Search found 36 matches

by Heptadecagon
Mon Jan 05, 2015 10:13 pm UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

Re: The infinite suburb and the teleporter breakdown.

I think the following is easier. The two moves are two sides of a triangle, the third side being of length 1. By the triangle inequality, those two sides cannot differ by more than 1. Therefore if one of the moves has integer length, the other cannot. Very nice, all the numbers I wrote out were rea...
by Heptadecagon
Wed Dec 31, 2014 6:35 pm UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

Re: The infinite suburb and the teleporter breakdown.

Nice solution of getting to every house exactly once! While that shows you can get to any house when a finite amount of houses are removed, can removing a finite amount of houses increase the potentially needed teleports arbitrarily high? Or is there an upper bound on teleports for that as well? Als...
by Heptadecagon
Mon Dec 29, 2014 10:18 pm UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

Re: The infinite suburb and the teleporter breakdown.

Lopsidation's equal-distance hop problem can't always be done in two jumps: any solution in two jumps from (0,0) to (x,y) would have to land on a line going through (x/2,y/2) perpendicular to the line from (0,0) to (x,y), and that leads to a linear Diophantine equation that doesn't always have solut...
by Heptadecagon
Mon Dec 29, 2014 8:21 pm UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

Re: The infinite suburb and the teleporter breakdown.

...(3,4) corresponds to a length of 5. So every hop is length 5. It's just that he took multiple hops in the same "direction", and you never specified each hop had to be in different directions. You don't need to use multiple hops of distance 5 on (3,4) ; you could rather do it all in one...
by Heptadecagon
Sun Dec 28, 2014 6:21 pm UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

Re: The infinite suburb and the teleporter breakdown.

Lopsidation, I think Nitrodon's variation, besides requiring the x and y distances to be relatively prime, continued with the restriction to integer distance jumps, so your solution would not work. Clearly there are a lot of variations on this problem, which is cool. Variations could also be made in...
by Heptadecagon
Sun Dec 28, 2014 4:17 pm UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

Re: The infinite suburb and the teleporter breakdown.

Nice solutions. Here is a proof that you can't get from (0,0) to (1,0) in two teleports : If it were possible, then there would exist a point (x,y) (which can be assumed to have x and y greater than 1) such that it is an integer distance from both (0,0) and (1,0). Bec...
by Heptadecagon
Sun Dec 28, 2014 3:21 am UTC
Forum: Logic Puzzles
Topic: The infinite suburb and the teleporter breakdown.
Replies: 26
Views: 7373

The infinite suburb and the teleporter breakdown.

Hello, here is an puzzle I thought of recently that I've only half-solved. Background: Suppose there is a suburb with houses occupying/centered on each integer pair of points (x,y) on an infinite plane. Each house has a teleporter at its center configured to teleport to any house an integer distance...
by Heptadecagon
Sat Dec 29, 2012 1:17 am UTC
Forum: Logic Puzzles
Topic: The truthteller and the politician
Replies: 65
Views: 26250

Re: The truthteller and the politician

Although now that I think about it, maybe the solution is to amend your rule to be: "I think if the politician acts as the truth teller would if the correct and incorrect roads were switched, and the truthteller and politician were switched, then there's no possible way to be sure which is whic...
by Heptadecagon
Fri Dec 28, 2012 8:50 pm UTC
Forum: Logic Puzzles
Topic: The truthteller and the politician
Replies: 65
Views: 26250

Re: The truthteller and the politician

example, if he always used the metric you describe, you could ask them 'What would the politician say if I asked him "Is the left road the correct road?"?' If left was the correct road, the politician could say ''The politician would say left was the correct road.'' , wouldn't that work a...
by Heptadecagon
Fri Dec 28, 2012 8:07 pm UTC
Forum: Logic Puzzles
Topic: The truthteller and the politician
Replies: 65
Views: 26250

Re: The truthteller and the politician

I think if the politician acts as the truth teller would if the correct and incorrect roads were switched, there's no possible way to be sure which is which.
by Heptadecagon
Fri Dec 28, 2012 5:28 am UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

Then indeed, every point on the unit circle would be real valued, but I want the unit circle to be the only place where the polynomial is real valued; a constant would leave the function real everywhere. --------------- Now I'll try and extend my work.... (recall Im(a+bi)=b) Suppose I wanted a polyn...
by Heptadecagon
Fri Dec 28, 2012 3:02 am UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

Thanks! I understand most of that except that I cannot yet work with the equations to project the sphere onto the plane and such, I'll have to learn that at some point.
by Heptadecagon
Fri Dec 28, 2012 2:43 am UTC
Forum: Mathematics
Topic: Proof that i=0 (not really but still)
Replies: 12
Views: 2714

Re: Proof that i=0 (not really but still)

e.g. x^2 = (x + x + x + ... + x) {x is being summed x times} d/dx(x^2) = d/dx(x + x + x + ... + x) {differentiating both sides wrt x} 2x = (1 + 1 + 1 + ... + 1) {power rule on each term on either side} d/dx(2x) = d/dx(1 + 1 +1 + ... + 1) {Differentiating both sides wrt x again} 2 = (0 + 0 + 0 + ......
by Heptadecagon
Fri Dec 28, 2012 2:32 am UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

I can visualize what is going on with rotating the sphere, and see how your solution works in that sense, but I am unaware as to how I can find a function for this. I wonder if there is anything fundamental about mapping the unit circle to the real line that always results in the interior and exteri...
by Heptadecagon
Thu Dec 27, 2012 11:55 pm UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

Hmmmm, sounds like quite an interesting function! I'll think about, maybe I'll think up a solution, perhaps it has some sort of polynomial expansion I could use, but it seems unlikely. edit: it seems i/(ln(-x)) maps the unit circle to the real line, has a pole at x=-1, and sends the interior of the ...
by Heptadecagon
Thu Dec 27, 2012 11:15 pm UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

Thanks! That's some very cool reasoning there, certainly rigorous enough for me! And as to the other poster, while the roots may be all on the unit circle, I also had wanted every real value on the unit circle as well; it turns out that doesn't work.
by Heptadecagon
Thu Dec 27, 2012 9:21 pm UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

I certainly don't want every point in the unit circle to be a root; I want every point on the unit circle to be real valued, and real valued only on the unit circle. So, all the roots must be on the unit circle, but not every point on the unit circle must be a root; every point must be real.
by Heptadecagon
Thu Dec 27, 2012 8:56 pm UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Re: Complex Polynomial with Certain Properties

That doesn't work, as expanding i(a+bi)^4-2i(a+bi)^2+i gives ia^4-4a^3*b-6i*a^2*b^2-2ia^2+4a*b^3+4a*b+ib^4+2ib^2+i, which has an imaginary part of a^4-6a^2*b^2-2a^2+b^4+2b^2+1=0 , which is certainly not a circle. Hopefully that also explains the process to go through... if you could get an imaginary...
by Heptadecagon
Thu Dec 27, 2012 8:24 pm UTC
Forum: Mathematics
Topic: Complex Polynomial with Certain Properties
Replies: 18
Views: 2577

Complex Polynomial with Certain Properties

Hello, I have recently been wondering if there exists a polynomial with complex coefficients, p(x) , such that the only x values where the imaginary part of p(x) is equal to zero lie on the unit circle in the complex plane.
by Heptadecagon
Sun Dec 09, 2012 5:14 pm UTC
Forum: Mathematics
Topic: Complicated Implicit Function Graph.
Replies: 1
Views: 1746

Complicated Implicit Function Graph.

Hello all, I recently found a very interesting looking graph, and am wondering if there is any sort of analysis that can be done on it. The equation for it is sin(x)+sin(y)=sin(x*y) . WolframAlpha graphs some of it, though this graphs it better: http://www.meta-calculator.com/online/ It's clear that...
by Heptadecagon
Tue Jul 31, 2012 6:42 am UTC
Forum: Mathematics
Topic: Frustrating Indefinite Integral
Replies: 16
Views: 5533

Re: Frustrating Indefinite Integral

I have it! After much mucking about with the Gamma function, I have a proof that our integral is indeed equal to -1/2 (log 2)^2. I've attached the proof as a pdf. That is absolutely awesome! It's tool late for me to go through it all right now, but it sure is cool! Thanks for all the effort put in!...
by Heptadecagon
Sun Jul 29, 2012 4:50 am UTC
Forum: Mathematics
Topic: Frustrating Indefinite Integral
Replies: 16
Views: 5533

Re: Frustrating Indefinite Integral

PS. I'm curious as to which infinite sum led to this integral... It's deals with of logarithms somewhat similar to to this thread http://forums.xkcd.com/viewtopic.php?f=17&t=87670 , and is in fact a more general continuation of my work there. If we can get this last integral maybe I'll post a f...
by Heptadecagon
Sat Jul 28, 2012 11:43 pm UTC
Forum: Mathematics
Topic: Frustrating Indefinite Integral
Replies: 16
Views: 5533

Re: Frustrating Indefinite Integral

-snipped math- Solving this differential equation for f(a) yields that f(a) = -log a log 2 + f(1), which is equivalent to the posted formula. Unfortunately I haven't yet found a way to prove that f(1) = -1/2 log² 2, but this will at least give you the value at every other point in terms of the valu...
by Heptadecagon
Sat Jul 28, 2012 5:01 am UTC
Forum: Mathematics
Topic: Frustrating Indefinite Integral
Replies: 16
Views: 5533

Re: Frustrating Indefinite Integral

...Actually these are equivalent, so all you need is (-ln(2) * ln(2a^2)) / (2a). Awesome, have you proved this or is this from piecing a pattern together from wolfram alpha results, or something like that. Still, I much would like a proof, but there is a small chance that by messing with some integ...
by Heptadecagon
Fri Jul 27, 2012 10:01 pm UTC
Forum: Mathematics
Topic: Frustrating Indefinite Integral
Replies: 16
Views: 5533

Re: Frustrating Indefinite Integral

I really would enjoy a closed form, for a=1 wolfram alpha give -((ln(2))^2)/2 , and also gives closed forms with logarithms for other positive integers, but does not prove its reasoning, although having the right answer helps for a start to work back to the proof...
by Heptadecagon
Fri Jul 27, 2012 2:37 am UTC
Forum: Mathematics
Topic: Frustrating Indefinite Integral
Replies: 16
Views: 5533

Frustrating Indefinite Integral

While evaulating a infinite sum, I have encountered the following improper integral as a function of 'a'. f(a) = \int^{\infty}_0 ln(x)/(1+e^{ax}),dx \ I have struggled in vain to evaluate this for any integer 'a' , but I am particularly interested in the case a=1 Any sort of ...
by Heptadecagon
Sat Jul 21, 2012 6:52 pm UTC
Forum: Mathematics
Topic: "Oh no! We forgot how to say... math... stuff!"
Replies: 294
Views: 92298

Re: "Oh no! We forgot how to say... math... stuff!"

A few common reasons are that the pole at zero rather than negative one is somehow more elegant, or that the recursion formula looks cleaner [compare Γ(n+1)=n*Γ(n) to (n+1)!=(n+1)*n!]. But then there's also the matter of some other formulas looking prettier, particularly the one involving the beta ...
by Heptadecagon
Sat Jul 21, 2012 6:20 pm UTC
Forum: Mathematics
Topic: An alternating sum
Replies: 7
Views: 3604

Re: An alternating sum

Cool, thanks for the help, it's interesting to see such different methods. I've already seen the -ln(2)*γ term, but not the other quite yet. I understand the proof you posted except to ''The second of these series approaches ∫12 ln x/x dx = (ln 2)2/2 . '' , but I have just woke up and haven't got ou...
by Heptadecagon
Sat Jul 21, 2012 8:25 am UTC
Forum: Mathematics
Topic: An alternating sum
Replies: 7
Views: 3604

Re: An alternating sum

Thank you, although I probably do not need too many digits, just enough to make me feel confident with my answer, once I get it. Wolfram-Alpha spits out hundreds of digits, although I'm sure it messes up quite quickly, who knows. All I know is that I have plenty of digits, thanks for the support! I ...
by Heptadecagon
Sat Jul 21, 2012 7:53 am UTC
Forum: Mathematics
Topic: "Oh no! We forgot how to say... math... stuff!"
Replies: 294
Views: 92298

Re: "Oh no! We forgot how to say... math... stuff!"

I don't know if it's been said already, but to me the gamma function should be put out of common use and be substituted with the Pi function used by Gauss, which to my understanding is more elegant, as the integral for it is simpler and it matches up with the factorial function, it just seems more n...
by Heptadecagon
Sat Jul 21, 2012 7:37 am UTC
Forum: Mathematics
Topic: An alternating sum
Replies: 7
Views: 3604

Re: An alternating sum

Thank you very much, while that probably won't help me find a closed form that I would like, it is very interesting, as Li s (-1) is related to the Dirichlet eta function (and is in fact it's opposite) , and right above references on this page: http://en.wikipedia.org/wiki/Dirichlet_eta_function is ...
by Heptadecagon
Sat Jul 21, 2012 5:38 am UTC
Forum: Mathematics
Topic: An alternating sum
Replies: 7
Views: 3604

An alternating sum

Hello, I have recently encountered the sum ln(1)/1 - ln(2)/2 + ln(3)/3 - ln(4)/4 + ln(5)/5 - ln(6)/6 ..... = sum n=1 to infinity -(-1)^n*ln(n)/n
It is clear it converges, but does anyone know the answer and have a proof? Thank you!
by Heptadecagon
Fri Jun 15, 2012 4:16 pm UTC
Forum: Mathematics
Topic: Riemann Zeta Function
Replies: 15
Views: 4458

Re: Riemann Zeta Function

It does seem right, all the steps are good, and I looked it up online , and it is a known fact. But a very cool thing to discover! Thanks for sharing!
by Heptadecagon
Sun Jun 10, 2012 12:23 am UTC
Forum: Mathematics
Topic: Seeming Coincidences
Replies: 1
Views: 1636

Seeming Coincidences

While working on a problem I figured that the sum n=1 to infinity (1/(2^(2n-1)*(4n-2)))*(-1)^(n-1) = (arctan(1/2))/2 , and checked it with wolframalpha, but forgot to put the '' *(-1)^(n-1) '' in the input. However I did not realize this, because the answer wolframalpha gave me was (arctanh(1/2))/2 ...
by Heptadecagon
Mon May 28, 2012 7:44 pm UTC
Forum: Mathematics
Topic: Fitting circles under a curve
Replies: 18
Views: 4111

Re: Fitting circles under a curve

That looks pretty tough, I think your best bet would be to try and see if the area of the circles forms a geometric series, which seems very unlikely, and then area would be related to the area of the first circle, being transcendental.. If it's not geometric then it is very unlikely you will come a...
by Heptadecagon
Mon May 28, 2012 7:20 pm UTC
Forum: Mathematics
Topic: The Meaning of Divergent Sums
Replies: 1
Views: 1336

The Meaning of Divergent Sums

I have in the past often heard about divergent sums such as 1+2+3+4... = -1/12 , but became very interested when I discovered my own proof using relatively simple methods. Is there any way to get a explanation/intuition about these things, or reason why so many different methods evaluate these sums ...

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