January 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Sherlock Holmes averts world war using mathematics
by hbaker1 09 Jan '12

09 Jan '12

Thanks very much for posting this article. Very interesting. However, based on what I've heard about codes in WWI -- e.g., the silly code that President Wilson used to communicate with his emissary -- the code described in the article below may be more sophisticated than typical for that time. Perhaps the deleted cuts will make it into the Numb3rs Edition DVD for $239.00. ;-) -----Original Message----- >From: Ray Tayek <rtayek(a)ca.rr.com> >Sent: Jan 9, 2012 10:49 AM >To: math-fun <math-fun(a)mailman.xmission.com> >Subject: [math-fun] Sherlock Holmes averts world war using mathematics > >http://www.newscientist.com/blogs/culturelab/2012/01/the-mathematics-behind… > >An evil mastermind is set on bringing about global war. Only one man >can stop him: Sherlock Holmes, with the help of his partner in >crime-solving Dr Watson. But in the latest Holmes flick, ><http://sherlockholmes2.warnerbros.com/>Sherlock Holmes: A game of >shadows, they don't just need their trusty revolvers and Holmes's >trademark prescient fight scenes, they also need to grasp some mathematics. > >The villain is Holmes's >nemesis, ><http://www.newscientist.com/article/mg12416962.200-james-moriarty--a-forgot…>James >Moriarty, a professor of mathematics and all-around evil genius. In >the book The Final Problem, he is described by Holmes himself as "a >genius, a philosopher, an abstract thinker. He has a brain of the first order." > >But behind the wit of the character in the film lies the mathematical >know-how of a team at the University of Oxford. ><http://www.maths.ox.ac.uk/contact/details/goriely>Alain Goriely and ><http://people.maths.ox.ac.uk/moulton/>Derek Moulton at Oxford's >Mathematical Institute have been hard at work behind the scenes >helping to formulate a believable mathematical villain. > >Initially, the filmmakers approached the mathematicians to ask them >to fill Moriarty's blackboard with equations. Not only did they have >to be real, they had to be historically accurate, based on a >19th-century understanding of the field. > >"When we did the equations on the blackboard, [the film-makers] got >excited," says Goriely. "Although they were quite secretive about the >story, they told us that Moriarty was a mathematics professor and >that they wanted us to help them add more meat to the script, which >was a little dumb and mostly incorrect." > >Goriely and Moulton ended up going beyond script-tweaking to develop >a secret code from scratch that Moriarty uses in the film to send >messages around a Europe on the brink of the war he is conniving. > >But how does one get into the mindset of a fictional evil genius from >the 19th century? Unfortunately, Arthur Conan Doyle's books were of >limited help, offering sparse details on Moriarty's interests, >Goriely says. "We do know that the character wrote two books - one on >binomial theorem and one titled The Dynamics of an Asteroid." > >To create a convincing code, the team started from the binomial >theorem. "Binomial theorem is linked to Pascal's triangle, so we >devised a secret code based on that," says Goriely. > >The code is hidden in Moriarty's red pocketbook, which is filled with >numbers. The numbers signal to the reader first which Fibonacci >p-code - a way to take digits from Pascal's triangle - to use. This >supplies another list of numbers, which are used to indicate which >page, line and words from a book to look up. Goriely reckons his code >is spot on for the character. "Moriarty was obsessed with Pascal's >triangle and Fibonacci's codes," he says. > >The pair also wrote an entire lecture for Moriarty based on his >interests in celestial dynamics. "I used elements of maths from >celestial mechanics at the end of the 19th century," says Goriely. >"It was a very hot topic at the time." > >The lecture discusses the n-body problem - a mathematical problem >that considers how moving celestial bodies interact with each other >as a result of their gravitational energy. Moriarty would have likely >had a particular interest in the theory, given its potential >implications for weaponry, says Goriely. "If you could build a >missile and throw it out of the atmosphere, it could re-enter with an >asteroid-like impact. It would be brought back by gravitational >forces," he says. > >While a disguised Holmes might have been party to the entirety of the >lecture, sadly only the smallest of snippets made the final cut for >the audience's edification. And while Holmes's fleeting glance of >Moriarty's blackboard proved key to his later success in foiling the >professor's evil plans, even the most beady-eyed mathematician >watching the film will find such a feat tricky. But perhaps therein >lies the attraction of Sherlock Holmes and his amazing powers of deduction. > >--- >co-chair http://ocjug.org/

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Re: [math-fun] Gamma function (formulas collection)
by Bill Gosper 09 Jan '12

09 Jan '12

wds>many formulas about gamma function are in my paper:http://rangevoting.org/WarrenSmithPages/homepage/gammprox.pdf _______________________________________________ Interesting stuff. Random remarks: P4 middle is garbled, at least in my Preview. Did you mean asymptosy vs asymptopia? P6, In[16]:= F[0, c] == Assuming[c > 0,Integrate[ x^(2*n + 1)/(1 + x^2)^(n + 1)*E^(-c*x) /. n -> 0, {x, 0, \[Infinity]}]] Out[16]= F[0,c] == (-Cos[c])*CosIntegral[c] + (1/2)*Sin[c]*(Pi - 2*SinIntegral[c]) In[12]:= G[0, c] == Assuming[c > 0, Integrate[ x^(2*n)/(1 + x^2)^(n + 1)*E^(-c*x) /. n -> 0, {x, 0, \[Infinity]}]] Out[12]= G[0, c] == CosIntegral[c]*Sin[c] + (1/2)*Cos[c]*(Pi - 2*SinIntegral[c]) In[14]:= H[0, c] == Assuming[c > 0, Integrate[ x^(2*n)/(1 + x^2)^(n)*E^(-c*x) /. n -> 0, {x, 0, \[Infinity]}]] Out[14]= H[0, c] == 1/c Eliminating G and H from the F recurrence, 4*(n-1)*n*F(n)=2*(n-1)*(6*n-5)*F(n-1)-(12*n^2-32*n+c^2+22)*F(n-2)+2*(n-2)*(2*n-3)*F(n-3) Mma gets only MeijerG for n>0 in all of these. P7 (40), see http://www.tweedledum.com/rwg/idents.htm near bottom of 1st screen. (42) and (43) raise the interesting question of evaluating Product[((1/(12*n)) + 1)*((n - 1/2)*(E^(1/n) - 1))^n,{n, 1, Infinity}] (i.e., prod(((n-1/2)*(%e^(1/n)-1))^n*(1+1/12/n),n,1,inf) --"cwg"

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Re: [math-fun] Lower Limit Found For Sudoku Puzzle Clues
by Tom Duff 09 Jan '12

09 Jan '12

James Cloos <cloos(a)jhcloos.com> writes: > The earlier report I read on this (possibly in arxivblog) said that they > had only proven that there were no 16-clue puzzles with unique > solutions, > and that <13 clue puzzles had been ruled out by others. > > Whether 13, 14, or 15 clue puzzles with unique solutions exist is still, > according to that report, an open question. OK, I'm not following something here. Doesn't the existence of a uniquely solvable 15-clue puzzle imply the existence of a uniquely solveable 16-clue puzzle? (Just fill in any blank.) So, contrapositively, the non-existence of 16s would imply the non-existence of 15s (and 14s and 13s, etc.) -- Tom Duff. This was just a preliminary test.

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[math-fun] Sherlock Holmes averts world war using mathematics
by Ray Tayek 09 Jan '12

09 Jan '12

http://www.newscientist.com/blogs/culturelab/2012/01/the-mathematics-behind… An evil mastermind is set on bringing about global war. Only one man can stop him: Sherlock Holmes, with the help of his partner in crime-solving Dr Watson. But in the latest Holmes flick, <http://sherlockholmes2.warnerbros.com/>Sherlock Holmes: A game of shadows, they don't just need their trusty revolvers and Holmes's trademark prescient fight scenes, they also need to grasp some mathematics. The villain is Holmes's nemesis, <http://www.newscientist.com/article/mg12416962.200-james-moriarty--a-forgot…>James Moriarty, a professor of mathematics and all-around evil genius. In the book The Final Problem, he is described by Holmes himself as "a genius, a philosopher, an abstract thinker. He has a brain of the first order." But behind the wit of the character in the film lies the mathematical know-how of a team at the University of Oxford. <http://www.maths.ox.ac.uk/contact/details/goriely>Alain Goriely and <http://people.maths.ox.ac.uk/moulton/>Derek Moulton at Oxford's Mathematical Institute have been hard at work behind the scenes helping to formulate a believable mathematical villain. Initially, the filmmakers approached the mathematicians to ask them to fill Moriarty's blackboard with equations. Not only did they have to be real, they had to be historically accurate, based on a 19th-century understanding of the field. "When we did the equations on the blackboard, [the film-makers] got excited," says Goriely. "Although they were quite secretive about the story, they told us that Moriarty was a mathematics professor and that they wanted us to help them add more meat to the script, which was a little dumb and mostly incorrect." Goriely and Moulton ended up going beyond script-tweaking to develop a secret code from scratch that Moriarty uses in the film to send messages around a Europe on the brink of the war he is conniving. But how does one get into the mindset of a fictional evil genius from the 19th century? Unfortunately, Arthur Conan Doyle's books were of limited help, offering sparse details on Moriarty's interests, Goriely says. "We do know that the character wrote two books - one on binomial theorem and one titled The Dynamics of an Asteroid." To create a convincing code, the team started from the binomial theorem. "Binomial theorem is linked to Pascal's triangle, so we devised a secret code based on that," says Goriely. The code is hidden in Moriarty's red pocketbook, which is filled with numbers. The numbers signal to the reader first which Fibonacci p-code - a way to take digits from Pascal's triangle - to use. This supplies another list of numbers, which are used to indicate which page, line and words from a book to look up. Goriely reckons his code is spot on for the character. "Moriarty was obsessed with Pascal's triangle and Fibonacci's codes," he says. The pair also wrote an entire lecture for Moriarty based on his interests in celestial dynamics. "I used elements of maths from celestial mechanics at the end of the 19th century," says Goriely. "It was a very hot topic at the time." The lecture discusses the n-body problem - a mathematical problem that considers how moving celestial bodies interact with each other as a result of their gravitational energy. Moriarty would have likely had a particular interest in the theory, given its potential implications for weaponry, says Goriely. "If you could build a missile and throw it out of the atmosphere, it could re-enter with an asteroid-like impact. It would be brought back by gravitational forces," he says. While a disguised Holmes might have been party to the entirety of the lecture, sadly only the smallest of snippets made the final cut for the audience's edification. And while Holmes's fleeting glance of Moriarty's blackboard proved key to his later success in foiling the professor's evil plans, even the most beady-eyed mathematician watching the film will find such a feat tricky. But perhaps therein lies the attraction of Sherlock Holmes and his amazing powers of deduction. --- co-chair http://ocjug.org/

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[math-fun] Probability of undecidability
by Dan Asimov 09 Jan '12

09 Jan '12

In number theory (as used in Gödel's undecidability proof) what is the probability that a statement is undecidable? Could almost all statements be undecidable? (E.g., look at the fraction of statements of length N that are undecidable, as N -> oo.) --Dan ________________________________________________________________________________________ It goes without saying that .

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[math-fun] probability of undecidability -- are Turing machines especially awesome?
by Warren Smith 09 Jan '12

09 Jan '12

the probability a length-N statement is undecidable practically surely goes to 0 or 1 when N-->infinity... but it depends on the precise language you permit for statements. E.g. if the statement was either an AND of many statements, or an OR of many statements, prob(decidable) would go to 1 and the truth value would almost surely be 0 or 1 respectively. So it is fairly difficult to cook up a language for statements in which this kind of obvious thing does not happen. A related question would be, given a random length-N input for some universal Turing machine, what is the probability it will halt when N-->infinity? In this case, I think the {0,1}-only law is avoided. This suggests that Turing machines actually are pretty special. For a lot of other things like Cellular Automata, tiling, diophantine equations, Kolmogorov optimal-compressedness, etc, I bet the limit probabilities of undecidability ---> 0, but for Turing machines I think it is positive.

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Re: [math-fun] Life's Garden of Eden
by Bill Gosper 08 Jan '12

08 Jan '12

acw> I read this, and it is, unfortunately, not the discussion of the proof that I remembered. Could it have been that Conway presented the proof at the talk I went to? I know it was discussed. His "parlor trick" around then was to challenge people to draw Life patterns, and he would construct a predecessor. He was uncannily good at it. Record small GoE: http://homepage.tudelft.nl/37b0y/ Just confirmed by Achim Flammenkamp. --rwg

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Re: [math-fun] min(x, y) and sorting without if()
by Phil Carmody 07 Jan '12

07 Jan '12

From: Joerg Arndt <arndt(a)jjj.de> > These (branch-avoiding) techniques are "well known", see (fxtbook) > section 1.11 "Avoiding branches", pp.15ff where a warning of the form > "Your compiler may be smarter than you thought" is given. Shouldn't the warning be more Nuddsian? In C, such expressions are either unnecessary or UB, pure and simple. Your compiler doesn't have to show smart behaviour at all - it may kill your cat if it wants to. Phil (with comp.{lang,std}.c hat on)

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Re: [math-fun] Puzzler
by N. J. A. Sloane 07 Jan '12

07 Jan '12

> > It is 72, I think, for x = 125 > Yes, you are right. It seems likely that 72 should suffice for all integers x. What about the sequence: a(n) = smallest k such that ... ? With a(125) = 72. Could you please submit it? (With keyword "base", of course) Neil

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[math-fun] Mental factoring
by Dan Asimov 07 Jan '12

07 Jan '12

Take a number like 2257 (say < 10000), easily checked to have no prime factors < 13. To find the least prime factor of such a number N, is there on average a faster way (after having excluded primes < 13) than the naïve method of dividing it by each prime in order <= sqrt(N) until one divides N evenly? --Dan ________________________________________________________________________________________ It goes without saying that .

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