August 2007 - mathfun@mailman.xmission.com

[math-fun] Tech question
by David Wilson 14 Aug '07

14 Aug '07

Can someone give me the a few terms of the power series of the inverse of exp(x)-x at 0? Exact coefficients would be preferred if rational.

3 2

[math-fun] Stirling number p-adic valuation
by Fred lunnon 14 Aug '07

14 Aug '07

I mentioned earlier (see Letter from Grozny) that I'd been looking at \ord_p s(m, n), the maximum power of p dividing a Stirling number of the first kind; after much head-scratching, I've come up with the following rather pretty lower bound. Define a "natural ceiling" function by \nace(x) = \max(0, \ceil(x)), where \ceil(x) denotes the usual ceiling function. Theorem: for 1 <= m <= n, \ord_p s(m, n) >= \sum_{j > 0} \nace(m/p^j - n) . The proof is fairly straightforward, via induction on m; the summation terms are nonzero only for j <= [(\log(m) - \log(n))/\log(p)]. Though only bounding the order below, this expression has the advantage of monotonicity in both m and n; meaning it can be used for example to bound the remainder of the infinite p-adic sum for (thankyou, Don) "negaBell" numbers B(-n) = \sum_m |s(m, n)| . Furthermore, it often does give the exact exponent: for example when n = m (zero), or n = 1 (Legendre's (m-1)! result in a rather different form); indeed apparently whenever for some k, (n-1)p^k < m <= n p^k. Such a simple result must surely be known already: yet any references to s(m, n) divisibility I have so far encountered have been restrictive to the point of feebleness. Fred Lunnon

1 0

[math-fun] Re: Death Valley marathon
by James Propp 12 Aug '07

12 Aug '07

Thanks to everyone who took the time to explain in more detail why the 7:1 split is right for both the Death Valley marathon version and the office building version. I wrote: >(I have a feeling that there's an >Arrow-like theorem lurking here that says that certain seemingly >compelling characteristics that fairness should have are actually >incompatible.) I totally retract that. There is in fact a complete first-order theory of this kind of fairness; it's called basic accounting, and the corresponding algorithm is called double- (or multiple-) entry bookkeeping. :-) Jim

1 0

[math-fun] Death Valley marathon
by James Propp 10 Aug '07

10 Aug '07

>Either way, this problem is suitable for a kindergarten entrance exam. With arithmetic like 5 minus 8/3 = 7/3? You've got to be kidding! If instead of $8 to be divided there were $24 to be divided, so that we could avoid fractions, this problem might be suitable for discussion in grade school. But I think there are economic/ethical issues of what constitutes fairness, in addition to purely mathematical issues. Without a clearer discussion of this, I don't think the puzzle has a single "right" answer. (Though I suspect that the 7-to-1 split is preferable. Maybe consideration of related scenarios with more extreme numbers could make this clearer.) If I ran an elite grade school and wanted to use this problem or some variant as the basis for admission, I'd change the story so that the third party pays $24 instead of $8, so that the use of fractions could be avoided. I'd ask the students how they thought the money should be divided, and why. If their answer involved a $15 and $9 split, I'd explain the reasoning behind a $21 and $3 split; and if their answer involved a $21 and $3 split, I'd explain the reasoning behind a $15 and $9 split. The kids I'd most want to admit to the school would be those who could see that both answers make a kind of sense, and who find this cognitive dissonance both frustrating and enjoyable. ("Hey, that's weird; hey, this is fun!") Speaking of which: Anyone out there care to help me resolve my own cognitive dissonance? Because I'm still unsure which way of splitting the money should be called "fair"! (I have a feeling that there's an Arrow-like theorem lurking here that says that certain seemingly compelling characteristics that fairness should have are actually incompatible.) Jim

3 3

[math-fun] Re: Death Valley marathon
by James Propp 10 Aug '07

10 Aug '07

Joshua Zucker writes: >I don't see the 15:9 split making any kind of sense, since the people >drank the original 8 units of water each too, so they GAVE 7:1 units >of water. What is the sense behind it? It seems to me to be the >solution you get if you just don't think through the problem >completely enough. Let's change the scenario: I invest $5,000,000 in a three-story office building and you invest $3,000,000. There are three tenants: Propp Enterprises, Zucker Enterprises, and Yoyodyne Incorporated, each of which occupies one floor. If the rent we charge Yoyodyne is $8,000 per month, how should you and I split it? Would you really settle for just $1,000 per month, even though you put up 3/8 of the money for the building? If this is different from the Death Valley marathon sccenario, what makes it different? (Maybe I'm being dense...) Jim

4 3

[math-fun] negafibonacci numbers and the hyperbolic plane
by James Propp 10 Aug '07

10 Aug '07

Did anyone go to Knuth's talk at MathFest this past weekend? It sounded interesting, but I wasn't able to go and couldn't find a write-up on the web; I'm hoping one of you can summarize. Here's what's on the web: PI MU EPSILON J. SUTHERLAND FRAME LECTURE NEGAFIBONACCI NUMBERS AND THE HYPERBOLIC PLANE Donald E. Knuth, Stanford University, Professor Emeritus of the Art of Computer Programming Saturday, August 4, 8:00 pm - 8:50 pm All integers can be represented uniquely as a sum of zero or more "negative" Fibonacci numbers F-1 = 1, F-2 = -1, F-3 = 2, F-4 = -3, provided that no two consecutive elements of this infinite sequence are used. The NegaFibonacci representation leads to an interesting coordinate system for a classic infinite tiling of the hyperbolic plane by triangles, where each triangle has one 90 degree angle, one 45 degree angle, and one 36 degree angle. Jim Propp

3 5

[math-fun] Invitation to connect on LinkedIn
by Marc LeBrun 09 Aug '07

09 Aug '07

Math, Cool! I'll see your Doostang and raise you a LinkedIn invite! Best regards, --Marc LeBrun Join Marc LeBrun's network: https://www.linkedin.com/e/isd/121323052/SaG5uDRP/ ------------------------------------------ "Increasingly, if you're not LinkedIn, you're left out." --Business 2.0 ------ (c) 2007, LinkedIn Corporation

2 1

[math-fun] social networking invitations banned from Math-Fun
by Schroeppel, Richard 09 Aug '07

09 Aug '07

I'm assuming that Andy Latto and Marc LeBrun have been the victims of some social networking virus. I've marked their posts for moderation until this mischief passes. Rich

1 0

RE: [math-fun] How we "do mathematics"
by David Gale 09 Aug '07

09 Aug '07

I wonder how many of you took a look at Eric Angelini's two lovely problems which I reproduce here. The first is a wonderful example of a Gardnerian Aha problem. Writing things down would be of no use here, -so much for my theory on the essentiality of pen and paper. The lovely and slightly surprising answer "came to me" a couple of hours after reading the e mail, sitting on a bench in the botanical garden of the town of Lund, Sweeden. Later while trying to get to sleep I worked out the general case where Mr. A brings a liters and Ms. B brings b liters where 2b>a>b. As for the second problem which I just finished, I defy ANYBODY to solve it without doing at least some writing. Remember the answer has to be a number.The paper in front of me contains four drawings, circles, coordinate axes, line segments with various labels, followed by a mess of symbols like x, y ,R, a, pi, integral signs, square root signs. I don't plan to have it framed. David . >A (very) old one I like : > >Two Death Valley marathonians friends complete their run >together and seek shadow in their tent nearby the finish >line. > >The first one had prepared a fresh 5-liter tank of water >in a cooling box and his friend a similar 3-liter one. > >They were about to start drinking when a journalist enters >the tent desperately asking for water. > >OK, the water is divided into three equal parts -- and drunk. > >The thankful journalist insists to pay his share and leaves >8 dollars to be split between the two friends. > >How? > >---- At 07:29 AM 8/6/2007, you wrote: > > it's the kind of thing that throws people because it > > sounds like there's not enough information. > >Old Boniface he took his cheer, >Then he bored a hole through a solid sphere, >Clear through the center, straight and strong, >And the hole was just six inches long. > >Now tell me, when the end was gained, >What volume in the sphere remained? >Sounds like I haven't told enough, >But I have, and the answer isn't tough! > >---- > > > There also is a meta-theoretic answer to this puzzle. > > Assume the puzzle can be solved. Then it must be solvable > > with a hole of any diameter, even zero. But if you drill a > > hole of zero diameter that is six inches long, you leave > > behind the volume of a six inch diameter sphere. > >http://www.faqs.org/faqs/puzzles/archive/geometry/part1/ > > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

3 2

Re: [math-fun] How we "do mathematics"
by Eugene Salamin 09 Aug '07

09 Aug '07

----- Original Message ---- From: David Gale <gale(a)math.berkeley.edu> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Thursday, August 9, 2007 3:17:48 AM Subject: RE: [math-fun] How we "do mathematics" ... >A (very) old one I like : > >Two Death Valley marathonians friends complete their run >together and seek shadow in their tent nearby the finish >line. > >The first one had prepared a fresh 5-liter tank of water >in a cooling box and his friend a similar 3-liter one. > >They were about to start drinking when a journalist enters >the tent desperately asking for water. > >OK, the water is divided into three equal parts -- and drunk. > >The thankful journalist insists to pay his share and leaves >8 dollars to be split between the two friends. > >How? > >---- Why is this problem so fascinating? The two friends are paid $8. The guy who contributed 5 l gets $5, and the guy who contributed 3 l gets $3. Or maybe the $8 goes towards paying for their campsite. I must be missing something. Here's another solution. I like this one better. The guy who contributed 5 l drinks 8/3 l, so he sold 7/3 l. The guy who contributed 3 l drinks 8/3 l, so he sold 1/3 l. So the money should be distributed $7 and $1. Either way, this problem is suitable for a kindergarten entrance exam. Gene ____________________________________________________________________________________ Pinpoint customers who are looking for what you sell. http://searchmarketing.yahoo.com/ _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ____________________________________________________________________________________ Choose the right car based on your needs. Check out Yahoo! Autos new Car Finder tool. http://autos.yahoo.com/carfinder/

2 1