May 2013 - mathfun@mailman.xmission.com

Re: [math-fun] "Ponder This"
by Adam P. Goucher 07 May '13

07 May '13

Mathematica 8 includes the ability to compile functions into C code and call them; this enables you to make the innermost loops as efficient as is possible with C. Indeed, most of the built-in functions are pre-compiled intensively optimised routines, so they'll generally run faster than most C code. Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Hans Havermann > Sent: 05/07/13 02:39 PM > To: math-fun > Subject: Re: [math-fun] "Ponder This" > > Keith F. Lynch: "I've heard good things about Mathematica. Too bad it's so expensive. I've been doing everything in C. I'm curious whether Mathematica is reasonably fast." > > I think Mathematica is a wonderful toy for the recreationalist (especially, with ready-made "black box" procedures for all manner of mathematical situations. For example, if I wanted to duplicate Bill Gosper's "representions of 31415 as sums of four squares" I'd use something like PowersRepresentations[31415,4,2]] and it might even be somewhat faster than his from-scratch procedure but, in general, I'll guess (based on C programmers trouncing my Mathematica-produced output in the past) that you can do things faster in C. > _______________________________________________ > 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] "Ponder This"
by Keith F. Lynch 07 May '13

07 May '13

Hans Havermann <gladhobo(a)teksavvy.com> wrote: > Keith F. Lynch wrote: >> It tried all 1,679,616 (6^8) possible ways of sticking zero or more >> plus signs, minus signs, times signs, divide signs, and decimal >> points into PONDER=THIS > ? thus arbitrarily excluding, I think, a decimal point preceding a > given string of digits. Right. No prefix plus, minus, or decimal point, and no parentheses. In fact, each equation alternated between digits and operations, to make my parser run faster. Concatenation -- creating multi-digit numbers out of digits -- used the comma (suppressed on printing) so that I could loop through consecutive ASCII characters: * + , - . / A typical string was: 3,8*0.1,2,5=7/4-6+9 which meant 38*0.125=7/4-6+9 >> The unique solution to the meta-problem is a single division sign >> and a single minus sign. Insert those into PONDER=THIS so that it >> has a unique solution. > A quick Mathematica hack suggests PON-DER=TH/IS meets the > requirement. Correct. I've heard good things about Mathematica. Too bad it's so expensive. I've been doing everything in C. I'm curious whether Mathematica is reasonably fast. In C, it took me about a month of CPU time to solve the meta-meta-problem, i.e. to find that a single division sign and a single minus sign was a solution. (I got lucky in that it's also the *only* solution.) I'm curious how long it would take Mathematica to find that. And what the code would look like. I only solved the six-four problem: ______=____. I'm curious if there are similar solutions with the equal sign elsewhere. Maybe I should get one or two dozen Raspberry Pis and split the problem between them. Of course it's easy to prove that there can be no *global* unique solution, i.e. if the equal sign can go anywhere. It can't be the case that something like "assemble -/0123456789= into a correct equation" has a single solution. Can you see why?

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Re: [math-fun] "Ponder This", Mma, four squares
by Bill Gosper 07 May '13

07 May '13

KeithL> I've heard good things about Mathematica. Too bad it's so expensive. I've been doing everything in C. I'm curious whether Mathematica is reasonably fast. In C, it took me about a month of CPU time to solve the meta-meta-problem, i.e. to find that a single division sign and a single minus sign was a solution. (I got lucky in that it's also the *only* solution.) I'm curious how long it would take Mathematica to find that. And what the code would look like. Well, here it is taking 10 sec to find the representions of 31415 as sums of four squares In[23]:= tim[Reduce[#.# == 31415 && LessEqual @@ #, Rest[#], Integers] &@{0, a, b, c, d}] During evaluation of In[23]:= 10.411966 Out[23]= Very Large Out[23] In[24]:= Length[%] Out[24]= 829 In[25]:= %%[[{1, 2, 3, -3, -2, -1}]] Out[25]= (a == 1 && b == 2 && c == 9 && d == 177) || (a == 1 && b == 2 && c == 99 && d == 147) || (a == 1 && b == 6 && c == 7 && d == 177) || (a == 78 && b == 81 && c == 83 && d == 109) || (a == 79 && b == 82 && c == 93 && d == 99) || (a == 81 && b == 85 && c == 85 && d == 102) (tim is my own timing function.) In[27]:= 31415.^(4/2) Out[27]= 9.86902*10^8 so a bruteforce search in C might have been faster. But there are scaling issues. And programmer time. --rwg I lied again: " (Digression, for a non-traditional, non-dyadic-rational plot that really needs memoizing, try, e.g., Graphics[Polygon[{Re[#],Im[#]}&/@Append[Hilbert/@Range[4^-6,1,5/2^11],1]]])" No, that just connects every 5th vertex of a polygon with dyadic rational vertices. For a legitimate (somewhat Mayan) example, try Graphics[Polygon[{Re[#],Im[#]}&/@Hilbert/@Range[0,1,1/320]]]

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[math-fun] Recreational math conference
by James Propp 06 May '13

06 May '13

FYI: ------ MOVES Conference, August 4-6 in New York City A research conference on the Mathematics of Various Entertaining Subjects Join MoMath, the National Museum of Mathematics, for its first-ever conference on recreational mathematics. Explore America's only museum of math in a night open exclusively to conference attendees and their families, then participate in two days of research talks on the mathematics of games and puzzles. Family and kids activities will run concurrently with the scientific research program for guests of conference attendees. Keynote speaker: Erik Demaine, MIT Registration deadline: May 15 or until at capacity For more information and to register, visit the MOVES Conference website at moves.momath.org. --- I've seen a list of the talks, which includes such intriguing titles as "The world's hardest elementary domino tiling problems" and "k-potent Groebner basis computations for Sudoku". MoMath hopes to hold this conference ever two years, so even though it's too late to submit a paper for MOVES 2013, there's plenty of time to start thinking about MOVES 2015! Jim Propp (member of the MoMath Advisory Council)

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Re: [math-fun] "Ponder This"
by Keith F. Lynch 06 May '13

06 May '13

This is a followup to my original April 13th message. After running for about a month, the program finally finished running yesterday. I'm pleased with what it found. It tried all 1,679,616 (6^8) possible ways of sticking zero or more plus signs, minus signs, times signs, divide signs, and decimal points into PONDER=THIS, to see which ones have a unique solution. (Infix only, no prefix plus or minus. No parentheses. Each letter represents a unique digit. Note that there are ten letters with no duplicate letters.) 1,019,744 of the 1,679,616 arrangements of operations had at least one solution. 1065 of them had exactly one solution. And just 1 of those 1065 had a set of operations not shared by any others. I could stop here and say find that one (a meta-meta- problem). But I won't be that cruel. The unique solution to the meta-problem is a single division sign and a single minus sign. Insert those into PONDER=THIS so that it has a unique solution. There's exactly one way to do this. Have fun. (There are an additional 5 ways to insert those character into PONDER=THIS that give multiple solutions. For instance POND/E-R=THIS has two solutions: 4608/3-7=1529 and 4608/3-9=1527) At the other extreme, there are 44 ways to insert a plus sign, a multiplication sign, two division signs, and two decimal points to get a problem with a unique solution. I've submitted this to the website, <http://researchweb.watson.ibm.com/ponder/>, where the problem that inspired this appeared, in hopes that they'll use it, with credit to me, in some future month.

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[math-fun] Archimedean dog bowl causes fire in Santa Rosa
by Henry Baker 06 May '13

06 May '13

FYI -- So much for MythBusters... I wonder what Arthur C Clarke would say about this: http://zebu.uoregon.edu/~probs/therm/sunstrok/sunstrok.html http://sanfrancisco.cbslocal.com/2013/05/02/shiny-dog-bowl-blamed-in-santa-… Shiny Dog Bowl Blamed In Santa Rosa Fire May 2, 2013 11:59 PM SANTA ROSA (CBS SF) Â A dogÂs water bowl was being blamed for starting a fire at its ownerÂs Santa Rosa home, according to fire officials. North Bay firefighters called it is one of the most unusual fire calls theyÂve ever had. Perry and Shay Weisbrich said they had to call 9-1-1 after their dog TobyÂs metal bowl reflected a concentrated beam of sunlight onto the side of their house Â a beam so powerful, the wood started to burn. ÂI just looked outside a minute and saw what i thought was steam.Â said Shay Weisbrich. ÂA big plume of it through the glass door.. So I went over there thinking, ÂthatÂs not right, not normal.Â And it was smoke!Â The Bennett Valley Fire Department responded before the fire got out of control. As for Toby, the Weisbrichs said heÂs getting a new dish: a plastic bowl.

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[math-fun] The 2012 USA presidential election
by Warren D Smith 06 May '13

06 May '13

> Warren, when you post a URL, could you add a sentence or two about > the contents? Your subject line is a good beginning, but a little more > information would help http://rangevoting.org/USA2012primary.html is an attempt to re-examine this election from the point of view of other voting methods which is possible thanks to the existence of over 100 polls conducted in approval and/or score voting style. It looks like approval & range voting would have made the election somewhat less insane.

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[math-fun] The 2012 USA presidential election
by Warren D Smith 05 May '13

05 May '13

http://rangevoting.org/USA2012primary.html

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[math-fun] Joerg Arndt
by Fred lunnon 04 May '13

04 May '13

I am hoping that somebody on this list is in regular contact with "Joerg Arndt" <arndt(a)jjj.de> and, if still able to reach him, would be kind enough to advise him that my replies to his emails are being blocked by rogue anti-spam software [apparently going under the name "grey-listing", though in my case "grey-hair-causing" might be more appropriate]. Fred Lunnon

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[math-fun] The Social, an possibly forgetful, mathematicians
by Victor Miller 04 May '13

04 May '13

I was reading the New York Times Numberplay blog, where they had a cute little puzzle: There are 64 mathematicians at a conference. Each of them knows his (or her) own unique theorem. At social time, the 64 pair up and exchange all of their knowledge, which takes an hour. At the end of the hour, they pair up again, exchange knowledge, etc. The question is -- how long does it take before every mathematician knows every theorem (presumably with an optimal selection of pairings)? There is really nothing special about 64. One can do this for an integer n >=2 (n=1 is uninteresting). If n is odd, there is just an "odd man out" for each pairing. This question is pretty easy (though I'll leave it to you to work out the solution), but a comment on the blog lead me to give the following variants: 1) Suppose that each pairing is chosen from all possible ones uniformly at random. The time until all mathematicians know all theorems, T_n, is now a random variable. What is its distribution? In particular, what is its expectation? 2) Suppose that there are m regular mathematicians and n forgetful mathematicians. The forgetful know nothing, and promptly forget everything they are told. However, there is no way to distinguish the regular from the forgetful by outward appearance. One can ask the same two questions: what is an optimal deterministic choice of pairings, and, in the random pairing case, what is the distribution of the waiting time until all m regular mathematicians know all m theorems? 3) Now suppose that each mathematician has a faulty memory. At first I suggested that there was a fixed probability, p, so that after exchanging the information, each piece of knowledge (theorem) contained in the memory of each mathematician is deleted with probability p, independently. However, after a discussion with my colleague Fred Kochman, I realized that there was some probability > 0 that a particular theorem might be deleted from everyone's memory, and so the expectation of the waiting time would be infinite. I thought of two possible ways to repair this: a) Every mathematician always remember his/her own theorem (i.e. it can't be deleted). b) More symmetrically, after mathematicians A and B exchange information, and before any deleting, they have the same information. For each theorem i that they know, with probability 1-p, they both remember it, with probability p/2 A remembers it and B forgets it, and similarly with probability p/2 A forgets it and B remembers. Thus, someone always will remember a theorem. Victor

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