July 2007 - mathfun@mailman.xmission.com

Re: [math-fun] Social golfer problem, 8 foursomes for 10 weeks
by Dan Asimov 31 Jul '07

31 Jul '07

Ed wrote: << 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 >> Aha -- so each person has played in 10 foursomes, i.e., with potentially 30 different other people, which in this solution are in fact all the 30 people other than themself & their spouse -- so this solution must be optional. Is there a combinatorial framework into which this kind of question falls (e.g., could it be "block designes" ?) ? --Dan

1 0

[math-fun] revitalized lazy eight
by Bill Gosper 31 Jul '07

31 Jul '07

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.

3 4

[math-fun] Substrings selfdescription
by Eric Angelini 30 Jul '07

30 Jul '07

Hello Math-Fun, The seq here: http://www.research.att.com/~njas/sequences/A046043 could be extended in a certain way, asking to self- describe substrings like "10", "11", etc. I found those: | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 | --------------------------------------------- | 5 | 3 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 | --------------------------------------------- | 6 | 2 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |10 |11 | ------------------------------------------------- | 5 | 4 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 2 | 1 | Are there more? Best, É. --- A longer explanation in french here: http://www.cetteadressecomportecinquantesignes.com/SubStrings.htm

1 0

[math-fun] Christian Boyer
by Eric Angelini 30 Jul '07

30 Jul '07

... is in the august issue of the french magazine "Science & Avenir" together with a magic square of his. Best, E.

2 1

Re: [math-fun] gefundenes Fressen (a cheap find)
by Dan Asimov 29 Jul '07

29 Jul '07

Wouter writes: << Imagine a necklace with n beads of up to n colours, say n=3 beads made of colours x[1], x[2] and x[3], then these are counted by: Table[NecklacePolynomial[n,Array[x,n],Cyclic],{n,3,3}] {x[1]^3 + x[1]^2*x[2] + x[1]*x[2]^2 + x[2]^3 + x[1]^2*x[3] + 2*x[1]*x[2]*x[3] + x[2]^2*x[3] + x[1]*x[3]^2 + x[2]*x[3]^2 + x[3]^3} . . . . . . >> Wouter, can you please remind us how to multiply colours (8-)> ? Dan

2 1

[math-fun] gefundenes Fressen (a cheap find)
by wouter meeussen 28 Jul '07

28 Jul '07

Imagine a necklace with n beads of up to n colours, say n=3 beads made of colours x[1], x[2] and x[3], then these are counted by: Table[NecklacePolynomial[n,Array[x,n],Cyclic],{n,3,3}] {x[1]^3 + x[1]^2*x[2] + x[1]*x[2]^2 + x[2]^3 + x[1]^2*x[3] + 2*x[1]*x[2]*x[3] + x[2]^2*x[3] + x[1]*x[3]^2 + x[2]*x[3]^2 + x[3]^3} now, change the colours into complex roots of 1 x[k] -> E^(2I Pi k/n) so that they do a 'complex cancellation' on the unit circle, and, hey presto, the whole caboodle collapses to EulerPhi[n]: EulerPhi[n] == NecklacePolynomial[n,E^(2I Pi Range[n]/n),Cyclic] which is nice... Wouter.

1 0

Re: [math-fun] ENIAC and maple : a simple test
by Eugene Salamin 27 Jul '07

27 Jul '07

Something is not right here. You say the ENIAC has 0.001 MHz clock, so the addition time should be more like 10 ms, and then in 1 s, only 100 numbers are added. Gene ----- Original Message ---- From: Simon Plouffe <simon.plouffe(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Cc: Jean-Paul Delahaye <delahaye(a)lifl.fr> Sent: Tuesday, July 17, 2007 11:13:05 AM Subject: [math-fun] ENIAC and maple : a simple test Hello, I was reading an article on the ENIAC, that monster could add 100,000 10 digits numbers in 1 second. At the time it was impressive in 1946 of course. I made this simple test, a naive test. I took maple 11, with a 3.0 Ghz PC pentium. ##################################### Digits:=10: som:=0: for i from 1 to 1e99 do: som:=som+1.0/i: if i mod 100000 = 0 then lprint(i,som) fi: od: ##################################### It displays the partial sum of the harmonic series every 100000 terms. this is not optimized, a lazy program. Well, this is the bad news. That loop made on a modern computer with one of the top programs goes at the SAME speed. It takes 1 second to display each step. Are we missing something ? of course, 1) 2 operations are made at each step. 2) the program is NOT optimized. 3) programmed in C, this would be a zillion times better. but still we are 60 years later and we hardly do better !!?? Does maple became a bloat-ware ? ps : the speed of the ENIAC was something like 0.001 Mhz... simon plouffe _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ____________________________________________________________________________________ Need a vacation? Get great deals to amazing places on Yahoo! Travel. http://travel.yahoo.com/

4 11

[math-fun] Chess Solved
by Robert Baillie 27 Jul '07

27 Jul '07

"Computer program can't lose at checkers" http://news.yahoo.com/s/ap/20070719/ap_on_hi_te/solving_checkers;_ylt=AoQVu… apparently, they analyzed all games with 10 or fewer remaining pieces. the correct website (not the one in the yahoo article) is: http://www.cs.ualberta.ca/~chinook/ warning: if you don't have the latest java, parts of the chinook site will hang your firefox browser!

10 13

[math-fun] Bell number question
by David Wilson 25 Jul '07

25 Jul '07

It looks as if B(p) == 2 (mod p) for prime p. Is this true, and are there "Bell pseudoprimes" in this respect?

5 14

[math-fun] ENIAC and maple : a simple test
by Simon Plouffe 21 Jul '07

21 Jul '07

Hello, I was reading an article on the ENIAC, that monster could add 100,000 10 digits numbers in 1 second. At the time it was impressive in 1946 of course. I made this simple test, a naive test. I took maple 11, with a 3.0 Ghz PC pentium. ##################################### Digits:=10: som:=0: for i from 1 to 1e99 do: som:=som+1.0/i: if i mod 100000 = 0 then lprint(i,som) fi: od: ##################################### It displays the partial sum of the harmonic series every 100000 terms. this is not optimized, a lazy program. Well, this is the bad news. That loop made on a modern computer with one of the top programs goes at the SAME speed. It takes 1 second to display each step. Are we missing something ? of course, 1) 2 operations are made at each step. 2) the program is NOT optimized. 3) programmed in C, this would be a zillion times better. but still we are 60 years later and we hardly do better !!?? Does maple became a bloat-ware ? ps : the speed of the ENIAC was something like 0.001 Mhz... simon plouffe

5 10