August 2010 - mathfun@mailman.xmission.com

[math-fun] Iterated square root puzzle
by Dan Asimov 15 Aug '10

15 Aug '10

Find a closed form for the the expression sqrt(1/2^(2^0) + sqrt(1/2^(2^1) + sqrt(1/2^(2^2) + ...))). --Dan P.S. No fair using electronic assistance. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Iterated square root puzzle
by Tom Karzes 14 Aug '10

14 Aug '10

Ok, this one I can do: n = sqrt(1/2^(2^0) + sqrt(1/2^(2^1) + sqrt(1/2^(2^2) + ...))) (1/sqrt(2))*n = sqrt(1/2^(2^1) + sqrt(1/2^(2^2) + sqrt(1/2^(2^3) + ...))) n = sqrt(1/2^(2^0) + (1/sqrt(2))*n) n^2 = 1/2 + (1/sqrt(2))*n sqrt(2)*n^2 - n - sqrt(2)/2 = 0 Discarding the negative solution: n = (1 + sqrt(5)) / (2*sqrt(2)) This is also the golden ratio divided by sqrt(2): n = phi / sqrt(2) Tom > Find a closed form for the the expression > > sqrt(1/2^(2^0) + sqrt(1/2^(2^1) + sqrt(1/2^(2^2) + ...))). > > --Dan > > P.S. No fair using electronic assistance. > > _____________________________________________________________________ > "It don't mean a thing if it ain't got that certain je ne sais quoi." > --Peter Schickele > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >

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Re: [math-fun] Iterated square root puzzle
by Dan Asimov 14 Aug '10

14 Aug '10

<< How do you suppose problem 8 was intended to read? I think the constant 2007 should be 2207, like all the terms in that positiion. 2207 = ((1+sqrt(5))/2)^8, so that should help. You get: x^8 + x^(-8) = (1+sqrt(5))^8 which is easily solved as a quadratic in y = x^8. But it feels as though the answer can be greatly simplified. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Strange-ish dilog sum
by Bill Gosper 13 Aug '10

13 Aug '10

Sum[(-Log[2])*Log[x] + 2^n*(Pi^2/12 + PolyLog[2, -x^(-2^(-n))]), {n, 0, Infinity}] -> -(Pi^2/12) + Log[x] - Log[x]*Log[2*Log[x]] + (1/2)*PolyLog[2, 1/x^2] Doesn't look too convergent, does it? Mma doubts it too. But In[389]:= MapAt[22 &, %, {1, 2, 3}] /. x -> .69 Out[389]= -0.0716747 -> -0.0716748 - 2.22045*10^-16 I I think there are 3^-n versions of this as well. --rwg

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[math-fun] The Stability of the Bicycle
by Henry Baker 13 Aug '10

13 Aug '10

As a mathematician/bike rider, if you've never read this delightful article, you should: http://ist-socrates.berkeley.edu/~fajans/Teaching/MoreBikeFiles/JonesBikeBW…

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[math-fun] sums of reciprocal multinomials
by Fred lunnon 12 Aug '10

12 Aug '10

Here's something which may be as new to some other people on the list as it was to me, though it must surely be well-known (and probably has some connection to do with those ghastly hypergeometric series that I keep trying to avoid). I illustrate with the tetranomial case --- the generalisation to any dimension is obvious: \sum_{0<=j<oo} p!q!r!j!/(p+q+r+j)! = p!q!r!/(p+q+r-1)(p+q+r-1)! Does anyone know of a reference for these and similar results --- for example, generalisation to summation over several indices? Can Mathematica, Macsyma etc. deduce them --- Maple 9 flunked! This was brought to my attention by an enquiry from a physicist: not only did I have to admit that I didn't recognise it, but neither could I tell him where to go look for it. Fred Lunnon

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Re: [math-fun] Commutapult: Catapulting Commuters
by Henry Baker 11 Aug '10

11 Aug '10

Solution: ICBP (InterContinental Ballistic Pizza), atmospherically cooked during delivery! At 05:04 PM 8/10/2010, Dan Asimov wrote: >The biggest problem is that outside of the NYC area, nobody knows how to season a pizza anywhere near as deliciously. It's just the right mixture of oregano and other spices that my entire life I've never had anywhere else. > >Even at the best pizza places elsewhere, whose pizzas are wonderful, they don't know how to season them. > >--Dan > >Rich wrote: > ><< >Pizza delivery! >Every home will have a pizza catcher on the roof. --Rich >>>

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Re: [math-fun] Commutapult: Catapulting Commuters
by Dan Asimov 11 Aug '10

11 Aug '10

The biggest problem is that outside of the NYC area, nobody knows how to season a pizza anywhere near as deliciously. It's just the right mixture of oregano and other spices that my entire life I've never had anywhere else. Even at the best pizza places elsewhere, whose pizzas are wonderful, they don't know how to season them. --Dan Rich wrote: << Pizza delivery! Every home will have a pizza catcher on the roof. --Rich >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Commutapult: Catapulting Commuters
by Henry Baker 10 Aug '10

10 Aug '10

So Werner von Braun's V2 was merely a precursor to Fedex gone horribly wrong? Who knew? Actually, with today's microprocessors & GPS & miniature drones/helicopters, it probably already is feasible to deliver mail robotically -- especially if everyone puts one of those "catcher" things on their roofs that the railroads used to use to transfer mail w/o stopping. Of course, the U.S. Postal system is quickly going the way of the telegram, so only high-time-value items (e.g., pizza & pot (Cal. only)) would make delivery economical. At 08:53 AM 8/10/2010, Victor Miller wrote: >And then there's "Rocket Mail" -- one of my favorite forays of the postal >service into (wildly impractical) technology: > >http://en.wikipedia.org/wiki/Rocket_mail > >On Tue, Aug 10, 2010 at 10:56 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Aha! Combo satellite dish & pizza catcher: >> order the pizza & Netflix together... >> >> Considering the fact that several centuries >> of math & computers were funded by calculating >> shell trajectories, we ought to be able to put >> these calculations to more peaceful uses. >> >> In New York & especially Chicago, around the >> turn of the (20th) century there was a pretty >> decent system of pneumatic tube transmission. >> I don't know if they were able to develop a >> roll-up pizza to deliver this way. >> >> http://en.wikipedia.org/wiki/Pneumatic_tube >> >> Perhaps pizza delivery won't work this way, >> but the "Canwich" would be appropriate to deliver >> either via a gun trajectory or a pneumatic tube: >> >> http://markonefoods.com/index.html >> >> At 08:00 PM 8/9/2010, rcs(a)xmission.com wrote: >> >Pizza delivery! >> >Every home will have a pizza catcher on the roof. --Rich >> > >> >--------- >> >Quoting Henry Baker <hbaker1(a)pipeline.com>: >> > >> >>FYI -- I had a similar idea a number of years ago to provide "cheap" >> >> L.A./SF flights. Relatively long acceleration & deceleration ramps >> >> required to keep passengers alive. Some significant problems in >> >>recovering the energy involved to reuse in next flight. Also, even >> >>minor changes due to air pressure/temperature can affect where the >> >>vehicle lands, so some steering mechanism is required. >> >> >> >>http://www.commutapult.org/

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[math-fun] Possible unfixable problem with recent proof of P=NP
by Noah Davidson 10 Aug '10

10 Aug '10

I don't understand this subject well enough to provide much insight myself, but here is a discussion some of you might find interesting. http://rjlipton.wordpress.com/2010/08/09/issues-in-the-proof-that-p%E2%89%A… -Noah Davidson

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