot of "2"s indices, which are the <span class="math">n[/spoiler] so that <span class="math">\lfloor(n+1)(2+\sqrt{3})\rfloor - \lfloor n(2 + \sqrt{3})\rfloor = 3[/spoiler], and those are all the <span class="math">\lfloor m(2+\sqrt{3})\rfloor[/spoiler], by your previous post. We don't have enough margin between two of these to put another <span class="math">2[/spoiler].

(Right?)

>>4323139
How do you know someone else didn't save the picture and repost it?

No ponies, please.

>>4323150
Hey there ponymath. Frustrated that you're a bit late today maybe? Wanna peer review the rest of the thread so that we know if we've done other mistakes?

>>4323157
Simple really. You go to installgentoo.net, find the image, search for images like it. If the image name is the same, it's same fag. installgentoo is going slow though. Also the faggot just admitted it.

>>4323150
reported for ban evasion. enjoy your permaban.

>>4323150
FUCK, really?

Now I don't even know what to do. Flip the inequality, and you get that the difference is bounded by 2, and so the difference in floors is 2, and so I have what I wanted originally, but I was already convinced that's impossible anyway.

>>4323169
I often save images under the same name.

/r/ing yesterdays problem

>>4323163
yes ponies please

>>4323165
>Frustrated that you're a bit late today maybe?
Well I already did the '93 test as a practice test not that long ago so I already know all these problems, (either by doing them or having looked up the answers).

>Wanna peer review the rest of the thread so that we know if we've done other mistakes?
I'm trying but it's too disorganized for me to follow ><

>>4323169
>Simple really. You go to installgentoo.net, find the image, search for images like it. If the image name is the same, it's same fag. installgentoo is going slow though.

Really? Spamming your dumb "REPORTED" messages is really that important to you?

>>4323181
Please.

>>4323181
>I'm trying but it's too disorganized for me to follow ><
I can give you that.

>>4323176
We didn't use that one post of yours in the end, so it doesn't matter, does it? Or is the inequality from the previous posts wrong?

seeems good to me, near as I can tell ^^

>>4323169
n < m(2+sqrt(3)) < n + 1
Okay, say the n+1 digit is a 2. Then <span class="math"> n = \lfloor m (2 + \sqrt 3)[/spoiler]. Then the n+1st 2 in the sequence came from expanding the n+1st digit, which expands to "233". Each 2 corresponds to a value of m, so we have to show that if <span class="math"> n = \lfloor m (2 + \sqrt 3)[/spoiler], which means that the n+1st digit is a 2, then the n+1st 2 and the n+2nd 2 are separated by two 3s, right?

The condition for that is <span class="math">\lfloor (n+1) r \rfloor - \lfloor n r \rfloor = 3[/spoiler]

If one implies the other, then the sequence created by r must obey the rules of the sequence created by string replacement.

>>4323239
Now, slowly and clearly, because I'm not too bright:

<span class="math">n = \lfloor m (2 + \sqrt 3) \rfloor[/spoiler]

means that
<span class="math">0 < m (2 + \sqrt 3) - n < 1[/spoiler]
<span class="math">0 < m - n (2 - \sqrt 3) < 1[/spoiler]
<span class="math">2n - m < n \sqrt 3 < 2n - m + 1[/spoiler]
<span class="math">\lfloor n \sqrt 3 = 2n - m [/spoiler]

>>4323179
Yesterday's problem --> >>4318469

>>4323255
But also,
<span class="math">0 < (n+1) - m r < 1[/spoiler]
<span class="math">0 < (n+1)(2-\sqrt 3) - m < 1[/spoiler]
<span class="math">0 < 2n + 2 - m - (n+1) \sqrt 3 < 1[/spoiler]
<span class="math">0 > (n+1) \sqrt 3 - (2n + 2 - m) > -1[/spoiler]
<span class="math">1 > (n+1) \sqrt 3 - (2n + 2 - m) + 1 > 0[/spoiler]
<span class="math"> \lfloor (n+1)\sqrt 3 = 2n - m + 1 [/spoiler]

What the FUCK YOU MATH PROBLEM.

>>4323293
Oh, but then <span class="math"> \lfloor (n+1)r \rfloor - \lfloor n r \rfloor = \lfloor (n+1)\sqrt 3 \rfloor + 2n + 2 - \lfloor n \sqrt 3 \rfloor - 2n = 3[/spoiler]

>>4323239
The other way:
<span class="math">k + 1 > (n+1) \sqrt 3 > k > n \sqrt 3 > k - 1[/spoiler]
<span class="math">2n - k + 1 < (n+1) (2 - \sqrt 3) < 2n - k + 2[/spoiler]
<span class="math">2n - k < 2n - n \sqrt 3 < 2n - k + 1[/spoiler]
and so <span class="math">n/r < 2n - k + 1 < (n+1)/r [/spoiler]

>>4323305
>>4323293
>>4323255
>>4323239
Finally, I think this holds up and makes sense.

ACHIEVEMENT UNLOCKED: Double ban!

>>4323416
I don't understand why pony = bad. Ponymath is actually pretty awesome.

>>4323431
>Ponymath is actually pretty awesome.

I agree.

>>4323438
Now don't y'all let it go to your head.

>>4323416
More interesting is "Achievement Unlocked: 2/2/2012 Putnam practice complete!"

Anyone can get banned from 4chan, no problem.

>>4323462
>>4323438
No more of this, please.

>>4323490
Does it matter that much if we're talking about math while we post ponies? Because I can post inane shit that isn't ponies and no one cares. You just have a problem with ponies.

>>4323500
It's a fact that pony hates are just guys who aren't comfortable with their sexuality. And I don't even like ponies.

>>4323500
>no one cares
Did I read that wrong or are you really this deluded?

>>4323513
No, I'm serious. If I post a Windows default picture of a sunflower and talk about math, no one cares. If it's a battleship, no one cares. If it's a dumbass scribble, no one cares. But now it matters because it's a cartoon pony.

>>4323534
That's not even a proper analogy. You and I both know it's not just "a cartoon pony".
You're scum.

>>4323570
It's a cartoon pony from a show by the writer of The Powerpuff Girls, which by all accounts is an awesome show.

>>4323510
>It's a fact that pony hates are just guys who aren't comfortable with their sexuality.
No. I hate people posting ponies because it reminds me that I'm talking to very strange people.

>>4323600
Reported for underage
Enjoy your ban

>>4323609
We're ALL very strange people. You'd rather we be boring? Boring people aren't hardly worth talking to.

>>4323613
I'd rather not know why you're strange.

>>4323612
I'm 22.

>>4323600
How old are you? Have your sanity evaluated please.

>>4323614
Oh, ponies or no, you still don't know why we're strange. You don't have the patience to find out anyway.

>>4323621
>Have your sanity evaluated please.

>>4323624
Yes. If you post ponies I do know why you're strange. This is a science and math board, not a "let's all post what makes us weirdos" board.

>>4323177
So you copy paste the original name? You do realize the new names are generated when you upload a picture.

>>4323630
Fuck you, you know in what way some of us are weird; you still don't know why.
>>4323650
You mean people don't normally name their images 1458290348129491038_391290383.jpg?
YES YOU STUPID CUNT I USUALLY REENTER THE NAME. If they thought it was a good name, why wouldn't I?

>>4323665
Well the people who post ponies are strange because of two reasons:

1) They watch a show for little girls.
2) They show it off online

Neither of these are that acceptable in real life situations

>You mean people don't normally name their images 1458290348129491038_391290383.jpg?

Most images that people post are numbers. It's very useful for spotting tripfags who post under anonymous though.

>>4323692
reported for trolling. enjoy your ban

>>4323684
People don't bother renaming their images, but they don't usually name them some string of numbers.

Also, protip: your parents probably watched kids shows with you.

>>4323696
>People don't bother renaming their images, but they don't usually name them some string of numbers.
They don't have to rename them. The site names the images.
>Also, protip: your parents probably watched kids shows with you.
Context escapes you.

>>4323698
I watched it with my little sister, so fuck you.

And yes, I do realize the website renames the images. That way, there are no filename collisions.

>>4323705
>I watched it with my little sister, so fuck you.
So? I don't understand what your point is. It's all about context. If you're watching it with your sister for bonding or similar reasons, that is socially acceptable in the real world. What is not is extending the show to become one of your own interests. Further is plastering it all over for the world to see.

>>4323714
Oh, my god, like intolerance is socially acceptable.

>>4323728
Yes in many cases that is true. I assume you would have learned this if you are older than 18.

>>4323733
moving_the_goalposts.jpg
SO ARE PONIES. Do you even LISTEN to yourself?

I know people in real life, and I have no idea whether they've ever been on 4chan, and yet they post ponies and no one has any problem with it.

>>4323741
Again it's all about context. In this case, it's the context of your audience.

>>4323733
Not in my social circle.

>>4323746
In this case, the context is 4chan, which has latched onto ponies like it's the fucking Beatles or some fucking thing.
>>4323747
I don't know anyone in your social circle, and I don't give a rat's ass about your social circle. Why should I?

>>4323747
You sincerely are not intolerant of anything?

>>4323755
>the context is 4chan, which has latched onto ponies like it's the fucking Beatles or some fucking thing.

No the context is /sci/. Your statement is untrue for the whole of 4chan.

>>4323747
Oh, sorry about the social circle thing. I assumed that you were talking about ponies.

>>4323750
There's a difference between what's acceptable and what's done. You can even do things you think are a bad idea. It's a revolutionary concept that's been around since about 1963.

>>4323760
/b/ > <span class="math"> \frac 1 2 \sum_{i \in 4chan} i[/spoiler].

>>4323763
>There's a difference between what's acceptable and what's done.
I don't know if you have followed along, but I don't care about what people do. I just don't want to know about it. There is no reason to bring your own peculiarities up in the context of science and mathematics.

>>4323772
and /b/ does not equal /sci/. Please read my post again if you are confused.

>>4323769
Don't really care what you want, either. It doesn't match up with the freedom of expression I expect on 4chan and in my country.

>>4323775
Your point is it doesn't apply on /sci/, but my point is that's not true; you only don't want it to apply on /sci/.

>>4323778
You should check out the rules on. Your freedom of speech values do not apply to a website. There are board specifically designed for trolling and cartoons.

>>4323784
It doesn't apply to /sci/. /sci/ is not a random board, nor a comics board. It is irrelevant what the size of /b/ is in comparison to /sci/, since /b/ is not /sci/.

>>4323784
Also, /sci/ can not and does not and will never exist as an island, completely separate from the rest of 4chan. 4chan has some homogeneity. If you don't like 4chan, find something else with the bits of /sci/ that you like.

You can't simply enter a group of people, form a smaller clique, then shun everything the other people do. You're no longer part of that group, and you can't resist all outside influences.

>>4323791
My values sure as fuck apply to an image board. They may not observe those values all the time, but they do a pretty damn good job. I didn't say they were legally obligated to let me post ponies; that's retarded and false. Why would you even think that's what I meant?

consider the ratio. of 2 and 3's. then the number r mustbe 2+sqrt3 otherwise the error would add and overflow.

>>4324166
Oh, shit, why didn't I see that? Thanks, anon.

>>4324166
nice

>>4326677
Oops, I'm retarded, this thread is over 11 hours old. Just checked to see if it was solved, and didn't realize i'd bump it all the way back for nothing.

>>4326686
I was going to answer about you not knowing what a sticky is, before I realize it wasn't stickied anymore. Having threads refreshing themselves in tabs doesn't help knowing which putnam is today's, I guess...

>>4326692
So it's not just me then. Good.