[math-fun] revitalized lazy eight
The early English editions, at least, of I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, were just phototypeset from the Russian. They included the rather pointless identity, (c37) (simpprod:false,'prod(gamma(z+k/3),k,1,8), %%=factcomb(mulfactorial(mulfactorial(apply_nouns(%%))))) 8 /===\ | | k (d37) | | Gamma(z + -) = | | 3 k = 1 3 8 %pi Gamma(3 (z + 2)) Gamma(3 z + 1) Gamma(3 z + 4) ----------------------------------------------------- 9 z + 19/2 3 gamma(z + 2) (or very nearly), but disguised as something nontrivial by virtue of the "8" being misprinted as infinity. In a later edition, someone had cleverly scissored out the oo and turned it upright! Can someone scan me in that bit of typographic rectitude? (If it looks like an ordinary 8, then your edition has been retypeset. And an editor missed an opportunity to elide a formula that never would have been in there, had it been printed correctly in the source.) --rwg NORTHUPITES SHORTEN IT UP CHAMPERTOUS CAME UP SHORT --------------------------------- Boardwalk for $500? In 2007? Ha! Play Monopoly Here and Now (it's updated for today's economy) at Yahoo! Games.
Recent solid angles analysis entailed some messy intermediate trig, but the most strenuous solution I've seen is for the phase parameter in the validity constraint for a bilateral generalization of Dougall's 7F6 identity, which I've tacked on to www.tweedledum.com/rwg/idents.htm The e^(i x) (EXPONENTIALIZE) sledgehammer gives you hideous radicals, which makes it a bit surprising the answer is as "nice" as it is. --rwg --------------------------------- Get the Yahoo! toolbar and be alerted to new email wherever you're surfing.
A die having five sides, 4,0,0,0,0, is paired with a seven sided die, 3,2,2,1,1,0,0. Construct a different pair of dice with the same statistics. --rwg PS, my Yahoo mailing miseries were due to a bad interaction with Mozilla. --------------------------------- Shape Yahoo! in your own image. Join our Network Research Panel today!
On Monday 30 July 2007, Bill Gosper wrote:
A die having five sides, 4,0,0,0,0, is paired with a seven sided die, 3,2,2,1,1,0,0. Construct a different pair of dice with the same statistics. --rwg
[answer and comments follow after spoiler space] ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... x^4+4 = (x^2-2x+2)(x^2+2x+2) [easy to find by considering (x^2+2)^2] x^3+2x^2+2x+2 is irreducible [by Eisenstein or, more elementarily, by seeing that there'd have to be a linear factor, etc.] so the only thing to do (other than boring things; see below) is to calculate (x^2-2x+2)(x^3+2x^2+2x+2), which turns out to be x^5+2x^2+4, all coeffs non-negative, yay. Hence {5,2,2,0,0,0,0} x {2,1,1,0,0} will do and nothing else will. If this was really a contest problem and was really stated as given, then it's dicey indeed, because an easier solution to the problem as stated is (e.g.) to replace the first die with one whose sides are 4,4,0,0,0,0,0,0,0,0. Or, almost as easy, to replace them with one all of whose sides are 0 and one with the obvious 35 sides. Given honest contestants, the question could have been rewritten as "Have you come across generating functions?" without much loss of discriminating power. -- g
Alejandro Aguado solved this longstanding problem. I'll rephrase it, then give his solution. 16 couples (A-P & a-p) play golf, in foursomes. Each day, all 32 of them play. At the end of 10 days, all of them have played exactly once with everyone else, except for their spouse. Day0 Day1 Day2 Day3 Day4 Day5 Day6 Day7 Day8 Day9 1pm ABCD AEIm Alof AciL AgNK AhJp AFde APbn AjMG AkHO 2pm EFGH BgcH Bhkm BjdK BElP BFin BoGJ BNap BIeO BMfL 3pm IJKL DFKp DNGi DkbJ DIoH DcmO DEaf DglM DhPL Djen 4pm MNPO kPdf FaLO FgoP hdiO INbf hlNc koce gaJm Flbm 5pm abcd Maei MbHK fGOp beLp leGK PiKm fHiJ cfKn IPcG 6pm efgh ChbG CPeJ CIla Cjfm CMod CHLn CEKO CFkN Cgip 7pm ijkl jNoL Ejcp EhMn FMcJ EgkL gjbO FhIj Eobi ENdJ 8pm mnop lJnO gIdn NeHm kaGn jPaH IkMp dGLm ldHp hoaK Best solutions for 28, 36, or 40 golfers are all unknown. http://www.cs.brown.edu/~sello/golf.html
participants (3)
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Bill Gosper -
Ed Pegg Jr -
Gareth McCaughan