December 2014 - mathfun@mailman.xmission.com

[math-fun] Unique factorization (connection to music?) question
by Warren D Smith 05 Feb '15

05 Feb '15

>Victor Miller victorsmiller at gmail.com >According to SAGE the field Q(2^{1/12}) has class number 1, and so does >have unique factorization. In addition Z[2^{1/12}] is the maximal order in >this field. --so: for which N does Q(2^{1/N}) have class number 1? http://oeis.org/A005848 = {1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84} claims to answer the same question for Q(Nth root of unity) duplicating the Main Theorem of J. Myron Masley and Hugh L. Montgomery: Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976) 248-256 which shows that all such N obey N<=84 and gives the full list. Amazingly enough, all the historically-used music-scale sizes {4,5,6,7,8,12} happen to be on this list (asterisk: M&M explain that if k is odd and on their list, then 2k automatically is also, so therefore they omit mentioning all 2k with k odd). Which actually is not so amazing since ALL numbers 1-12 are on the M&M list.

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[math-fun] Microwave ovens
by James Propp 02 Feb '15

02 Feb '15

Why do microwave ovens alternate between rotating the food-carousel clockwise and rotating it counterclockwise (or, as the Brits would say, anticlockwise)? Jim Propp

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[math-fun] Fwd: Countdown
by Bill Gosper 05 Jan '15

05 Jan '15

This is starting to make me want one of those "Ten fingers are overrated" T shirts. ---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh(a)gmail.com> Date: Wed, Dec 31, 2014 at 5:36 AM Subject: Re: Countdown To: Bill Gosper <billgosper(a)gmail.com> Hans Havermann> > > If we insist on only one the four operators (+-*/) as inbetweens, 2016 is > already impossible. > The cheat is to allow concatenation as well. This gives us up to (but not > including) 2102, > even if the concatenation is disallowed after the 10 (to force the saying > of 'ten'). > Except for 1992: the only way to get it is 1098+765+4+32+1. Julian

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[math-fun] Double moves
by James Propp 01 Jan '15

01 Jan '15

Has anyone studied variants of standard combinatorial games (such as Nim) in which each player makes two standard moves in succession on each turn instead of just one? (This is the way my 8-year-old son likes to play tic-tac-toe against me: he puts down two X's, then I put down two O's, then he puts down two X's and wins. Fair, right?) Note that this form of two-player game can be seen as a four-player game in which players #1 and #2 collaborate and players #3 and #4 collaborate. Jim Propp

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[math-fun] 2015 -> 11111011111 (binary)
by Stuart Anderson 01 Jan '15

01 Jan '15

The most interesting property of the number 2015 that I could find is that in binary it is the palindromic string with a single zero 11111011111. Considering such palindromic numbers starting from 101, 11011, 1110111, 111101111, 11111011111 gives integers 5, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 33550335, .... Using the formula a(n)= 2^(2n+1) - 2^n - 1 So the next time this happens is in 8127. This would be a nice opportunity for the NYE firework operators on the Sydney Harbour Bridge to display 2015 in binary - 5 large jets on each side, along with a circular firework from the center of the arch. Stuart

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[math-fun] "The Imitation Game" -- review
by Henry Baker 31 Dec '14

31 Dec '14

Yesterday, I was dragged, kicking & screaming, to The Imitation Game by my wife. It was extremely painful to watch this movie, and I've been having a hard to trying to figure out why. Perhaps the best analogy would be a movie about Mozart by someone who was born deaf. No matter what anyone told this deaf movie-maker about Mozart's music, it would always be completely foreign hearsay, and could never convey Mozart's musical genius to someone who does know & love Mozart's work. ("Amadeus", of course, is the complete opposite, as it was made by people who really, really enjoyed & understood Mozart's music.) Turing's life was totally tied up in his head, and he must have spent an hour or two running every day as his form of mental yoga (his marathon running times were world-class, and you can't keep that level of fitness without running at least one or two hours per day). Although the movie shows him running, the movie completely misunderstands the point of his running -- it is time alone when he can contemplate & work out some of his theories. "Aspies" like Turing find too much external stimulation painful and distracting, so Turing's marathon runs were a welcome respite against this unwanted external stimulation. "I have such a stressful job that the only way I can get it out of my mind is by running hard; its the only way I can get some release." Perhaps a key point that The Imitation Game wanted to make, but really screwed up, was the fact that mathematicians like Turing were so highly motivated by the challenge of the puzzle, that they wouldn't have worked any faster (or any slower) on solving the Enigma problem knowing that many, many lives were at stake. It isn't because Turing didn't care about people's lives; it's just that whether he cared or not wouldn't make any difference in how he attacked the problem. "Caring" in this context simply became another distraction to be ignored. The Cairncross character is as completely gratuitous as it is unhistorical; there is no evidence that Cairncross ever even met Turing, much less worked with him. It is sad that The Imitation Game has to stoop to such low depths to find intrigue; Turing never made any secret about his homosexuality, and as a result, any attempt to blackmail him by threatening to out him would not have worked. Turing's actual real life is far more interesting than the stereotyped "Big Bang Theory"/"Bones" Yet Another Aspie Geek (YAAG) in popular media-type character. Unfortunately, The Imitation Game never gets beyond these stereotypes. https://en.wikipedia.org/wiki/Asperger_syndrome_in_popular_culture Some interesting things about Turing that I hadn't realized: * Turing did a dissertation on the Central Limit Theorem * Turing himself invented Turing computability with "oracles" * Turing worked on encrypted speech technology * Turing rode his bicycle 60 miles to the Sherborne School when he was 13 * Turing _rowed_ instead of running while at Cambridge * Turing's best marathon time would have placed him perhaps 15th in the 1948 Olympics * Turing invented LU matrix decomposition (!!) https://en.wikipedia.org/wiki/Alan_Turing http://www-history.mcs.st-and.ac.uk/Extras/Turing_running.html

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Re: [math-fun] Ten-finger discounts
by Henry Baker 31 Dec '14

31 Dec '14

IBM's Power8 chip: http://www.linleygroup.com/mpr/article.php?id=11088 "The DFU efficiently supports binary-coded decimal (BCD) math" The BCD hardware math accelerator looks interesting&fun: Can anything clever/cool be done with it? At 01:58 PM 12/31/2014, Bill Gosper wrote: >This is starting to make me want one of those "Ten fingers are overrated" T shirts.

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[math-fun] Fwd: 2015 with the other six digits
by Bill Gosper 31 Dec '14

31 Dec '14

I believe Knuth sketched an algorithm for this in his latest Christmas Tree Lecture. ---------- Forwarded message ---------- From: Zack Chroman <zacchro(a)gmail.com> Date: Wed, Dec 31, 2014 at 6:10 AM Subject: Re: [math-fun] 2015 with the other six digits To: Bill Gosper <billgosper(a)gmail.com> Using 1,2,3,4,5,6,7,8,9 in that order (with concatenated digits), it's possible up to 6964. I have some Mathematica code for this because of this problem <https://projecteuler.net/problem=259>, but it's being annoying when I try to get it to go 10-1 instead of 1-9. -Zack On Wed, Dec 31, 2014 at 4:37 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > Hans Havermann>[...] > > Don't forget to count down on New Year's Eve: > > 10*9*8*7/6/5*4*3-2+1 > -------- > > When's the soonest year this becomes impossible? --rwg >

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Re: [math-fun] "The Imitation Game" -- review
by Warren D Smith 31 Dec '14

31 Dec '14

> From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] "The Imitation Game" -- review > Message-ID: <E1Y6NNI-0006Lv-1l(a)elasmtp-kukur.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > Yesterday, I was dragged, kicking & screaming, to The Imitation Game by my > wife. > > It was extremely painful to watch this movie, and I've been having a hard to > trying to figure out why. > > Perhaps the best analogy would be a movie about Mozart by someone who was > born deaf. No matter what anyone told this deaf movie-maker about Mozart's > music, it would always be completely foreign hearsay, and could never convey > Mozart's musical genius to someone who does know & love Mozart's work. > ("Amadeus", of course, is the complete opposite, as it was made by people > who really, really enjoyed & understood Mozart's music.) > > Turing's life was totally tied up in his head, and he must have spent an > hour or two running every day as his form of mental yoga (his marathon > running times were world-class, and you can't keep that level of fitness > without running at least one or two hours per day). Although the movie > shows him running, the movie completely misunderstands the point of his > running -- it is time alone when he can contemplate & work out some of his > theories. "Aspies" like Turing find too much external stimulation painful > and distracting, --The whole notion Turing had Asperger syndrome, likely was made up / exaggerated by the makers of this movie, I do not think there is evidence for it. Anyway the two book bios of him, do not claim he had Asperger. He apparently did have a minor speech problem something like a stutter, which may have affected his sociality or lack thereof. It also has been popular, perhaps as part of an attempt to make Asperger-afflicted people feel better by association, to claim that e.g. Mozart, Einstein, Newton, Tesla had it. Well, maybe, but I'm dubious of such retrospective diagnoses. P.A.M.Dirac strikes me as quite likely, though. http://en.wikipedia.org/wiki/Retrospective_diagnoses_of_autism > so Turing's marathon runs were a welcome respite against > this unwanted external stimulation. "I have such a stressful job that the > only way I can get it out of my mind is by running hard; its the only way I > can get some release." > > Perhaps a key point that The Imitation Game wanted to make, but really > screwed up, was the fact that mathematicians like Turing were so highly > motivated by the challenge of the puzzle, that they wouldn't have worked any > faster (or any slower) on solving the Enigma problem knowing that many, many > lives were at stake. It isn't because Turing didn't care about people's > lives; it's just that whether he cared or not wouldn't make any difference > in how he attacked the problem. "Caring" in this context simply became > another distraction to be ignored. > > The Cairncross character is as completely gratuitous as it is unhistorical; > there is no evidence that Cairncross ever even met Turing, much less worked > with him. It is sad that The Imitation Game has to stoop to such low depths > to find intrigue; Turing never made any secret about his homosexuality, and > as a result, any attempt to blackmail him by threatening to out him would > not have worked. --well... I do not think Turing made a huge secret of it, but it obviously was not tremendously well known either, and in fact when it did become known to the wrong people, Turing paid heavily for that. So yes, he could in principle have been blackmailed. --there happen to be at least 2 book biographies of Turing, both pretty good, (by Andrew Hodges and by B Jack Copeland), as well as a "docudrama" about his life "Codebreaker, the sort of Alan Turing" on DVD, which, amazingly, I'm about to see tonight at some party.

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Re: [math-fun] 2015 with the other six digits
by Bill Gosper 31 Dec '14

31 Dec '14

Hans Havermann>[...] Don't forget to count down on New Year's Eve: 10*9*8*7/6/5*4*3-2+1 -------- When's the soonest year this becomes impossible? --rwg

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