I yield my turn. I am a bit busy right now. Go ahead Galathaea
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I yield my turn. I am a bit busy right now. Go ahead Galathaea
Well, I was working on this one problem the other day, and it actually surprised me when I figured out the answer, so maybe some of you would like to check it out.
Let n be a given positive integer. How many solutions are there in ordered positive integer pairs (x, y) to the equation
In other words, finds some equation based on n and / or properties of n that calculates the number of solutions to this equation. Both x and y are positive numbers, and the fact that the pair is ordered means that when x and y are different (x, y) is counted as a separate solution from (y, x).Code:xy
----- = n
x + y
Number of pairs = (number of divisors of n) * 2 - 1Quote:
Originally posted by galathaea
finds some equation based on n and / or properties of n that calculates the number of solutions to this equation.
e.g. if n=12, divisors are 1, 2, 3, 4, 6, 12. There are 6 of them. So there are 11 ordered pairs (x,y) solving xy/(x+y)=12.
There are more than 11 solutions to the n=12 subproblem, so your solution is off. I would give the total for n=12, maybe even enumerate them, but I'll give some more time first for you track down your error (because I already think this will be solved soon from your post :)).
Oh that's right... there was one part where I had if this then do that, and if the other thing, I'll come back to that later but i forgot ;).
Anyway, I already threw out the paper I did it on, so maybe I'll go through my steps again later.
Hmmm, k let's try this:
first, prime factor n:
n=a^i*b^j*c^k*d^l*... and so on.
Now it should be:
(let NS[n] denote "the number of paired solutions for n")
NS[n]=NS[a^i]*NS[b^j]*NS[c^k]*NS[d^l]*...
=(2i+1)*(2j+1)*(2k+1)*(2l+1)*...
This means there are 15 (not 11!) solutions for n=12, for example:
12 = 2^2*3^1
NS[12]=NS[2^2]*NS[3^1]
=(2*2+1)*(2*1+1)=(5)(3)=15.
Here's a new question:
good luck!Code:a+b+c=-6
ab+ac+bc=2
a^3+b^3+c^3=3
a=?, b=?, c=?, abc=?
If I'm correct, that is not soluble over integers (a check over the possibilities mod 7), so are you looking for solution over a radical field?
And yes, your previous solution was correct even though I don't quite see your reasoning... :D
That's right, there are no integer solutions.Quote:
Originally posted by galathaea
If I'm correct, that is not soluble over integers (a check over the possibilities mod 7), so are you looking for solution over a radical field?
My mind works in ways even I can't comprehend.Quote:
Originally posted by galathaea
And yes, your previous solution was correct even though I don't quite see your reasoning...
Well, a quick expansion of (a+b+c)^3 leads me to abc = 61. This then gives me the fact that a, b, and c can be found as the roots to x^3 + 6 x^2 + 2 x - 61 = 0 (using either the cubic equation root formula or elliptic function methods can give the precise form of the radical -- or something like Mathematica -- but from here it is just rote). Unless I made an error in my multinomial expansion...
I'm sorry, I meant to ask for just abc=?. You got it correct, with 61. It's clear from the problem set-up that a, b, and c are symmetrical or interchangeable (what's the correct term for this?), so obviously there is no single correct answer, but perhaps it would still be possible to find numbers that fit the bill. Although, trying to find three numbers whose product is prime may be quite annoying.Quote:
Originally posted by galathaea
Well, a quick expansion of (a+b+c)^3 leads me to abc = 61. This then gives me the fact that a, b, and c can be found as the roots to x^3 + 6 x^2 + 2 x - 61 = 0 (using either the cubic equation root formula or elliptic function methods can give the precise form of the radical -- or something like Mathematica -- but from here it is just rote). Unless I made an error in my multinomial expansion...
In a completely different and strange universe, there exists an earth divided into three human races or tribes. These tribes have been forever in epic battle, and are fearful and hating of each other. One group is known as the Hashishim and are ruled by an enigmatic man known as Hassan-i-Sabbah. They claim that nothing is true, and that everything is therefore permissible. Another group is known as the Bilderbergers and is under the control of a man named John Birch who vehemently denies the views of the Hashishim and instead declares that everything is true, yet nothing permissible. The final (and least populous) group calls themselves the Discordians and, although technically not ruled, pay direct homage to their spiritual leader Korzybski. The Discordians make no claims and just generally feel that reality is really confusing.
Long ago, 3 obtuse-triangular fragments of an ancient and enigmatic metal alloy engraving were found, with strange words in a strange language enscribed upon them. These 3 fragments fell one each into the hands of the tribes. Each tribe soon developed its own mythology as to the origins of the great engraving. The Hashishim felt it was a part of the Necronomicon, composed from the elemental universe by Abdul Alhazrad on one of his trips to Sirius. The Bilderbergers felt it was a Merovingian engraving detailing plans to arrive to Sirius. And the Discordians felt it was a direct gift from the peoples of Sirius to the Dogon, who passed it on to the Discordians for their continuing support. All myths led to the conclusion that the race of people that finds "the number of true and distinct meaning words" and utters this number at a huge obelisk at the north pole will be granted eternal life and happiness, and they all agree on this because the Great Obelisk of the North has
this written on it in a common tongue of the peoples. It also mentioned "horrible destructions" for error, and so no person was allowed near in fear of error.
There was a famous philosopher of ancient times, called Hermes Trismegistus and of unknown tribal origin, who laid out the basics of the analysis of the fragments. He first pointed out that there are 5 syllables from which all the words are constructed: mo, lo, me, le, and ni. Using his great philosophy, he determined that the syllable ni was meaning-neutral. By this, he meant that the words lomo and lomoni have exacly the same meaning, as do nilomo, loninimo, and so on. Therefore, although there were thousand of words enscribed upon the fragments, not all of the words were distinct. It was realized that this was the real task at hand.
Soon after the philosophies of Hermes were published, the three tribes became suspicious of each other and no longer allowed public analysis of the fragments. All copies and notes were burned in a worldwide conflagration of paranoia. A popular front of like-minded peoples from all three tribes attempted to "return the knowledge to the masses" and was able through their secret geometries to discover some other meaning-neutral sequences of syllables, which eventually became popular knowledge. They discovered that "mome", "memo", "lole", and "lelo" all were meaning neutral when found in words. Eventually, though, all members of the popular front were either killed or called very bad names, and the front disbanded and became only history.
Each of the three tribes pursued their analyses in private. The Hashishim perfected cryptanalysis and applied it to their fragment. They discovered that "momo" has the same meaning as "me". The Bilderbergers used the ancient analysis of kabbalah to discern that "lo" has the same meaning as "le". And finally, through application of their advanced methods in pattern recognition, the Discordians were able to find out that "molomolomolo" is meaning-neutral. Among the three of them they had enough information to speak unto the obelisk the correct number of distinct words in the fragments, but they dared not share their information for fear the others would learn the answer before themselves.
Then, after many years of secrecy, a magical little girl Guzel was born, the daughter of three parents (one from each of the tribes). Seeing that she was of all races, and thus could benefit all equally, the three tribes gathered at a great convention known as the Trilateral Commission, and gave the little girl the secrets they had learned. Then Guzel and her pet Xeon went forth on a great journey to the Great Obelisk of the North during which she had to use the knowledge to determine what to speak. What number of distinct words should she determine there are in the fragments?
mmm...Quote:
What number of distinct words should she determine there are in the fragments?
Can be that she could not determine nothing because she was eaten by her pet? :D ;)
3!Quote:
Originally posted by galathaea
It also mentioned "horrible destructions" for error, and so no person was allowed near in fear of error.
...
What number of distinct words should she determine there are in the fragments?
Hey, you said there would be horrible destruction... where is the horrible destruction!
Couldn't you just say "momolomomolomomolo...(etc.)", meaning it is an infinite number? And also, do the people have anything to do with this?
There are no infinite words as the engravings are finite and they would not have been able to analyze them otherwise (for example, a count of words is mentioned, and they could not count words if there were infinte info on the engravings -- well without a convenient black hole around for the Church-Turing trick :)). The names are meant to add color to an otherwise interesting problem :D.
Well yes I realize that, but if the following are in fact words:Quote:
Originally posted by galathaea
There are no infinite words as the engravings are finite and they would not have been able to analyze them otherwise (for example, a count of words is mentioned, and they could not count words if there were infinte info on the engravings -- well without a convenient black hole around for the Church-Turing trick :)).
momolo
momolomomolo
momolomomolomomolo
momolomomolomomolomomolo
momolomomolomomolomomolomomolo
...
Then how would you know when to stop? Or is there some information that I have missed that would prevent this impossibility?
But molomolmolo is the same word as ni. They are both meaning-neutral (as the Discordians discovered).
But nowhere in the words listed above is "molomolomolo". Does this mean that those syllables need not be consecutive to be neutral?Quote:
Originally posted by galathaea
But molomolmolo is the same word as ni. They are both meaning-neutral (as the Discordians discovered).
Sorry for the misread... you are correct that sequence counts. But if you start applying the rules, you will notice that some of the words you listed are equivalent to words of a smaller size. The rules are the trick in this problem.
Now, to make it clear. I have checked this problem out several times while I was making it up to make sure that the rules were complete (unlike the farokabarwonhul problem -- I have that word forever etched my psyche), so I think everything is in order. If something is not, I will eat crow as soon as possible, but for the examples you listed, I know of some equivalent words, so that is not a problem here.
Good Luck!
As an addition on how to apply the rules, note the following which I should definitely have clarified. Syllable sequences define the meaning of words. In other words, since "momo" and "me" have equivalent meanings to each other, then "momolo" has equivalent meaning to "melo" (but not necessarily equivalent meaning to "momo"). And similarly, if "blah-blah-bloo" (made up here) is equivalent to "blah-bloo-blech-bloo", then "blah-blah" is equivalent to "blah-bloo-blech" (ie. you can remove appended syllables from a sequence on equivalence, but you cannot remove from inside a sequence except if the subsequence is meaning-neutral -- sequence is what matters here). I had tried to make that implicit, but I see that I may have done a poor job with that one. I'll add that to the accomplishments of Hermes...
Oh I get it... the appending thing helped,.
So now
lolo=molomolomolo
<=>
lo=molomolomo
So
momolo=momomolomolomo=lomolomo.
momolomomolo=momomolomolomomomomolomolomo
=lomolomolomolomo=lomo.
momolomomolomomolo=momomolomolomomomomolomolomomomomolomolomo=ni.
which makes a huge difference.
One final question, is "ni" a word?
Yes, ni counts as a word. It is meaning-neutral, but there is a meaning-neutral word. Good question, though...
you are asking
|{x,y: x^3=e, y^2=e, (xy)^3=e}|=?
Well, if you want to remove all of the great storyline and everything...
Oh, sorry. The story line was great. :D
uuuhhh... Hi guys sorry for being away such a long time but I was really sick... seems that I've missed a lot... What's been up! who's asking? what's the question?
If you haven't seen it yet, the last question was asked by me (its title is "a new one") just a few posts above. You should read the clarifications below it (as always, its one of those questions that needed some clarification) and souldog has distilled all the math out of it (well, along the group theoretical formulation, but there are of course others). I'll give it another day or two before I post a solution, since it is a pretty fun problem (in my opinion).
Too tired, can't think. Don't feel like counting either. G'night.
I was going to post the answer tonight but I'm gonna be lazy and wait till tomorrow. That will give everyone one more day to do something else :D
Okay, okay! Enough suspense! Now for the result that no one has been waiting for...
Well, I guess souldog's distillation sums up the best way to look at the problem in my opinion, though you could look at it as a sequence enumeration problem (where you look for sequences with no meaning-neutral subsequence and map them over the relations), but it would be a real pain to work out. Now, there is a canonical method to solve for the order of groups described through free generators and a presentation of relations known as the Todd-Coxeter coset enumeration, but I never seem to use it (its really straightforward, but it seems to me to be as much work as the alternative for small cases like this).
The answer was 12. Here's how I would solve it. First, as souldog points out, there are two relations on one generator: momomo = ni and lolo = ni. But if you look at the other relationship in the presentation you will notice that you can get some really nice reduction formula from it by slowly moving syllables over to the other side. You get
Those come in really handy. Now since we don't need to look at anything with lolo or momomo in it, we will just start listing everything without them. Let's list everything up to 5 syllables to start withCode:molomolomolo = ni
molomolomo = le = lo
lomolomolo = me = momo
molomolo = lomomo
lomolomo = momolo
molomo = lomomolo
lomolo = momolomomo
molo = lomomolomomo
lomo = momolomomolo
where I have gone through and used whatever obvious set of relations from those derived above to get the equalities. All the equalities are to words of lesser number of syllables. Since all 5 syllable words have lesser syllable equivalents, a simple induction shows that all larger words are equivalent to a word of four syllables or less (since they contain 5 syllable subsequences). So the ones that don't have lesser equivalents are the only possibilities. There are twelve of them. At that point, I just did a multiplication table and brute forced out the 144 possibilities. No two rows or columns were equivalent (you actually don't need to do anywhere near all of the table to find this out, but I was having too much fun...), and therefore none of the words were equivalent. Thus the answer.Code:ni
mo
lo
===========================
momo
molo
lomo
===========================
momolo
molomo
lomomo
lomolo
===========================
momolomo
molomomo
molomolo = lomomo
lomomolo = molomo
lomolomo = momolo
===========================
momolomomo = lomolo
momolomolo = molomomo
molomomolo = momolomo
molomolomo = lo
lomomolomo = molomomo
lomolomomo = momolomo
lomolomolo = momo
Okay... someone else's turn!
To change, i'll post an easy one:
A doctor goes to a party among a lot of other guests. Just after the punch was served, he receives an emergency call. He immediately finishes his drink and leaves. When he comes back, some time later, every one is dead. The analysis reveals the poisoned punch is the reason of their death.
Why didn't the doctor die?
was the doctor's drink actually punch?
The doctor finishes his drink just after the punch was served. But you don't say if 'his drink' is punch or not. I suppose the doctor's drink is not punch because he leaves just after the punch is served.
Is this the reason why the doctor did not died?
The doctor has been drinking punch as well.
Doctor Luz. Are you giving in Simon666's like avatar?
Noooo.
It's me after trying to solve the last galathaea's question. ;)
[this is a joke]
Could this have been a kind of posion that dissolves under a great deal of pressure (that the doctor must've felt with his patient whereas the calm other guests died) ?
The doctor puts the poisson into the punch?
And fakes the call...Quote:
Originally posted by Doctor Luz
The doctor puts the poisson into the punch?
I want a logical answer not some... urban fairy tail that can be explained in a million ways. If it's like that I'm not counting it as a point.
you still don't have the answer. But I've never heard of a poison that dissolves into stress! :D:D
I think its a very logical possibility. But surely there are more than one possible explanation (more ore less credible).Quote:
Originally posted by SeventhStar
And fakes the call...
I want a logical answer not some... urban fairy tail that can be explained in a million ways. If it's like that I'm not counting it as a point.
OK
For me there are only 2 possibilities
First the punch was poisoned after the doctor left but thats just a guess.
Second the doctor was poisoned too but eventually he took some kind of antidote maybe because of working wiyh an ill patient or he was just checked for infections when he got to the hospital...
Anyway I couldn't figure out a straight answer but maybe the second one will satisfy you
Nope, that's not it.
It's on the rocks
Solarflare, what do you mean by that?
The poison is inside the ice cubes?
Sorry, I meant the poison is in the ice - the doctor drank his so quick that it didn't have time to melt, while others could take longer so more poison dissolved into the drink.Quote:
Originally posted by Elrond
Solarflare, what do you mean by that?
("Give me a ____ on the rocks" = "Give me a ____ with ice")
Yes, the poison is inside the ice cubes, and as the doctor drank it right after is was served, it didn't have enough time to melt and being toxic.