/sci/ - Science & Math

my erdos number is 2 u mad bro

the lowest erdos number that's possible on /sci/ is probably 4. anyone who says otherwise is obviously a mental patient visiting 4chan on the hospital computer.

bump 4 erdos

>>3403609
pics or it didnt happen

I'm grothendieck. sup

>>3403666
pic of u with fields medal or it didn't happen
and datestamp

Erdos number 4

Kevin Bacon number 3

u jelly?

puzzle
step one: picture a flat washer O (circle with circle removed from the middle). find the area of the washer with only a single straight line measurement.

step two: do the same thing with finding the volume of a sphere that's had a sphere taken out from the middle.

morphy number 6

u jelly?

ITT: Godtier individuals who arent fucking retarded.

Needs more euler though..

>>3403684
draw tangent to smaller (removed) circle. call the point of tangency A.
call the points where it hits the larger circle B and C.
measure AB.

can you generalise this to n dimensions?
for example in 2 dimensions we have v-e=0
for n=3 we have v-e+f=2

puzzle

derive pi

>>3403696
What is the proof for that?

>>3403706
pi is old skool man. tau is cooler!!

>>3403707
call the small radius r and the big radius R
the area we're trying to find is pi (R^2-r^2)
now make a right triangle with legs r, AB and hypotenuse R
by Pythagoras AB^2 = (R^2-r^2)
so
area= pi (R^2 - r^2) = (AB^2) pi

>>3403719

sexy as fuck. this is why i love math.

btw whats your major

x, y > 1
1. let m^m = x^y; solve for m
2. let n^n = y^x; solve for n
3. replace x with m and y with n
4. repeat unless x = y
Does this algorithm always end?

>>3403732
not in college yet... in high school
I just do a lot of math
and im OP btw

>>3403719
cool proof but that doesn't work for the sphere!
volume is 4/3 pi (R^3 - r^3)
AB^3 <> (R^3 - r^3)

>>3403738

good job man. you have a bright future ahead of you. Any math books you can recommend for not so mathematically gifted community college student?

>>3403734
let y=1 and x=4
it wont end... y is always 1 and x keeps on decreasing

>>3403750
get The Act and Craft of Problem Solving by Paul Zeitz
it is the best book every written. It changed my life.
there is no math book out there that is even close.

>>3403747
since you know the area of the washer that you get with the intersection of a plane and the sphere (the plane goes through the center), you can use pappus theorem to get the volume.

>>3403751
>x, y > 1

prove that the product of four consecutive integers cannot be a perfect square

>>3403776
check the number of 2's in the prime factorization

bump!

>>3403734
someone solve this

problem by erdos:

prove that in any subset of n+1 elements of {1,2,3...2n} there exist an a and b in the subset such that a divides b.

bumping with euler

>>3403738
>high schooler on /sci/
>actually smart and not retarded

this is as amazing as a girl on /b/ during 2004

prove that the power set of any set A has cardinality larger than lAl.

>>3403978
OP here again... i think ive seen something similar before...

if the set is finite, its trivial since 2^n>n
otherwise, suppose there exists a bijection f.
now consider the set S (a subset of A) such that for all s in S f(s) does not contain s.
now consider f^-1(S)
it can neither be in S nor in S complement
so were done.

>>3404021

wow kid I bet you no more about math then most engineers here

>>3404021

wow kid I bet you know more about math then most engineers here

>>3403910
Assume the opposite is true, i.e. that there is a subset without a and b such that a divides b. Let x be such a subset. Let f(x) be the smallest number in x. Let y be a subset such that f(y) is maximal among all f(x), i.e. there is no valid subset z with f(z) > f(y). 2f(y) is not an element of y, since otherwise a=f(y) and b=2f(y) would disqualify y. Since any number that nontrivially divides 2f(y) is smaller than or equal to f(y), the only such number that is an element of y is f(y). Hence, the set y with f(y) replaced by 2f(y), say y', also contains no a and b such that a divides b. However, f(y') > f(y) since f(y) is not an element of y'. Since we assumed that y was chosen such that f(y) was maximal, this is a contradiction.

>>3404047
engineers don't do real math

I got a math number theory puzzle for you guys:

prove that 3N^4+2N^3-3N^2-2N is divisible by 24 for any integer where N is greater than 0.

It is rather difficult so don't be too ashamed if you can't do it.

>>3404083
just check all residues mod 3
and then all residues mod 8.

there might be a prettier solution... but this works too

>>3403910
>>3404053

nice solution
here's mine:

we partition the set into subsets in the form {a, 2a, 4a, 8a...} for odd a
we will have n sets (one for every odd number<=2n)
clearly if we have 2 elements in the same subset then one of them divides the other, and since we have n subsets and n+1 elements then we are done.

im OP again btw

and send more problems /sci/fags

>>3404083
A proof by simple induction is straightforward. But I'm guessing that's not what you had in mind?

>>3404083
(N - 1)N(N + 1)(3N + 2)

The first two guarantee the result is even.
The first three guarantee the result is divisible by 3.
Therefore the result is divisible by 6.
If N is odd, the first and the third guarantee it's divisible by 4.
If N is even, the second and the fourth guarantee it's divisible by 4.
So, it's divisible by 24 .

>>3403734
someone's gotta solve this

Prove that for any sequence of integers a1, a2, ... , an (not necessarily distinct), there is some nonempty subsequence whose sum is a multiple of n.

>>3404162

OP here again... send harder problems...
consider the sequence of sums (mod n)
a1, a1+a2, a1+a2+a3, a1+a2+a3+a4...
we have n such sums, meaning either there is a sum that is 0 mod n or there are two sums S1 and S2 with S1=S2 (mod n).
If there is a sum that's 0 mod n we're done.
Otherwise, we take S2-S1 to get a sum that's divisible by n.

for example if we have
a1+a2+a3+a4+a5+a6=m (mod n)
and
a1+a2+a3+a4=m (mod n)
then we have
a5+a6=0 (mod n)

Prove that any even integer > 2 can be expressed as a sum of 2 primes

Here's a differential equation I've been having trouble with recently:
<span class="math">
\frac{d^2x}{dt^2}=-\frac{k}{x^2}
[/spoiler]
Any suggestions would be greatly appreciated.

Haha OP I bet you can't solve this!
> Generic unsolved problem I just found in Wikipedia.

>>3404187
f(x) = k ln(x)

...Wrote 'me own Tan function just today...
Testing so far looks like (+/-) 0.1% Accuracy =)

...I'll Call it 1% just to be safe ;)
Anyone wanna look?

>>3404186
OP again...
>be in school
>history class
>write sums for goldbach for 2<n<=1000
>continued doing that for 45 mins everyday until n=15000
>fuck history

Let A_n be the matrix filled in left to right, top to bottom with integers 1 to n^2 in order. Show that if det(A_n) = -2 that n=2

>>3404214
That's nice... I'm looking for x(t).

>>3404274
send harder problems
so subtract the last column from two other columns
these two columns are multiples of each other.

Uses two less accurate Approximations(?) to make one good(?)-ish one...

...Its still not perfect though =(
Half a degree out probably not close enough eh....

But here you go...
Using Side a / side b // angle A
(Assumes side a>=b // Haven't tested using a<b)
A1 = [a / ( a + b)] * 90
A2 = [ a*a / (a*a + b*b)] *90

Angle A = (A1 * 2 + A2) / 3

Pretty simple hey?

>>3404279
Wait... nevermind. I read that completely wrong, as d^2y/dx^2 = -k / x^2

>>3404350
It's all good... it's just like 3a.m. here, so I'm a bit crabby. I tried this:
<span class="math">
\frac{d}{dt}(\frac{dx}{dt})=\frac{dx'}{dt}=\frac{dx'}{dx}\frac{dx}{dt}=x'\frac{dx'}{dx}
[/spoiler]
which makes it separable, but by the time you work it out it is ridiculously complicated (not to mention the 2nd time derivative is NOT -k/x^2).

Show for integers a,b a<b that if a^b = b^a that a=2 and b=4

>>3404377
Haha, I'm on EST too. And don't mind me, I've never taken differential equations.

>>3404379
theres -4, -2...
but except that
if you solve it then u get that (x,y)= (1 + 1/a)^a, (1 + 1/a)^(a+1)
for some a
and if x,y are rational then a is rational
and that clearly clearly can never be an integer

send moar problems

>>3404450
This one still needs a solution:
>>3404187
halp pl0x

>>3404479
sorry can't do much calculus...
the only calculus i know is from deriving myself and reading wiki...

>>3404492
If you're in highschool shouldn't you be doing Calculus by now? I recently passed AP Calc (just finished highschool), it was hella fun. Free university credit for scoring a 5 on the AP exam.

>>3404502
well i do know the stuff from AP calc BC, and i took the AP and got a 5. They don't really teach much there.
and i know some multivariable stuff...
but i mean i don't know differential equations or anything...

>>3404311
Wait you know matrices in high school? Your educational system is like fucking awesome...

>>3404535
nah... i taught myself... my school system is as shitty as all other ones...

>>3404535
Matrices are simple, why wouldn't they teach them in highschool? I was taught them here in highschool.

>>3404187
>>3404187

Try this:

x''(t) = -k/x(t)²
x''(t)*x'(t) = -k*x'(t)/x(t)²
x'(t)² = 2*k/x(t) + A (A is a constant)
dx/dt = sqrt(2*k/x + A)
dx/sqrt(2*k/x + A) = t

Then you gotta integrate this shit, too lazy to do it myself

>>3404549
Well they seem simple now, but I certainly wasn't taught about them in high school. I don't live in the USA, maybe that's why.

>>3404549
Yup, same here. We did Gaussian elimination on, like, 8-9th grade.

>>3403613
My erdos is 4.

...In true trial & error style, should I be putting Pi in there?

Angle A = (A1 * 2.14 + A2) / 3.14...

...Seems to even out its accuracy a bit =)
0.3 Degree's out on angle 67.5 (gives 67.8 instead of 68 // wants to be 67.5 really)

...Is this awesome or am i stupid?
=D either way....

bumping for more problems!!!!

What is the sum of every possible combination of 3 numbers 1 through N multiplied by eachother? In terms of N.

A.K.A. if N=3, the sum is 1*2*3=6
if N=4, the sum is 1*2*3+1*2*4+1*3*4+2*3*4=50

if this is too difficult, you can try the sum of every combination of two (aka if N=3, 1*2+2*3+1*3)

>>3404723
you use induction... it trivializes it...

erdos number 5
u jelly?

>>3403704
Yes, it can. Look for Gauß-Bonnet theorem.

bumpin for interest

anyone have a relatively simple proof showing for the irrationality of pi?

>>3404723
>>3404723
Basically it's
<span class="math">\sum_{i=1}^{n}\sum_{j=i+1}^{n}\sum_{k=j+1}^{n}ijk = \sum_{i=1}^{n}i\sum_{j=i+1}^{n}j(\frac{n(n+1)}{2}-\frac{j(j+1)}{2}) [/spoiler]

and you calculate this shit and so on

I tried to find its exact expression but holy shit that takes too long. It will be like N^5 or something.

>>3403734

Ok I solved it. If you start with y > x > e, it won't terminate in finite time. Nice problem - where'd you get it?

>>3403704
I am confused as to how you want it generalised.
Take R^n, add a point to get a n dimensional sphere. Triangulate using simplicial complexes and take the homology complex. Taking dimensions and adding along the chain gives you what you want I think.

>>3403734

Proof:

Say y > x. We have m^m = x^y, so y > m > x, because z^z is an increasing function (for argument z > 1, which we're assuming here), and x^x < x^y < y^y. Same goes for n. So, at each step of the algorithm, the numbers get closer together.


Say we start at y_0 > x_0, with x_0 > e (you'll see why soon). If it finishes after finite steps, then let's call the last values before it terminates (x,y), with y > x. That it terminates means x^y = y^x = m^m.

Let A = m log m = y log x = x log y. Since m > x, then if A > m, we have A > x. If m > e, then A = m log m > m, so let's suppose that x_0 > e => x > e => m>e => A > m => A > x.

From y log x = A, we can write y = A / log x, and substituting this we have A = x log y = x log [A / log x]. One solution to this is x = y = m, or A = x log x, but that's not the one we're looking for, because we need one with y > x. So, are there any others (with A > x)?

Let's differentiate x log[ A / log x] - A with respect to A. The derivative is x/A - 1, and since we're restricting to A > x, it's negative. So, there can't be multiple zeroes on A > x, which means there's only the one.

So, if x_0 > e, it won't finish. QED

Question: Does the gap between x and y shrink to zero asymptotically? I think so, but I don't know how to show it

>implying Erdos ever actually touched a woman.

>>3405759
>implying this is important

>>3404535
I did matrices in high school. (British further maths)

>>3405759
Erdos had 13 children, 5 of which outlived him.

>>3405765
>implying op's pic is irrelevant to the discussion

>>3405786
lolwut? you got some sauce there, chief?

>>3405794
>mfw I'm stoned and I read and wrote euler instead of erdos.

>>3404535

Canada fag here
You learn of matrices and things along those lines depending on which grade 12. math course you take.
You learn it in the simpler one knows as applied.

Wait what? Most people in the US math system learn matrices a little bit in 8th/9th grade, then cover them more in depth in 10th/11th grade

>>3405794
>>3405786
Erdős never married and had no children.

http://en.wikipedia.org/wiki/Paul_Erd%C5%91s

>>3405944
he had some problem where his genitals hurt if he got hard...
but who needs sex when u got math and meth?

moar math!

>>3405954
Oh yeah, the fucker was on amphetamines, he was challenged to a 500$ bet that he we addicted, and couldn't do a month without them. Motherfucker won the bet and then resumed the use.

>>3406117
erdos afterwards told the dude who bet him that he put his math back a month...
something about how he couldn't get ideas without it...

>>3405616
I made it up, thinking about how to find the exponential mean of two numbers. Someone in a different thread suggested I just solve log log m + log log m = log log x + log log y, which made sense and avoids ambiguity, but I still thought the original problem was interesting.

bump

bamp

>>3403609
mine is 0...that's right...I'm not dead....

there are 3 people, one always speaks the truth, one always lies, and one guesses randomly. How can you find out which is which with only 3 questions?

>>3403704
Yes, I know how to do this.
V - E + F
Okay, Let D(n) be the number of n-1 dimensional 'pieces', so D(1) = V, D(2) = E, D(3) = F.

For an N dimensional shape the number
D(1) - D(2) + D(3) - D(4) + D(5) - ... + (-1)^N*D(N) is a constant. Not sure which one though, maybe N-1 or something.

>>3408299
Whoops, that constant looks like it might be 2N - 2.

>>3408306
aw god damnit, the formula is wrong too:
D(1) - D(2) + D(3) - D(4) + D(5) - ... + (-1)^(N-1)*D(N)
is the right one.

find the infinite sum
1/4+1/9+1/25+...+1/p^2+...
of the reciprocals of primes squared

prove fermat's last for specific exponents.

>>3408384
>>3408384
>implying this is possible

>>3408450
it is very possible

find all positive integer solutions to
a^2+b^2=3c^2

a^2 + b^2 = c^2.
solve for A.

>>3408384
so we know the sum of reciprocals of squares is pi^2/6=a
and the sum were trying to find is B
a=B^0+B^1+B^2+B^3+B^4...
and this is a geometric series we can easily solve...

>>3408498
a = sqrt(c^2 - b^2)

Sup, bitches? Where's my boy Perelman?

/sci/ needs moar math

>>3408225
>>3408490
>>3408415
these aren't solved yet

Flip a coin repeatedly. You "win" if more than 2/3rds of the flips are heads. What is the probability that you win?

>>3409489
You win as soon as the total heads > (2/3) number of flips?

>>3409518

Oh, apologies for the ambiguity. Yes, the moment 2/3rd of your flips are heads, you win. If you get a head on your first flip, you win.

A part a one inch long needle is placed randomly on 2x2 surface. If another one inch needle is placed in the same manner, what is the probability that the two needles overlap?

>>3409524
> or >= ?
>=, then you win before the first flip.

Definitions:
Let X and Y be metric spaces and f: X -> Y be a map:
The map f is proper if its inverse image of a bounded set is bounded.

The map f is (uniformly) bornologous if for every R>0 there exists S>0 such that if d(x,y) < R then d(f(x), f(y)) < S for all x,y in X.

The map f is coarse if f is proper and bornologous.

Two maps g,h from a set X to a metric space Y are close if d(g(x), h(x)) is uniformly bounded in X.

Problem:
Check that two maps f,g are close if and only if they arise from a coarse map X x [0,1] -> Y by evaluation at 0 and 1.

bumpity

>>3409489
>>3409524
so i did an experiment. it looks like you'll win about 69% of the time. sounds good to me!

>>3409489
You'd be amazed how many people are utterly convinced the answer's 100%. /sci/'s had a couple shitstorms over this one. Josef was in the 100% camp for a while, I think.

>>3409874
>>3409914
The exact answer, by the way, is (5 - sqrt(5)) / 4.

>>3409935

Proof? I got the same answer, but did it in a hamfisted way.

>>3409914
But the game doesn't specify when you lose.

is there a way to prove that math isn't just made up bullshit like the rules of chess?

>>3403776
>prove that the product of four consecutive integers cannot be a perfect square

<span class="math">0 x 1 x 2 x 3 = 0^2[/spoiler]

>>3410662

When you don't win, of course.

>>3410667

http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Nope. That doesn't imply that our Mathematical System is flawed, but we can't prove that it's not.

postin' in a cool thread

Me and my friends had this little math going going where you get three of the same positive integral and the common operators as well as the square root, the factorial and parentheses and have to arrive at 3. It's my turn at 11 and I'm now trying to prove it's not possible. To reduce the cases to a finite number, I'm trying to eliminate the square root and the factorial, as they are the only ones that can be used infinitely. What I'm asking now is: Is there any natural n greater than 1 so that sqrt(n!) is a natural number again?
I remember reducing the problem to 'Is there a prime k so that there are no primes p so that k<p<=2k', but don't recall the exact procedure.

Me and my friends had this little math thing going going where you get three of the same positive integer and the common operators as well as the square root, the factorial and parentheses and have to arrive at 3. It's my turn at 11 and I'm now trying to prove it's not possible. To reduce the cases to a finite number, I'm trying to eliminate the square root and the factorial, as they are the only ones that can be used infinitely. What I'm asking now is: Is there any natural n greater than 1 so that sqrt(n!) is a natural number again?
I remember reducing the problem to 'Is there a prime k so that there are no primes p so that k<p<=2k', but don't recall the exact procedure.

Found this gem in a philosophy book.

"Nevertheless, there are many unexpected connections in mathematics that are difficult to explain. For example, pi turns up in all kinds of quite unrelated places, such as the solution to Buffon's needle problem [infinite set of parallel lines one unit apart, one unit needle dropped, what's the probability of the needle landing on a line?]... The answer to this problem turns out to be [...], but why pi should turn up in this context is a complete mystery. At a deep level, then, there remains something mysterious about mathematics. "

>>3410787
Not sure why that's so hard to explain, since the probability of it crossing a line depends on the angle between the needle and the lines. You take the integral over the possible angles and there you are. At least I guess that'd be how it works

>>3410921

Yep, just take the expected value of the sine function over the interval (0, 1/2pi).

For a more interesting problem, replace the parallel lines with an infinite set of 1 unit radius circles tangent to 4 other circles.

mystery in math = you don't understand the material

>>3410662
You lose if you flip forever and never get up to the 2/3 ratio.

>>3410659
I never figured out the proof, although I've seen at least one posted. It involved recurrence relations, I think.

>>3410787
Pi shows up there because you put it in there. It'd be like an Object-oriented programmer saying it's a problem to skeptics that object oriented programming languages are so perfect for describing things. It's because real math is Turing complete, and every language is just as productive, though less effective in various carious, for solving problems.

>>3410671
positive integers was implied

>>3410748
you need an even number of each prime
so say the biggest prime you have is p
now to have it again you need 2p,
but there is another prime between p and 2p, a contradiction that p is the largest prime you have...

OP here btw and holy shit the threads still alive :D

>>3413344

>6 hours since last post

>thread is 'alive'

wat

>>3413349
when i got here it was front pg
cause of >>3413313

>>3409489
if the first flip is heads u win, if not then use recursion (so then if it happens again after that tails you need 2 heads in a row to win, if u get HT or TH after then u do recursion again, if TT then recursion a bit differently)
this trivializes it

>>3408225
number the people 1,2,3, then ask if the number assigned to random+1=the number assigned to true (mod 3)
the rest is simple

>>3408490
consider solution with smallest a
but a,b,c all divisible by 3, so you can get a smaller a by dividing all variables by 3, a contradiction

>>3413594
and why is b divisible by 3 exactly?

>>3413623
for all n, n^2=0 or 1 (mod 3)
if b is 1 then there are no solutions since
1+0 or 1=0 (mod 3) cant happen

>>3413638
ty

bumping a win thread

evaluate:
1/2+1/2^3+1/2^6+1/2^10+1/2^15...

>>3413804
>implying this is possible

Prove that P=NP

>>3413833
... yeah

>>3413804
That's
<span class="math">\sum_{n=0}^{+\infty}\frac{1}{2^{\frac{n(n+1)}{2}}} [/spoiler]

>>3413638
I may sound stupid but why is n² = 2 (mod 3) impossible?

>>3403704

Hey , this looks like Gibbs phase rule...
any reason for that ? or it's just a coincidence ?

http://en.wikipedia.org/wiki/Gibbs_phase_rule

>>3413344
Yeah, that was probably the way I reasoned. Any idea on how to solve the actual problem? I doubt there is a prime with those properties, but I don't have much of a clue about how primes work.

this got me a 2.7/10 on physical chemistry.

We need more threads like this on /sci/ good work mentalmen.

I have just finished a degree in maths, so don't take me to be a crank but I don't accept the solution to the Monty-Hall problem.
http://en.wikipedia.org/wiki/Monty_Hall_problem


Here's why. Instead of doors, have boxes with gold or coal.
After he opens the first coal box, you close your eyes and the two that are left are moved around so you don't know which is which. now you choose one, and have a half chance to get the gold. If you choose the same door as you did before, you STILL have a half chance to win.

So staying and switching give the same result, half a chance to win.

If anyone can explain why this is wrong please do because it really pisses me off.

>>34139289
well, 0^2=0, 1^2=1, 2^2=1 (mod 3)

>>3415140
experiment... take 3 cups, put a coin under one of them, get someone to play and do like 100 rounds. it should give you the intuition.

>>3414937
its well known that there's always a prime between n and 2n.

erdos (i think) wrote this poem about the theorem:

ill tell you once
and ill tell you again
there is always a prime
between n and 2n

>>3413916
i think he meant in a closed form...
as in no infinite sum...

>>3415376
It's called Bertrand's postulate.

>>3415140
well, if you know which one you chose earlier it doesn't matter that you shuffled them...

toasting in an epic bread
/sci/ needs more threads like this one.

>>3415385
Yeah, I meant they are indistinguishable obviously.

>>3413928
Try it. There are only 3 numbers mod 3. square them

>>3415140
When you first guess you have a 1/3 chance. Then the host basically lets you swap that door, for the other two. Of those two at least one will have a goat and he reveals it. If you switch you get 2 doors.

>>3415394
when you pick randomly then if you pick the one you had before there's a 1/3 chance, if not then 2/3.
there's a 1/2 chance you will pick either one, and 1/2(1/3)+1/2(2/3)=1/2
so the probability is 1/2, but if you pick the one you didn't have its 2/3

how do you prove pi is irrational?

I can prove God exists using binary logic. Umad, mathfags?

bump for moar problems

Are the mods even deleting math threads now? I was posting in the pure math thread just a few minutes ago but now it's vanished. It had some discussions relating to stock market speculation, I wonder if that's what caused it to be zapped?

>>3415953
they probably deleted it for a different reason

PUZZLE:
You have a set S of 12 variables. Each is a real number. One variable has a value that is not equal to one; the other eleven variables all have a value of one.

You do not know which variable is the unusual one, nor do you know if it is greater or less than the others.

Define a function f: P -> R,
where P is the set of all subsets of S, and R is the set of real numbers. f(X) is the sum of all elements in X.

You have an oracle, which knows the value of each variable. However, you can only communicate with the oracle in the following way: You can present the oracle with two disjoint subsets of S, denoted X and Y, and the oracle will tell you which of the following is true:
f(X) > f(Y)
f(X) < f(Y)
f(X) = f(Y)

You may communicate with the oracle in this way three times. From this, you must deduce which variable has a unique value.

Note: This is entirely possible. If you try to prove that it is not, I will call you a troll.

Pic is broomrape.

>>3416002
Like the naked girl in the first post.

>>3416006
>>3416006
first question, you take variables 1,2,3,4 and 5,6,7,8.
if they are the same, then you know one of 9,10,11,12 is the unusual one, so you compare 1,2 and 9,10
if 1,2,3,4, and 5,6,7,8 are different, suppose WLOG that the sum of variables 1,2,3,4 is greater. now compare 9,10,11,12 with 1,2,5,6.
if they are different and WLOG 9,10,11,12 is larger, you know 5 or 6 is the different one.
if they are the same then one of 3,4,7,8 is different. so compare 4,7 with 9,10.

>>3416065
What if:
You compare 1,2,3,4 against 5,6,7,8 and 1,2,3,4 is greater.
Then you compare 1,2,5,6 against 9,10,11,12 and they are the same.
Then you compare 4,7 with 9,10 and find that they are the same.
Then the unique variable must be either 3 or 8 but you are out of tries!

I don't think your solution works in all cases.

>>3416114
>>3416065
you should compare on the second step 9,10,11,12,8 and 3,4,5,6,7

>>3416006
>>3416006
>>3416006
>>3416006
>>3416006
>>3416006
>>3416006
>>3416006
>>3416006
moar liek this

Has anyone got a good way of remebering the basic graph transformations? (Stretchs, translations)
I'm 20 and still have to stop and think when i'm doing them

>>3416051

ConspiracyCat is pleased with your answer

>>3416403
you mean the names of the transformations?

>>3416430
i dont think so(unless they have formal names i dont know), the stuff you learn when you're about 15/16ish. Like f(x+2) is a horizontal shift of a graph 2 units to the right.
Mostly the reflections are what trip me up

>>3415383
Thanks, I had no idea.

>>3416440
oh... i learned that when i was very young (6-7) so its intuitive for me.
but this is kind of how I think about it.
the trick is usually to find where its f(0)

so if you have f(ax+b), (i wont do anything more difficult than linear transformations cause its similar) then this is the same as
f(a(x+b/a)), which means you dilate by a with the center (f(0)) at -b/a.
so you put f(0) at x=-b/a and then you just put the graph, but on your axis, instead of intervals of a, you put intervals of 1.

hope this makes sense...

>>3413804
>>3413804
someone solve this

Find the coordinates of stationary points and nature of them of y=(2x-3)^6

i dy/dx this and i got 12(2x-3).

i solved for x and got 3/2 but when i sub it back in after i d2y/dx^2 i get 0 which means its a point of inflection ntot a minimum like in the answer.s any help pls?

>>3416545
shouldnt the first derivative be 12(2x-3)^5?

>>3416545
>>3416545
if the 2nd derivative is 0 it doesn't imply its a point of inflection. you have to check the 3rd derivative, and if it is 0 it may still be a min.

I'm having trouble interpreting OP's image.

Does it mean
>Have sexual intercourse with women and prove theorems

Or

>Disregard women and prove theorems.

>>3416555
sorry i meant that

>>3416553
are you sure? ive never heard of finding the 3rd dy/dx never used it before...

>>3416559

the latter

>>3416559
erdos (guy in pic) had some problem and he couldn't have sex...
so he did math instead
and meth

>>3416560
take y=x^6 at x=0.
the 2nd derivative is 0, but it clearly is a minimum.

>>3415771
http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
-.-

A cylindrical hole is drilled straight through the center of a solid sphere. The length of hole in the sphere (i.e. of the remaining empty cylinder) is 6 units. What is the volume of the remaining solid object (i.e. sphere less hole)?

>>3416545
just take lim z-->0 of f(3/2+z).
since this is always greater than f(3/2) then its a min

>>3416594
reword the problem please. I don't get what you're asking.

>>3416594
this is simple with calculus.
any discrete way to do it?

I like this one.

There are n balls rolling along a line in one direction and k balls rolling along the same line in the opposite direction. The speeds of the balls in the first group and in the second group are equal. Initially the two groups of balls are separated from one another and at some point the balls start colliding. The collisions are assumed to be elastic. How many collisions will there be?

By the way, if you are ever bored or want to waste several hours:

http://mathoverflow.net/questions/29323/math-puzzles-for-dinner

>>3416664
>>3416664
this is really simple... just think about the collisions as the balls passing through each other...
so each one of the n hits each of the k 1 time so its just nk

>>3415140
picture a scenario in which there are 100 doors instead of 3, with 1 having a car behind it and 99 having goats behind them.
You choose your initial door, and Monty opens 98 doors, showing you the goats behind them. Would you consider it beneficial to switch to the only other remaining door?

>>3416673
By your logic, wouldn't it be min(n,k)?

>>3416680
no. try k=1. clearly there wont be just 1 collision.
k=1 n=2 is easy to visualize.

>>3416675
If I closed my eyes and forgot which door I chose and choose randomly I have a half chance of getting the right door, right?

Flipping a coin to choose whether you stick or not gives you a half chance of being right no?

>>3416691
>>3416691

explained here:

>>3415749
>>3415749
>>3415749

There's a simple problem I've been having trouble with. I'm new to combinatorics, but I still feel a little dumb.

There are 45 white balls and 5 black balls distributed uniformly at random into 10 bins such that each bin has exactly 5 balls.
How many ways are there for the first two bins to have all white balls?

Answer 1: There are 45 white balls and 10 being picked from them, so nCr(45,10)

Answer 2: The first 10 balls are fixed, and there are 35 white balls and 5 black ones left, so it's nCr(40,5)

Obviously, one of these is wrong. Can someone help me out?

>>3416691
No, that's the thing. Think about your chances of picking the door with the car vs your chances of picking a door with a goat in the first place.

>>3416696
>>3415749

Why is there a 1/3 chance if I pick randomly the one I chose at first. The previous decision has no bearing on this one. Now there are only two things to choose from and I choose one at random. The fact that earlier I chose something has had no effect on where the car is and so cannot effect the probability of which door it is behind.

>>3416701
>>3416701
suppose the balls are distinguishable.
so number all the balls so that 1-45 are white and 46-50 are black.
now in bin 1 we chose 5 white ones so 45C5
in bin 2 we choose 5 whites, but there are only 40 left since 5 are in bin 1, so 40C5
in the next bin anything is fine, and there are 40 balls left, so 40C5
next one is 35C5
and so on
so its

(40C5)(35C5)(40C5)(35C5)(30C5)(25C5)...(5C5)

if they are indestinguishable, its just how many different ways can you put the 5 black balls in 8 bins, which you can do cases:
case 1: all 5 balls in same bin
case 2: 4 balls in 1 bin, 1 ball in another
and so on

>>3416710
>>3416710
suppose you pick the one you chose 1st
now we know you picked it first (we are supposing that)
and then the probability is 1/3.

>>3416710
Your previous decision does, in fact, alter the situation.

In the beginning, you have a 1/3 chance of choosing the car. Monty opens a goat door. Now if you switch, you lose, and if you stay, you win.

You have a 2/3 chance of choosing a goat initially. Monty opens the one other goat door. Now if you switch you win, and if you stay you lose.

Therefore, your chance of winning by switching is 2/3 and your chance of winning by staying is 1/3.

Here's a handy diagram I drew for a thread a few weeks ago

>>3416754
>>3416754
>>3416741
Thank you. I accept the reasoning and will never bring it up again.

My problem was this: I can get 50% chance of winning by flipping a coin to switch or not.

Answer is: You gett 66% chance of winning by always swapping.

I'm an idiot. Thank you /sci/

MOAR PROBREMS

>>3416800
Hard to know if you've seen these before but this is always a good one:

In a room there are some people who shake hands with each other. No one shakes hands with themselves and some pairs of people don't shake hands.

Show that an even number of people shook an odd number of hands?

>>3416830
>>3416830
easy... the sum of the number of handshakes of everyone must be even (since each handshake involves 2 people), and if an odd number shook an odd number of times then the sum is odd.

POST HARDAR PROBREMS

>>3416771
nah you're not an idiot this 'monty hall problem' is famous and alot of very intelligent people get it wrong at first.

Two players play a game where on their respective turns, each player says a number between 1 and 10 higher than the current score, which starts at 0. Whoever says 100 first wins. Does the first player have a winning strategy?

>>3416899
yes, he picks 1, and then picks 11-(what his opponent said last turn).
that way first player says all numbers in the form 11n+1, and 100=11*9+1

i'm disappointed /sci/... post harder stuff

I'll just leave this here.

>>3416846
2 people shake hands once. Then the number of handshakes is 1. ??

>>3417009
but we count the sum of the number of handshakes of each person.
person 1: 1 handshake
person 2: 1 handshake
1+1=2 which is even

>>3416989
Cauchy's proofs for the Cauchy Schwarz inequality and the AM-GM inequality are two of the most beautiful discrete mathematical gems.

Here's a hard one: Show that the probability of two positive integers chosen at random being relatively prime is 6/(pi^2)

>>3417035

Isn't the proof for the Cauchy Schwarz inequality just tiresome algebraic manipulation? I've never seen a different proof.

>>3417039

by random integers i believe you mean random between 1 and n as n goes to infinity...
anyways...
the probability that they are both divisible by some prime p is 1/p^2, and we need to have that they are not both divisible by any prime.
from here its simple... (i solved the calculations earlier here: >>3408384 >>3408543)

>>3417045
the prettiest one is this:
(a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_nx+b_n)^2= f(x)
now since this is always >=0, its descriminant is <=0

>>3417045

you can also prove it with vector dot products, and how cos(x) is between 1 and -1
but the proof with the quadratic is prettier

>>3417064
yes 1,2,3...., i said "positive integers".

You've skipped too many steps, how did you show that the prob that they are both divisible by some prime p is 1/p^2?

>>3417108
well, if we want them to be divisible by p,
the prob that the first one is is 1/p
same for the second
so we multiply 1/p*1/p=1/p^2

>>3417108
and what i meant was that you cant pick a random integer... you can only pick a random integer in an interval
so we have to define the interval for our random variable N to be between 1 and n, and then we can take lim n-->infinity

>>3417064
actually its easier to do the calculation like this:
the product over primes of (1-1/p^2)=product 1/(1/(1-1/p^2))= 1/(the infinite sum 1/1^2+1/2^2+1/3^2...)=6/pi^2

Science & Math?
In my /sci/?

http://chanarchive.org/request_votes

shit thread is shit

win thread is win

BUMPIN FOR SOME HARD PROBREMS

>>3413558
>recursion
>recursion a bit differently
>this trivializes it

>>3417466
>>3417466
it does...
then theres some algebra and shit... you can let the candyass engineers do that

IF YOU CAN SOLVE THESE, THEN +1000 Internetz TO YOU AND I WILL BLOW YOUR JOB!

(+100 Internetz for a good hint as well)

check them dubs

Betty and Sam are both given a natural number less than twenty.

Betty is instructed to subtract two from her number, divide the result in half, and then add two to the new result.

Sam is instructed to divide her number in half, add two to the result, and divide the new result in half.

Before carrying out their instructions, Betty and Sam secretly swap numbers. Their results should have been the same, but instead they differ by six.

What numbers were given to Betty and Sam?

>>3417578

Betty has the number 8 and Sam has the number 16.

<span class="math"> \frac{b-2}{2}+2=\frac{b+2}{2} [/spoiler]
<span class="math"> \frac{\frac{s}{2}+2}{2}=\frac{s+4}{4} [/spoiler]
<span class="math"> \frac{b+2}{2}=\frac{s+4}{4} [/spoiler]
<span class="math"> 2(b+2)=2b+4=s+4 [/spoiler]
<span class="math"> 2b=s [/spoiler]

Now when you switch them the difference is 6, so:

<span class="math"> \left | \frac{s+2}{2}-\frac{b+4}{4} \right |=\left | \frac{2s-b}{4} \right |=6 [/spoiler]
<span class="math"> \left | 2s-b \right |=24 [/spoiler]
<span class="math"> \left | 2(2b)-b \right |=\left | 3b \right |=24 [/spoiler]
<span class="math"> \left | b \right |=8 [/spoiler]
<span class="math"> b=\pm8 [/spoiler]

Because b<20 this gives b = 8 and therefore s = 16. You can plug those numbers back into the original equations and see that it satisfies everything.

>>3417500
I keep seeing this problem on this board. Does it have a canonical name?

>>3417723
>>3417723

the math looks correct, but my question is this:

since they differ by six, why is b=8 and s=16. that's a difference of 8. what's the deal? =|

>>3417782

The problem doesn't state that the numbers Betty and Sam get differ by six. It says that after they switch numbers with each other and do the appropriate calculations the resulting numbers differ by six.

Let n be any finite sequence of digits abc...k, show that there exists an n such that 2^n = abc...k....m

ex: pick the sequence 3, then 2^5 = 32, so n=5 works.

>>3417847
sorry mistake:
>Let abc...k be any finite sequence of digits,

>>3417876

Still not totally sure what it is you're asking.

>>3417918
If you give me any finite sequence of digits, say 1531513553634, then i can always find some integer n such that 2^n = 1531513553634(somemore digits)

like if you picked 409, I'd say n=12 since 2^12 = 4096

show that i can always do this.

>>3417733
It emerged in my research.

>>3417940
So, for any natural number X, X*10^a + b = 2^n for some n.

bump

>>3417723
how do you display math nicely here in the forum?

>>3418377

put it in [m ath] [/m ath] tags (without the spaces).

bump

bump
captcha: godel

>>3403706
pi
therefore pi

>>3417500
why are you using vartheta and varphi?

>>3417500
somehow looks like cauchy-riemann-eqn ^^

help with this please

4x over (x-6) times 2 over (x+6)

This thread seriously lacks geometry.

>>3420256
all the problems in your post are easy as fuck

>>3420268
Feel free to post some problems that are of higher difficulty so that others like yourself may enjoy solving them. :)

>>3420272
prove that if 2 internal angle bisectors of a triangle are congruent, then the triangle is isosceles.

>>3420272
prove that all euclidean constructions with a compass and straightedge can be done with only a compass.

wait, this thread is actually 4 days old. I'm proud, /sci/

>>3420287
OP wins an internet for starting this

In a triangle ABC, prove <div class="math">tan(A)tan(B)tan(C) = tan(A) + tan(B) + tan(C)</div>.

>>3420304
(tan A+ tan B)/(1-tan A tanB)= (tan 2PI- tan C)/(1+tan 2PI tan C) = -tanC
(tan A+ tan B)= -tan C(1-tan A tan B)

>>3420304
I've never even paid attention to that.
>>3420344
wtf? this shows tanA+tanB+tanC=tanA*tanB

>>3420354
Look at that last line again. Multiply tan(C) through both terms of the parenthesis.

>>3420354
oh I'm sorry, I must be really high today! my bad!

>>3420304
tg(A+B) = tg(Pi-C)
given the fact that tg(Pi - x) = -tgx, we have
tg(Pi-C) = -tgC
which means that:
tg(A+B) = (tgA+tgB)/(1-tgA.tgB) = -tgC

<=> tgA + tgB = -tgC (1-tgA.tgB)
<=> tgA + tgB + tgC = tgA.tgB.tgC

Suppose <span class="math"> k [/spoiler] runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, is there a time at which that runner is distanced at least <span class="math"> \frac{1}{k} [/spoiler] (along the track) away from every other runner?

>>3420268
Could you help me out on #23 then?

>>3420577
Shit, I mean the second #23, the one with 2 semicircles.

>>3420579
notice that the answers don't depend on the position of the center of the circles.
If the point is at the center of the square, you have a single circle (the two halves are the same)
Therefore, you have a circle with R=1
The surface is then: <span class="math">\pi[/spoiler]

>>3420579
The answer is B.

Refer to picture and what I have labeled as s.

The radius of the small circle is s/2.

The radius of the big circle is s*root(2)/2.

The area of the shaded region is 3s^2*pi/8.

The sum of the radii is 2.

s/2 + s*root(2)/2 = 2

Solve for s and substitute in 3s^2*pi/8 for the area.

It's messy, I'm sure there's an easier way to do it, but it works.

>>3420622
The problem explicitly states they are different sized semicircles.

>>3420635
the problem doesn't prevent you from taking a "limit". (the surface is a continuous fonction, since it is constant, according to the possible solutions)
I can then assume that the point is at the exact middle of the diagonal, can't I? (I'm really asking, maybe I'm wrong but this is the reason that makes me think I'm not)

>>3420642
No, you can't assume that. The answer is pi when the centers match the center of the square, but not otherwise. Assume the radii are distinct and calculate it.

>>3420642
here
>>3420627 plus I think there is a mistake: the radius of the small circle isn't s/2! (what you refer at in the drawing isn't a diameter)

>>3420656
so you deny that the sum of the two disctinct sufaces is a continuous function? I can't agree :/

>>3420657
What? Yes it is. Draw the rest of the square, clown.

>>3420693
oh, you're right.
However, tell me:
you say "solve for s"
Which means that the way you reason allows a single value for s. What about the surface for a different value of s?

>>3420642
It's not a function dude. There is only one possible situation where there are two semicircles of different radius inside a square. You can't just pick any point on the diagonal and draw semicircles in there. It's a specific point. There's no limit to take because the function is not continuous for there are infinite points on the diagonal where such semicircles cannot be constructed.

>>3420699
oh right, I didn't pay attention to that. Thanks for highlighting it!

Why is combinatorics very difficult? I found those geometry questions fair but problems with combinations and anything discrete leave me with a blank stare.

>>3420777
People need to learn how to count. A lot of people lack conditional probability skills, and results in probability are sometimes counter-intuitive.

"How the hell am I supposed to count ALL those things?"

Such as the Iraq.

Calculate sqrt(2 + sqrt(2 + sqrt(2 + ... )))

>>3421049
Let that number be equal to x.

Square both sides.

x^2 = 2 + x

Solve.

x = 2 or x = -1.

2 works.

another challenge for you guys:
show in two different ways that
<span class="math"> \lim_{n\to +\infty} n^2 \int_0^{1/n} xf(x) \,\mathrm{d}x = \frac{f(0)}{2} [/spoiler]
(one of them is obviously a change of variable, but I'm struggling with finding a second method)

>>3421305
actually, people who have these exercises to do don't have access to series expansion (which works too, before integrating), so if someone has a more elementary solution, it is welcome!

In pentagon ABCDE we have the following:
AB || CE
AC || DE
AD || BC
AE || BD
BE || CD

Prove pentagon ABCDE is a regular pentagon.

>>3421049
well phi (1.61...) in nested radical form is sqrt(1 + sqrt(1 + sqrt(1 + ... )))

This thread is inspiring me to open up my Calculus book and stop dicking around.

>>3420627
Thanks a lot man, this thread is quite inspiring. Although I'm not bad in math (for my age and location) all these people make me realize that I really don't know shit, and that I need to learn math asap.

Cheers.

>>3421647
>emulation
>beautiful
>synonyms

>>3421663
wut

>>3417500

I spent about 10minutes looking at this.

If you work out the derivatives (ie quotient rule on the left, product rule on the right) and then square and sum both of those equations you'll get an equation without any cos(f) or sin(f) that looks like

(df/dy)^2 = [cos^2(x) + sin^2(x)*(df/dx)^2]/sin^4(x)

I don't have much knowledge of PDE's and it doesn't seem to be solvable by separation of variables.

gl bro