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/sci/ - Science & Math


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[ERROR] No.3751105 [Reply] [Original]

Original problem is in the pic.

You have an n episode tv series. You want to watch the episodes in every order possible. What is the least number of episodes that you would have to watch?

Over lapping is allowed. For example, in the case of n=2, watching episode 1, then 2, then 1 again, would fit the criteria.

The orders must be continuous. For example, (1,2,1,3) does NOT contain the sequence (1,2,3)

>> No.3751122

Archives of previous threads for reference:
http://4watch.org/superstring/
http://archive.gentoomen.org/cgi-board.pl/sci/thread/3734198

>> No.3751147

103

>> No.3751167

n^13 or n^14?

Non-math, gayfag guess.

>> No.3751172

14! derp

>> No.3751177

>>3751172
this! lol

>troll is too stupid to include "randomly out of all possible episodes"
too easy

>> No.3751180

>>3751172
Nope, we already have the (trivial) lower bound n+n!−1.

>>3751177
>this! lol
You have only once again reconfirmed my suspicion that /sci/ is one of the boards with the largest proportion of underage b&.

>> No.3751188

14!/2

>> No.3751189

What we know at the moment is that it can be done in <span class="math">\sum_{i=1}^n i![/spoiler] episodes (where n=14 for the original problem).

Here's a Python implementation of the currently known algorithm:
http://pastebin.com/aNwANugC

And some explanation of what's going on:
http://math.stackexchange.com/questions/15510/what-is-the-shortest-string-that-contains-all-permutat
ions-of-an-alphabet

Another way of generating the same string:
http://www.notatt.com/permutations.pdf

However, as of yet it seems no one has found a proof that the algorithm is optimal.

More links with discussion of the problem:
http://www.reddit.com/r/math/related/foj1l/the_shortest_string_containing_all_permutations/
http://oeis.org/A180632
http://stackoverflow.com/questions/2253232/generate-sequence-with-all-permutations/2274978
http://forums.xkcd.com/viewtopic.php?f=17&t=68643

>> No.3751197

I think I have a proof of the lower bound n! + (n-1)! + (n-2)! + n-3 (for <span class="math">n \geq 2[/spoiler]). I'll need to do this in multiple posts. Please look it over for any loopholes I might have missed.

As in other posts, let n (lowercase) = the number of symbols; there are n! permutations to iterate through.

The obvious lower bound is n! + n-1. We can obtain this as follows:

Let
L = the running length of the string
<span class="math">N_0[/spoiler] = the number of permutations visited
<span class="math">X_0 = L - N_0[/spoiler]

When you write down the first permutation, <span class="math">X_0[/spoiler] is already n-1. For each new permutation you visit, the length of the string must increase by at least 1. So <span class="math">X_0[/spoiler] can never decrease. At the end, <span class="math">N_0 = n![/spoiler], giving us <span class="math">L \geq n! + n-1[/spoiler].

I'll use similar methods to go further, but first I'll need to explain my terminology...

>> No.3751199

Edges:

I'm picturing the ways to get from one permutation to the next as a directed graph where the nodes correspond to permutations and the edges to ways to get from one to the next. A k-edge is an edge in which you move k symbols from the beginning of the permutation to the end; for example,

1234567 -> 4567321

would be a 3-edge. Note that I don't include edges like

12345 -> 34512

in which you pass through through a permutation in the middle (in this case 23451). This example would be considered two edges:

12345 -> 23451
23451 -> 34512

From every node there is exactly one 1-edge, e.g.:

12345 -> 23451

These take you around in a cycle of length n. A 2-edge moves the first two symbols to the end. A priori, it could either reverse or maintain the order of those two symbols:

12345 -> 34521
12345 -> 34512

But the second, as already stated, is not counted as a 2-edge because it is a composition of two 1-edges. So there is exactly one 2-edge from every node.

>> No.3751204

1-loops:

I call the set of n permutations connected by a cyclic path of 1-edges a 1-loop. There are (n-1)! 1-loops.

The concept of 1-loops is enough to get the next easiest lower bound of n! + (n-1)! + n-2. That's because to pass from one 1-loop to another, it is necessary to take a 2-edge or higher. Let us define:

<span class="math">N_1[/spoiler] = the number of 1-cycles completed or that we are currently in
<span class="math">X_1 = L - N_1 - N_2[/spoiler]

The definition of <span class="math">N_1[/spoiler] is a bit more complicated than we need for this proof, but we'll need it later. You might ask, isn't <span class="math">N_1[/spoiler] just one more than the number of completed 1-cycles? No! When we have just completed a 1-cycle, it is equal to the number of completed 1-cycles.

In order to increment <span class="math">N_1[/spoiler], we have to take a 2-edge, which increases L by 2 instead of 1. Therefore <span class="math">X_1[/spoiler] can never decrease. Since <span class="math">X_1[/spoiler] starts out with the value n-2, and we have to complete all (n-1)! 1-loops, we get the lower bound n! + (n-1)! + n-2.

>> No.3751205

2-loops:

Suppose we enter a 1-loop, iterate through all n nodes (as is done in the greedy palindrome algorithm), and then take a 2-edge out. The edge we exit by is determined by the entry point. The permutation that the 2-edge takes us to is determined by taking the entry point and rotating the first n-1 characters, e.g.:

12345
is taken by n-1 1-edges to
51234
which is taken by a 2-edge to
23415

If we repeat this process, it takes us around in a larger loop passing through n(n-1) permutations. I call this greater loop a 2-loop.

The greedy palindrome algorithm uses ever-larger loops; it connects (n-k+1) k-loops via (k+1)-edges to make (k+1)-loops. But I haven't been able to prove anything about these larger loops yet.

The tricky thing about 2-loops is that which 2-loop you're in depends on the point at which you entered the current 1-loop. Each of the n possible entry points to a 1-loop gives you a different 2-loop, so there are n*(n-2)! 2-loops, which overlap.

>> No.3751207

And now for the proof of the n! + (n-1)! + (n-2)! lower bound...

To review:
n = alphabet length
L = running string length
<span class="math">N_0[/spoiler] = number of permutations visited
<span class="math">X_0 = L - N_0[/spoiler]
<span class="math">N_1[/spoiler] = number of 1-cycles completed or that we are currently in
<span class="math">X_1 = L - N_1 - N_2[/spoiler]

In order to increase <span class="math">N_1[/spoiler], you must jump to a new 1-cycle -- having completed the one you are leaving. That means the next permutation P' in the 1-cycle (following your exit point P) is one you have already visited. Either you have at some point entered the 1-cycle at P', or this is the second or greater time you've visited P. If you have ever entered the 1-cycle at P', leaving at P by a 2-edge will not take you to a new 2-cycle; you will be in the same 2-cycle you were in when you entered at P'.

So these are the available ways to enter a 2-cycle you've never been in before:
* take a 3-edge or higher
* take a 2-edge but don't increase <span class="math">N_1[/spoiler]
* take a 2-edge from a permutation P that you were visiting for the second or greater time

In the first two cases, <span class="math">X_1[/spoiler] increases by 1 in the step under consideration. In the third case, <span class="math">X_1[/spoiler] must have increased by 1 in the previous step. Because of this third case, it is convenient to regard any series of edge traversals that takes you through permutations you've already visited as a single step. Then if we define

<span class="math">N_2[/spoiler] = number of 2-cycles visited
<span class="math">X_2 = L - N_0 - N_1 - N_2[/spoiler]

the quantity <span class="math">X_2[/spoiler] does not decrease in any step. Since 2-cycles are n(n-1) long, you must visit at least (n-2)! 2-cycles. <span class="math">X_2[/spoiler] is initially n-3, giving us the lower bound

<span class="math">L \geq n! + (n-1)! + (n-2)! + n-3[/spoiler].

>> No.3751212

Guise!
Itsa trick question!

>>What is the least number of episodes you would have to watch?

I can watch each episode 1,000 times but I've still only watched 14 episodes.

So the answer is just 14. (no factorial) :p I win!

>> No.3751227
File: 13 KB, 259x195, 1.jpg [View same] [iqdb] [saucenao] [google]
[ERROR]

>>3751212

>Ops face when he realizes how idiotic his attempt to answer the question is

>> No.3751240

>>3751212
>>3751227
you knew what I mean. I could say if each ep was 30min, then how many hours would you have to watch, but that would be just semantics

>> No.3751241

Someone in another thread wanted the two n=5 strings, so reposting from last thread:

The "standard" solution given by the algorithm we've seen expressed several ways now which I think are equivalent:
(I'm just listing the cycles, so 01234 means it goes through the permutations 01234,12340,23401,34012,40123.)
01234 12304 23014 30124
12034 20314 03124 31204
20134 01324 13204 32014

10234 02314 23104 31024
02134 21304 13024 30214
21034 10324 03214 32104

The "alternate" solution:
01234 12304 23014 30124
12034 20314 03124 31204
20134 01324 13204 32014

10243 02413 24103 41023
02143 21403 14023 40213
21043 10423 04213 42103

The alternate solution deviates from the standard one in the central step where the next permutation only shares one symbol with the previous permutation. Where the standard solution hits the permutation 10234, the alternate one uses 10243. It continues on in the same manner as the standard one except that the 3 and 4 are reversed.

>> No.3751251

So I've come to the conclusion that http://www.notatt.com/permutations.pdf is complete bullshit. Anyone disagree?

>> No.3751264

>>3751251
It would be nice to hear some reasoning for that conclusion. I don't see any problem with what he wrote.

>> No.3751288

Greetings again, good sir.

Let's get all the 14! and 14*14! derps out of the way now so the big boys can talk math

>> No.3751313

Alright, after that pretentious statement, I'll acknowledge that I'm not a big boy at math at all, I'm quite a novice.

Here's my thoughts, from the ground up:

Initially, intuition sets the upper bound at 14*14! (1.22 * 10^12), for obvious reasons. Extending beyond that is redundancy. That gentleman from the other night showed that a more workable upper bound is the sum of the factorials (9.39 ^ 10.)

That's a huge step forward. What about the lower bounds? Intuitively, 13 + 14! would be the initial lower bound, being the 13 necessary as filler for the first permutation, and the possibility of a sequential permutation for each episode added.

We know that this isn't true-- that you can't add one episode and get one new permutation continuously. That much is intuitive. How would one go about proving that?

I'm trying to look at some sort of pattern in the attainability of new permutations for n different episodes, but I wouldn't even know where to start.

>> No.3751318

>>3751251
how is it bullshit? its very solid

>>3751288
ha. /sci/ has improved a great deal when it comes to that in these threads. The first one was full of 14!*14 /thread comments, people telling them

>> No.3751326

>>3751313
9.39 * 10^10, I meant to say.

>> No.3751343

>>3751313
this is my line of thinking. In the most brute force way of doing the problem is just the episode numbers. A slightly more sophisticated way of doing that is looking at the permutations. Even better than that, We discovered we could use cycle groups, i.e. each group represents its permutations, and only the order of the cycle group matters.

This is an important innovation because the order of cycle groups is much more succinct and generalizable than just looking at permutations.

Two questions must be asked:
In an efficient string, must the permutations of a cycle group be listed consecutively? i.e. for n=6, does each efficient string contain 1,2,3,4,5,6,1,2,3,4,5? If the answer is yes, that means a cycle groups only appear once.

Is there a way to generalize the algorithm based on cycle groups?

>> No.3751366

>>3751313
The moves where you add a string of length 1 to get the next permutation take the symbol from the start of the last permutation and appends it to the end. This rotates the symbols in the permutation, and after n such moves you're back where you started:

12345
23451
34512
45123
51234
12345

You can only go in between the n-cycles with a move that adds a string of at least 2. Since there are (n-1)! - 1 such n-cycles, that gives the n! + (n-1)! + n-2 lower bound.

I think I've proven n! + (n-1)! + (n-2)! + n-3 as a lower bound also, but my reasoning needs to be checked by other eyes:
>>3751197
>>3751199
>>3751204
>>3751205
>>3751207

>> No.3751370

>>3751197
Ok, I'm going to read through your posts carefully. I like that you're looking at the places where you change cycle groups, but one thing I'm thinking is, how do we know that NOT completing a cycle graph would lead to a better solution. i.e. maybe we could start with 1,2,3,4,5,6,7 then put 2,3,4,5,6,7,1 somewhere later down the road

>> No.3751373

OK, I'm almost...
not quite, but almost...
ready to conjecture that the number of different ways of achieving this (likely optimal) sequence length is:
the product from 1 to n-3 of k!

>> No.3751384

>>3751343
After seeing these cycle groups mentioned many times, I think I'm starting to get it..

The 123451234 makes sense as starting with the natural pattern and obtaining 4 additional permutations promptly, and from there you'd take the 234 at the end and start 15, then repeat the first four for 234152341?

123451234
-----------234152341
And so on, until you get back to 123451234? That makes sense, but what does this accomplish? Does it just give us a pattern to run with?

>> No.3751403

>>3751370
>how do we know that NOT completing a cycle graph would lead to a better solution
I assume you mean wouldn't. At the moment, we don't know that. But leaving a cycle early and coming back to finish it later has an associated cost. And what I'm doing is trying to carefully quantify that cost. Doing so only gives you a lower bound. It might be that when n is large you can avoid the really long moves of the standard algorithm by making a few strategic early departures from the smaller loops.

>> No.3751404

>>3751373
Oh damn. This is one question I asked last time. I don't think the world is ready for that madness. in those cases where only one element overlaps, ALL (n-1)!/2 cycle groups are up for grabs. luckily, for n=4 and n=5 this only happens once in the middle, but how do we know this doesn't happen again.

Also I think I have figured something out. We only need to prove that the first half is efficient, because these are all palindromes. i.e. the proof you make with the first (n-1)!/2 cycle groups is guarentteed to work withe the second half

>> No.3751407

>>3751370
This was my initial thought, too.

So with the cycle groups, it's 1 addition per permutation until switching between cycle groups, at which point it's 2? Are there larger gaps in between there, like moving from one set of cycle groups to the next?

And would I be asking too much if you explained
Since there are (n-1)! - 1 such n-cycles" why this is true?

Thanks a bunch mr. mathematicians

>> No.3751421

>>3751407
>>3751366
Sorry, that should be
>Since there are (n-1)! such n-cycles
(n-1)! - 1 is the number of times you have to move between n-cycles. And the reason there are (n-1)! n-cycles is simple: n! permutations, n in each n-cycle, so (n-1)! n-cycles.

>> No.3751441

>>3751384
Yes. But I'm still not sure how to choose the groups. Because sometimes you have more than one option. Check the google doc for an explanation of that. Please comment on it if you want changes or additions

>>3751403
Oops, you're right. I mean it wouldn't. Yeah it's all a cost benefit thing. Staying in a cycle means we'd be giving up the chance to have a 1-edge as the other poster calls it. And the total amount of these should be low as possible. But what if sacrificing a 1-edge for a 2-edge, meant we could avoid higher edges along the way?

Like one thing that keeps appearing is the n-1 edge, i.e. when only one digit overlaps between cycle groups. Those points are very high cost, and occur in the middle of n=4 and n=5. how do we know those can't be avoided with incomplete cycle groups

>> No.3751474

>>3751407
when you switch between cycle groups, there is variable cost. For example in n=4, sometimes there needs to be an addition of 2, sometimes 3. check the google doc listed at the top for an explanation.

In our algorithm that produces a palindrome sequence, we'd always been prioritizing low edge numbers. i.e. cycle groups are ALWAYS completed (you'd never have 12345124, because that interrupts the group.) then you always try and get a 2-edge. The way I see it, its all about making the sum of edges numbers low as possible. The derp 14!*14 way of doing it would involve n-edges, i.e. no overlap

>> No.3751476

>>3751441
>Those points are very high cost, and occur in the middle of n=4 and n=5. how do we know those can't be avoided with incomplete cycle groups
For large n (meaning n>4), we don't know that yet. For n=4, according to the n!+(n-1)!+(n-2)!+n-3 argument, it may be possible to avoid taking the 3-edge, but not without doing something else of equal cost, specifically either repeating a permutation you've already been over, or taking a 2-edge that doesn't gain you anything.

>> No.3751492

>>3751441
>>3751474
I think you either forgot to post the Google doc, or your post didn't go through, because I don't see it.

>> No.3751503

>>3751476
do we know any alternative sequences for n=4? Hopefully we can show that repeating a permutation is never going to be const efficient
>>3751492
oops, https://docs.google.com/document/d/1AXXx1516LLq3wpiIZT_FayLnp6Q7xw8JlmFqsyyoy8Y/edit?hl=en_US&pl
i=1

>> No.3751507

Guys, I think this was way easier than we thought.
The proof of
>>3751373
(or finding a different answer for it) seems much harder.

>> No.3751522

>>3751507
Easier than we thought? Most people, including me thought this was gonna be a snap at first. But 5 minutes into it, I realized there are so many paths and options to account for. Even worse, is that there are multiple efficient sequences.

But finding out how many relies on that initial question. I think if we truly understood why the sum of factorials works, we'd know how many options there are. My body isn't ready for that second question though.

>> No.3751541

>>3751204
That's a great way to think about it, only looking at the gaps. With your permission, I'd like to add this to my google doc under the tag Anonymous or with any tripcode you might use

>> No.3751556

>>3751441
>Yes. But I'm still not sure how to choose the groups. Because sometimes you have more than one option. Check the google doc for an explanation of that. Please comment on it if you want changes or additions

The simple rule is that whenever you have to add n symbols to the string to get the next permutation, they should be the n symbols from the beginning of the previous permutation, in reverse order.

This gives you a series of nested loops, each of which rotates the first certain number of characters of the string.

For n=4:
http://pastebin.com/TgPzZeZu

Given any permutation, you can figure out where it will be found in this cycle by rotating the 4 until it's at the end, then rotating the first 3 characters until the 3 is at the end, and so on. This guarantees you'll hit all permutations.

>> No.3751570

>>3751541
Feel free to do so.

Actually, what we might want is a small wiki for this sort of stuff. By which I mean collaborative work on this and future math problems.

>> No.3751588

>>3751556
Er, but sometimes there are 2 or more cycle-groups to choose from.
see >>3751241

>> No.3751621

Answer comin up.
Damn it took a long time to type.

>> No.3751625

Thanks for showing me the google doc and bringing me up to speed. I'll ponder my feeble mind on it throughout the week.

I doubt I'll be able to provide any insightful answers, but as someone who's intensely interested in understanding what's going on, I might stumble upon the right questions to ask.

>> No.3751626

>>3751570
doesn't /sci/ have a wiki? If not it should have one for explorations in math
>>3751621
answer to?

>> No.3751627

>>3751621
cant wait

>> No.3751630

>>3751588
It's a rule, not the only rule. I don't pretend to know all possible strings of sum-of-factorial length.

But as far as generalizations go:

I think the standard algorithm can be generalized by replacing the step that moves the first k symbols to the end of the permutation in the reverse order, e.g. (with k=4, n=5)

43201 -> 10234

with any step that moves the kth symbol to the end, followed by the (k-1)th symbol, followed by the first (k-2) symbols in arbitrary order, e.g.

43201 -> 10243.

This is the sort of generalization that gives you the second n=5 string.

Another observation:

In both the standard algorithm and the generalized one, the moves in each k-loop preserve the cyclic ordering of n-k+1 of the symbols, specifically the n-k+1 symbols at the beginning of the permutation at the entry point of the loop, and also at the entry point of each (k-1)- subloop. This is important to verifying that the generalized algorithm hits all permutations. Don't have an explanation of this ready-typed, though, and I'm a bit tired.

>> No.3751636

Yes, easier than we thought. Hopefully I didn't make a mistake and embarrass myself

Let a k-iteration be the process of adding k numbers in a row without making a new permutation, such that the next number will make a new permutation (the other guy called it a k-edge)
so 1234 -> 2341 would require a 0 iteration
1234 -> 3421 would require a 1 iteration and some 0 iterations


Assume x is the highest number of permutations you can get with j0 0-iterations, j1 1-iterations, j2 2-iterations, up to jk k-iterations. Now, we need to move up to a*x permutations, where a is an integer. Clearly, we must use at least one permutation of order higher than k, and if there is a solution using (a*j1) 1-iterations, (a*j2) 2-iterations, up to (a*jk) k-iterations, along with (a-1) k+1 iterations, it is the most efficient way to get 2x permutations.
Applying this to the problem, we know that the most efficient way to get n permutations (starting from a permutation) is to use n-1 0-iterations. Therefore, the most efficient way to get n*(n-1) permutations is to use n-2 1-iterations and (n-1)*(n-1) 0-iterations, as we have seen.

>> No.3751642

>>3751636
Now I just have to show that those iterations actually add up to sum(k!)

>> No.3751646

>>3751636
>Clearly, we must use at least one permutation of order higher than k
Why is this?

>> No.3751652

>>3751636
Or when you say
>Assume x is the highest number of permutations you can get with j0 0-iterations, j1 1-iterations, j2 2-iterations, up to jk k-iterations.
do you mean for the <span class="math">j_i[/spoiler]s to be arbitrary, and x is the highest you can get with any values of the <span class="math">j_i[/spoiler]s? Because you can get to any permutation using only 1- and 2- iterations.

>> No.3751663

>>3751646
Hmm, I was trying to do it a bit differently but it became a pain so i wrote that up. Perhaps I should go back to try the other way.

>> No.3751685

>>3751636
what do you mean by move up to a*x permutations.
I'm not actually sure how this lower bounds the problem

>> No.3751691 [DELETED] 

>>3751652
>Because you can get to any permutation using only 1- and 2- iterations.
Are you sure? Its a pretty strict restraint to have to resume making more permutations after every couple digits

>> No.3751718

>>3751691
1-iterations allow you to rotate the string.
Using a 2-iteration followed by (n-2) 1-iterations, we can exchange the first two elements:

12345
34521
45213
52134
21345

And by conjugating that with a rotation, we can exchange arbitrary adjacent elements. And from there you can generate any permutation by moving the element you want first up to first place, the element you want second into second place, and so on.

>> No.3751754

>>3751630
An implementation of the generalized algorithm:
http://pastebin.com/D16PrjvR

>> No.3751769

well shit
I'll blame the fact that its 3am

>> No.3751773

>>3751626
Found it:
http://sciref.wikispaces.com/message/list/home

>> No.3751795

>>3751769
it's ok, we'll continue this. I'm still blown away by the follow up problem. My mind is broken, it's time for me to watch anime.

>> No.3751809

>>3751626
If /sci/ has a wiki I couldn't find it. In either case, I propped one up real quick, it should be easy enough to maintain for this sort of stuff. If a better one exists then dropping this one's easy enough.

http://mathsci.wikia.com/wiki//sci/_-_Math_%26_Science_Wiki

>> No.3751825

using the iteration notation, if each new element yielded a new permutation, then the string would have n elements from the first permutation + (n-1)!-1 element for the remaining permutation. so S=n+(n-1)!-1

but since you can't use only 0-iterations, this isn't the case. each i-iteration has an added cost of i. you use <span class="math">j_i[/spoiler] i-iterations. the maximum of i is n-2, because that is the number of elements you'd add if the overlap was only 1 element. So the goal would be to minimize
<span class="math">\sum_{i=1}^{n-1} i*j_i[/spoiler]

>> No.3751849

>>3751809
ah cool joining. Next week I'll post another, easier but quite peculiar exploration. It involves polynomials, series, and pascals triangle.

>> No.3751860

/sci/ - Math & Science

Yep.

>> No.3751869

>>3751860
just like it was meant to be.
can you add the google doc?

>> No.3751897

>>3751869
Aye, I just plopped it all on there. I'll organize it into something intelligible tomorrow, Kaiji just ruined my brain.

http://www.youtube.com/watch?v=OeLxUVACQF0

>> No.3751914

So something that could be important:
a 0-iteration switches the order of no elements (it just rotates the cycle without moving the cycle around). A 1-iteration can switch 2 elements around. A 2-iteration can switch 3 elements around. Basically, a k iteration can switch k+1 elements around in the cycle.
Example: A 1-iteration can take (1 2 3 4 5) to (1 2 4 3 5), and a 3 iteration can take (1 2 3 4 5) to (4 3 2 1 5) (this is allowing any number of 0 iterations). The important thing is that a k-2 iteration, combined with some number of 0 iterations, can do things that (k-3) 1-iterations + an arbitrary number of 0 iterations CANNOT do (for example, to rotate k-1 elements).
For example, a 3 iteration could take (1 2 3 4 5) to (4 1 2 3 5), but 2 1-iterations could not.
This seems like it could be generalized without too much difficulty to state that iterations with a total weight <= k-3 are weaker than a single k-2 iteration.

>> No.3751932 [DELETED] 

>>3751914
oops, i think k should be n for most of that, I don't think its true for k < n-2

>> No.3751949

>>3751914
k should be n for most of this

>> No.3751950

>>3751914
So I guess you're defining adjacency for the cycles. If a new permutation requires a 1-iteration, the cycle group it belongs to is adjacent to the previous. Adjacency is neither reflexive or transitive.

i.e. (1 2 3 4) is adjacent to (1 4 2 3) which is adjacent to (1 2 4 3). But (1 4 2 3) is not adjacent to (1 2 3 4) which is not adjacent to ( 1 2 4 3)

>> No.3751953

>>3751949
would you mind reposting? Also does /sci/ have an archive? because when I tried searching for thread I, I only found the opening post

>> No.3751965

>>3751914
>For example, a 3 iteration could take (1 2 3 4 5) to (4 1 2 3 5), but 2 1-iterations could not.
You only need a single 1-iteration (combined with 0-iterations) to do that:
1 2 3 4 5
2 3 4 5 1
3 4 5 1 2
4 5 1 2 3
1 2 3 5 4 (the 1-iteration)
2 3 5 4 1
3 5 4 1 2
5 4 1 2 3
4 1 2 3 5

>> No.3751979 [DELETED] 

>>3751953
The gentooman archive does archive things, but um. I can only ever retrieve individual posts, not the entire threads.

>> No.3751980

>>3751953
I'll just leave it at this - this is what happens when you start typing before you're done thinking about stuff:

So something that could be important:
a 0-iteration switches the order of no elements (it just rotates the cycle without moving the cycle around). A 1-iteration can switch 2 elements around. A 2-iteration can switch 3 elements around. Basically, a k iteration can switch k+1 elements around in the cycle.
Example: A 1-iteration can take (1 2 3 4 5) to (1 2 4 3 5), and a 3 iteration can take (1 2 3 4 5) to (4 3 2 1 5) (this is allowing any number of 0 iterations). The important thing is that a k iteration, combined with some number of 0 iterations, can do things that (k-1) 1-iterations + an arbitrary number of 0 iterations CANNOT do (for example, to rotate k+1 elements).
For example, a 3 iteration could take (1 2 3 4 5) to (4 1 2 3 5), but 2 1-iterations could not.
This seems like it could be generalized without too much difficulty to state that iterations with a total weight < k are weaker than a single k iteration.


Anyway, the goal would be more to show the triangle inequality with distance than to define adjacency. It doesn't seem like it would be that hard, but I'm too tired atm.

>> No.3751984

>>3751953
Thread 1 happened at the same time as changes in 4chan's HTML code that broke the archiver for a while, but see
>>3751122
for a copy.
But Thread II and this one are showing up on archive.gentoomen.org just fine.

>> No.3751995

>>3751965
Hmm
Make it: (1234567) -> (4123567)

>> No.3751996

>>3751718
should be 0-iteration and 1-iteration.
Was confused since the k-iteration is the (k+1)-edge from before.

>> No.3751997

>>3751953
I think I understand your point, that sometimes the amount of overlap will be 1,2..n. The only case I've looked at extensively is n=4. In that case, you'd have 0-iterations (obviously) as well as 1-iterations and 2-iterations (maximum). I wonder if for case n, if you must have a non-zero number of 0,1,2...n-2 iterations in the efficient process.

>> No.3752034

I wonder if there are any situations where a k iteration could be strictly stronger than every sum k set of iterations.

>> No.3752047

>>3752034
nevermind thats dumb. My intelligence has fallen enough that I'll force myself to sleep.

>> No.3752047,1 [INTERNAL] 

cevap 1 :D

>> No.3752047,2 [INTERNAL] 

>>3751197
nice

>> No.3752047,3 [INTERNAL] 

im gay #wow #whoa

>> No.3752047,4 [INTERNAL] 

niggers

>> No.3752047,5 [INTERNAL] 

put me in the screencap

>> No.3752047,6 [INTERNAL] 

Put THIS in your research paper:
BRAAAAAAAAAAP

>> No.3752047,7 [INTERNAL] 

*tink* *tink* *tink*
Attention everyone! Honestly, I have just one thing to say: Fuck mods and fuck jannies. Long live frogposters and funposting. Cheers.

>> No.3752047,8 [INTERNAL] 

Cite my anus

>> No.3752047,9 [INTERNAL] 

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>> No.3752047,10 [INTERNAL] 

I miss moot

>> No.3752047,11 [INTERNAL] 

daily reminder to stay hydrated

>> No.3752047,12 [INTERNAL] 

Gas the kikes fuck off normalfags

>> No.3752047,13 [INTERNAL] 

put me in the research paper

>> No.3752047,14 [INTERNAL] 

Sup niggas this is some sick math shit yo

>> No.3752047,15 [INTERNAL] 

Ebin suomi :D

>> No.3752047,16 [INTERNAL] 

based

>> No.3752047,17 [INTERNAL] 

I think I solved it.

First calculate the max string which is the number of permutations multiplied by n.
Then divide the max string by n - 1
Then add to the answer the fibonacci number located at 2(n-3)

int max_string = n! * n;
int answer = max_string / (n - 1);
answer += fib(2(n-3));

Example for n = 5;

max_string = 5! * 5 = 600
answer = 600 / 4 = 150
answer += fib(2(5-3)) = fib(4) = 3
answer = 153

Here is the first 14 using this method:

2 -> 3
3 -> 9
4 -> 33
5 -> 153
6 -> 872
7 -> 5901
8 -> 46135
9 -> 408384
10 -> 4032377
11 -> 43909467
12 -> 522549784
13 -> 6745945965
14 -> 93884331311

>> No.3752047,18 [INTERNAL] 

Coma back when you found a solution to the nigger problem in the US

>> No.3752047,19 [INTERNAL] 

Congratulations, you newfags have discovered that new comments on archived threads are not locked down.

>> No.3752047,20 [INTERNAL] 

Very nice problem solving a true display of autism I could not be prouder of you autistic niggers

>> No.3752047,21 [INTERNAL] 

sage goes in every field goes in every field

>> No.3752047,22 [INTERNAL] 

WE DID IT REDDIT

ps bring back ghostposting on /jp/

>> No.3752047,23 [INTERNAL] 

im kinda confused with some of the terminology seeing as im only taking geometry rn so can someone explain edges/iterations/loops

>> No.3752047,24 [INTERNAL] 

Can someone implement this in a C-like lang so I can understand what the fuck is going on?

>> No.3752047,25 [INTERNAL] 

>>3751197
okay but what about the part where Jews make up 2% of the American population but 50% of Harvard graduates?

>> No.3752047,26 [INTERNAL] 

gewr

>> No.3752047,27 [INTERNAL] 

Ohohohaha ohohohehee ha.
Hello me old chum.
I'm gnot a gnelf.
I'm gnot a gnoblin.
I'm a gnome.
And you've been gnomed.

>> No.3752047,28 [INTERNAL] 

toasting in an epic (archived) bread

>> No.3752047,29 [INTERNAL] 

NIGGERS (this goes in the paper)

>> No.3752047,30 [INTERNAL] 

>>3752047,24
>the absolute state of cs ni/g/gers

>> No.3752047,31 [INTERNAL] 

Daddy’s 🧔 cummies, 💦 nice 😊 and yummy 😋
Thick 😫👌and gooey, feel like honey 🍯
That sweet 🍭 milk, 🥛 oh-so-tasty 🤤
Daddy, 🧔 Daddy, 🧔 please be hasty!💨
My tongue 👅 swirls 😛 round 🔁 and round 🔁
While Daddy 🧔 gives my ass 🍑 a great big pound 🤜💥
Over 😑 in the corner, wrapped 🎁 in chains ⛓
Mommy 👩 huddled over, screaming 😱 in pain 😪
“Shut up, bitch! 👩 Stay on the floor!”
Mommy 👩 sobbing 😭 louder, I 👧 call 🗣 her a whore
She reaches behind her 👩 for her gun 💀
While Daddy gropes 👋 and tickles 🤗 my sweet 🍬 buns 🍑
Puts the barrel 😛 between 👉👈 her teeth 👄
And Daddy’s 🧔 semen 💦 begins to seep ☔️
Gunpowder, blood, 💉 brains 🧠 and gore 😎
Mommy’s 👩 lifeless 👻 corpse slumps ⬇️ to the floor 😂
Cummies, 💦 cummies 💦 filling my throat
My pussy 🐱 is completely 😍 soaked 💦
But Daddy’s cock 🍆 just gets bigger 😳😲
Over near 😯 the drawers, he grabs the scissors ✂️
Cuts 🔪 a hole 🕳 in Mommy’s 👩 stomach
In her hand, 🤲 a gin and tonic 🍺
He 🧔 begins to unravel her strings of intestines 😝
“Look 👀 closely, you’ll learn 👨‍🏫 a lesson”
Ties a noose, 😔 rigs it tight 😫
“Sweetie, 👧 sweetie, 👧 no need to fight” 👊💥
Puts her 👩 guts 🤤 around 🔄 my neck
“Looks like everything’s good 👍 and set” 👌
Lets me fall down ⬇️ about 🤔💭 a yard
Face 👧 turning blue, choking 🤭 hard 💪
All the while, 🕑 Daddy’s 🧔 stroking his cock 🍆
And for a moment, 😳 our eyes 👁 lock 🔒
Tears 😭 of joy 😊 stream down my face 👧
I’m going to 😵 a better ✨ place~

>> No.3752047,32 [INTERNAL] 

Bill Clinton is a rapist! INFOWARS.COM

>> No.3752047,33 [INTERNAL] 

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>> No.3752047,34 [INTERNAL] 

me bone me jo

>> No.3752047,35 [INTERNAL] 

toasting in epic bread

>> No.3752047,36 [INTERNAL] 

lo

>> No.3752047,37 [INTERNAL] 

okay so basically im monky (paper reference)

>> No.3752047,38 [INTERNAL] 

isn't it strange that 13% of the population commits over 50% of crimes?

>> No.3752047,39 [INTERNAL] 

>>3752047,38
It's not strange, it's genetics. Blacks are considerably more violent than most races.

>> No.3752047,40 [INTERNAL] 

cum

>> No.3752047,41 [INTERNAL] 

Animes are fucking gay and they can suck my proverbial dick
……………………………………….¸„„„„
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………….…………………… („-~~--„¸_….,/ì'Ì
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……………………¸„„-^"¯ : : : : : : : '\¸„„,-"
**¯¯¯'^^~-„„„---~^*'"¯ : : : : : : : : : :¸"
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: : : : : : :, : : : : : :/
"-„_::::_„-*__„„~"

>> No.3752047,42 [INTERNAL] 

>>3752047,38
Nah man, it's poverty and racism.

Let's say a black dude is born in a poor area with bad public infrastructure, underfunded public education, and very few job opportunities. And even if he applies for good jobs in other areas, he will be discriminated against because of his race, which can make upward mobility or leaving a bad area a very difficult thing to do. So there's a cycle of poverty that is almost impossible to escape.

Then, let's say this guy is impoverished and is having difficulty in finding employment, so in an act of desperation, he steals some food from a gas station. Then the media will jump on him and make it sound like he's lazy and evil and therefore deserves to be poor. But that's not true at all.

Of course, that's not to say that all people are in poverty, or that all black people steal. What I am saying is that crime stats are skewed because vulnerable populations, such as those who are poor or discriminated against, might be more likely to commit crime because of their bad life circumstances, and if you were in a similar situation, you would do something similar too. Even poor white people are more likely to commit crimes than wealthy white people. Financial stability really helps.

It's easy to dehumanize other people if you don't have to face the same kind of adversity, but have some empathy and try to understand where other people are coming from.

You think crime is an issue of race. I think crime is an issue of poverty. Bring people out of poverty and they will stop committing crimes. You can pretend that success is all about an individual's actions, but a lot of it is actually about circumstances out of your control, such as the environment you grow up in.

>> No.3752047,43 [INTERNAL] 

>>3751197
now this is EPIC

>> No.3752047,44 [INTERNAL] 

math is fucking stupid yall are fucking virgin's

>> No.3752047,45 [INTERNAL] 

I can't stop farting. Farting on combinatorics algorithms!

>> No.3752047,46 [INTERNAL] 

>>3752047,44

Well you are just a nigger who can't do math. kys

>> No.3752047,47 [INTERNAL] 

Here come the desperate journalists, thanks to glorious standupmaths.

>> No.3752047,48 [INTERNAL] 

nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger nigger

>> No.3752047,49 [INTERNAL] 

If you read this you're a redditor

>> No.3752047,50 [INTERNAL] 

Matt Parker is a soy boy cuck

>> No.3752047,51 [INTERNAL] 

He did it for free

>> No.3752047,52 [INTERNAL] 

Yeah but what if I want kill every stinky nigger and jew in the US of A and use the least amount of money and gas, how would I do it??????

>> No.3752047,53 [INTERNAL] 

NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER

>> No.3752047,54 [INTERNAL] 

>>3752047,38
This really makes u think

>> No.3752047,55 [INTERNAL] 

NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGENIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER NIGGER

>> No.3752047,56 [INTERNAL] 

PARKER SQUARE
A
R
K
E
R

S
Q
U
A
R
E

>> No.3752047,57 [INTERNAL] 

29² 1² 47²
41² 37² 1²
23² 41² 29²

>> No.3752047,58 [INTERNAL] 

>people have invaded the archive since the video...

straal is gay

>> No.3752047,59 [INTERNAL] 

PS: Frank rocks btw

>> No.3752047,60 [INTERNAL] 

Dear intellectual elite, I appeal to you, the elite of society. Tell the fool where you ordered the essay?

>> No.3752047,62 [INTERNAL] 

Oh no the spammers are here.

>> No.3752047,63 [INTERNAL] 

REDDIT REDDIT REDDIT r/woosh r/tech r/4chan

>> No.3752047,64 [INTERNAL] 

>>3751197
FUCK NIGGERS

>> No.3752047,66 [INTERNAL] 

cmkasms

>> No.3752047,67 [INTERNAL] 

I love anime women, guns, and India.

>> No.3752047,68 [INTERNAL] 

BASED

>> No.3752047,69 [INTERNAL] 

>>3751197
Hi mom I'm on a mathematic theorem!
Also Kurisu is best girl and jews belong in the oven

>> No.3752047,70 [INTERNAL] 

>>3752047,64
log off kid

>> No.3752047,71 [INTERNAL] 

WE DID IT REDDIT

>> No.3752047,72 [INTERNAL] 

I hate anime women, guns, and India.

>> No.3752047,73 [INTERNAL] 

nigger

>> No.3752047,74 [INTERNAL] 

>>3751197
despite being 13% of the population

>> No.3752047,76 [INTERNAL] 

>>3752047,75
eat my penor you fuckin spic

>> No.3752047,77 [INTERNAL] 

asd

>> No.3752047,78 [INTERNAL] 

>>3751197
There are 26 bicycles and you want to find out how many different ways you can steal them all.

>> No.3752047,79 [INTERNAL] 

yes

>> No.3752047,80 [INTERNAL] 


Put me in the screenshot senpai

>> No.3752047,82 [INTERNAL] 

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>> No.3752047,89 [INTERNAL] 

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>> No.3752047,91 [INTERNAL] 

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>> No.3752047,129 [INTERNAL] 

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>> No.3752047,144 [INTERNAL] 


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>> No.3752047,155 [INTERNAL] 

>thread unironically filled with ad bots
Glad I was here before the second Matt Parker video.

>> No.3752047,158 [INTERNAL] 

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>> No.3752047,165 [INTERNAL] 

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>> No.3752047,170 [INTERNAL] 

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>> No.3752047,175 [INTERNAL] 

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>> No.3752047,182 [INTERNAL] 


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>> No.3752047,186 [INTERNAL] 

Facing issue with Microsoft Office we are here to help ou with step by step instructions.

>> No.3752047,192 [INTERNAL] 

Norton antivirus helps to keep away the viruses from the device, and it also protects them from malware, adware and other things that attempt access and tries to damage the system. If you install Norton antivirus or other Norton products on your device, you can protect your data and save your computer from additional damage.protect and create

>> No.3752047,197 [INTERNAL] 

AOL Tech Support Phone Number is a customer care helpline that is offered by one of world's best mail service provider. The experts at this toll-free helpline renders all your forget password, password recovery and other related issues. Reach them anytime of day or night at AOL Tech Support whenever you need their help.

>> No.3752047,198 [INTERNAL] 

Canon Printer Support is a team of experts who takes care of your canon device issue. If face any problem with your device like your device is crashed , device is not working properly or others then contact Canon Support and get instant help.

>> No.3752047,211 [INTERNAL] 


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>> No.3752047,216 [INTERNAL] 

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>> No.3752047,219 [INTERNAL] 

Get the support of our best technicians if you have any issue related to Lexmark Printer, you can solve your problems by just following the few steps guided by our technicians over the phone. Even if you do not have any knowledge of computer or printer, our printer specialists will guide you in such a way that it would look like a walk in the park for you.

>> No.3752047,220 [INTERNAL] 

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>> No.3752047,226 [INTERNAL] 

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>> No.3752047,233 [INTERNAL] 

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>> No.3752047,240 [INTERNAL] 

HP PRINTER customer service– There are different means of communication and every way of communication has its benefits and restrictions. HP PRINTER is pondered as a modern way of communicating with your friends, family members and colleagues. It has an easy and sleek interface and has over 300 million active users every month throughout the world. However, since it has a great base of users and many features, its users sometimes experience different tech glitches while utilizing HP PRINTER . To fix these issues, you must get in touch with the official HP PRINTER support number which is round-the-clock available through email and toll-free number.

>> No.3752047,242 [INTERNAL] 

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>> No.3752047,244 [INTERNAL] 

>>3752047,42
>niggers
>human
choose one

>> No.3752047,247 [INTERNAL] 

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>> No.3752047,248 [INTERNAL] 

www.norton.com/setup - Sign in to manage, renew, download or setup your Norton. 1. Visit norton.com/setup. 2. Enter a new product key and 3. Install Norton

>> No.3752047,251 [INTERNAL] 

GARMIN MAP UPDATES FREE is one of the leading service given in todays world. We are providing every cruicial information so let's get connect to each other.

>> No.3752047,253 [INTERNAL] 

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>> No.3752047,262 [INTERNAL] 

Encountering issues or errors while using Lexmark printer? Dial the Lexmark printer toll-free phone number and get instant 24/7 support from the Lexmark printer customer service.

>> No.3752047,277 [INTERNAL] 

To get started with your Microsoft Office Installation we can help you with our entire process to setup office product online.

>> No.3752047,281 [INTERNAL] 

Garmin.com/Express - Register, Update or Sync Device with Garmin Express at www.garmin.com/express. Now you may Start doing Garmin Update using Garmin Express Software.

>> No.3752047,302 [INTERNAL] 

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>> No.3752047,306 [INTERNAL] 

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>> No.3752047,319 [INTERNAL] 

Get 24/7 service by calling on Brother printer customer support toll-free phone number and troubleshoot all sorts of Brother printer related issues. Get the best help from experts.

>> No.3752047,322 [INTERNAL] 

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>> No.3752047,323 [INTERNAL] 

If you do not like to accept such ban in emailing interface anymore, then it would be better option to file your problem to our technical engineer through dialing Outlook Support Phone Number. We do not take any problem very lightly and offer the instant solution. In this way, your business production cannot halt anymore. For knowing more information, you can surf our web portal.

>> No.3752047,325 [INTERNAL] 

Productivity tools have always proven to be a boon for expanding businesses that need to tackle the increasing workflow. It is needless to say that they aim at sustaining the quality while enhancing the quantity of output. Moreover, these are not just limited to our work life.

>> No.3752047,327 [INTERNAL] 

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>> No.3752047,333 [INTERNAL] 


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>> No.3752047,337 [INTERNAL] 

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>> No.3752047,340 [INTERNAL] 

Norton designed and developed some other anti-virus by using their users such as NU16 and sometimes perhaps Utility 16 when it comes to some utility program, that works on the device.

>> No.3752047,350 [INTERNAL] 

TomTom has designed a variety of navigation and mapping products to provide you the correct route using the satellites. There are some other products as well which have developed by the company like camera, watches, and location based products etc.

>> No.3752047,358 [INTERNAL] 

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>> No.3752047,359 [INTERNAL] 

Are you looking for TomTom map update? You tried to update it but it displays an error.GPS Contact number could be the best help for your GPS technical issues. Dial +1-888-480-0288 (USA/Canada) and +44-800-041-8324 (UK).

>> No.3752047,369 [INTERNAL] 

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>> No.3752047,370 [INTERNAL] 


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>> No.3752047,378 [INTERNAL] 


TomTom Home is an essential tool for any TomTom user. However, it enables you to manage their device hassle-free. It is a useful tool to keep your TomTom up to date. Finally, it has easily install maps and services. It also, prepare trips searches shops for more services.

>> No.3752047,381 [INTERNAL] 

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>> No.3752047,397 [INTERNAL] 

TomTom Update - Now Update Your TomTom Navigation Devices utilizing TomTom MyDrive Connect and Manage the Device utilizing TomTom Home Management Tool. update tomtom | how to update tomtom | Tomtom home

>> No.3752047,403 [INTERNAL] 

Bitdefender login offers a wide range of products making it a truly versatile security solution. While some are dedicated to specific operating systems. Yet others are designed for cross-platform compatibility.

>> No.3752047,404 [INTERNAL] 

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>> No.3752047,409 [INTERNAL] 

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>> No.3752047,428 [INTERNAL] 

HP provides many high-quality printers with extreme advance features to the customer. But like other electronic components, a user may face many issues with this device also; you can directly Call HP Printer Support. This number will directly connect you to our technical assistant. We have an experienced HP Support Team which provides an adequate solution to all our customers. HP Printer Telephone support team provides round the clock services; you can dial our toll-free HP Printer support Number and get the most reliable working solutions.

>> No.3752047,436 [INTERNAL] 

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>> No.3752047,443 [INTERNAL] 

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>> No.3752047,444 [INTERNAL] 

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>> No.3752047,445 [INTERNAL] 

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>> No.3752047,456 [INTERNAL] 

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>> No.3752047,468 [INTERNAL] 

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>> No.3752047,471 [INTERNAL] 


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>> No.3752047,496 [INTERNAL] 

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>> No.3752047,513 [INTERNAL] 

If you forgot or lost your Pandora account password and you facing a problem in opening your account, to pandora password reset you'll use these steps, first, you have to visit the Pandora home page, there you'll get the choice of submitting the email/phone number that's linked to your account. Submit that info so click on the Next button. After this, you'll be receiving the recovery code to your email address that is linked with your Pandora account. Enter that code in the space provided and click on the Next button. currently, you'll be allowed to create a new password for your account, now click on the confirmation button and your password will be reset. By doing this, you'll be able to reset your Pandora account password quickly.

>> No.3752047,522 [INTERNAL] 

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>> No.3752047,530 [INTERNAL] 

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>> No.3752047,531 [INTERNAL] 

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Where responsive design relies on changing the design pattern to fit the real estate available to it, adaptive design has multiple fixed layout sizes. When the site detects the available space, it selects the layout most appropriate for the screen. So, when you open a browser on the desktop, the site chooses the best layout for that desktop screen; resizing the browser has no impact on the design.
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>> No.3752047,4252 [INTERNAL] 

What is Adaptive web design?
To reference an article from interaction designs Mads Soegaard:
Adaptive Web Design was introduced in 2011 by web designer Aaron Gustafson in his book, Adaptive Web Design: Crafting Rich Experiences With Progressive Enhancement. It is also known as a progressive enhancement of a website.
Where responsive design relies on changing the design pattern to fit the real estate available to it, adaptive design has multiple fixed layout sizes. When the site detects the available space, it selects the layout most appropriate for the screen. So, when you open a browser on the desktop, the site chooses the best layout for that desktop screen; resizing the browser has no impact on the design.
Choosing Between Responsive and Adaptive Design
Responsive design is easier and takes less work to implement. It affords less control over your design on each screen size, but it’s by far the preferred method for creating new sites at this moment. This may also have something to do with the large number of cheap templates available for the majority of Content Management Systems (CMS) such as WordPress, Joomla, etc. — after all, who wants to reinvent the wheel?
Responsive designers create a single design to be used on all screens and will generally start in the middle of the resolution and use media queries to determine what adjustments will be made for the lower and higher end of the resolution scale. This tends to make users happy because that familiar web design seems guaranteed to translate across to any device’s screen. Uniformity and seamlessness are crucial considerations in providing a good user experience.
It’s important to keep an eye on the visual hierarchy of responsive design projects; you want to try to maintain this as your elements shuffle around the screen. That means a lot of testing with many different devices to be certain that you’re delivering the goods. If the design for a site is relatively simple, it will translate well across device screens, flowing like a liquid from the container to container.
SEO is another big argument for using a responsive design. Sites that use responsive design (i.e., ones with a URL that serve all devices) are currently more search engine friendly.
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