May 2016 - mathfun@mailman.xmission.com

[math-fun] Fwd: [LifeCA] No Grandparents!
by rcs＠xmission.com 07 May '16

07 May '16

Dave Green reports that 'mtve' has solved the Grandfather Problem for Conway Life. GP is to find a pattern with a one-generation predecessor, but no two-generation predecessor. ----- Forwarded message from LifeCA(a)yahoogroups.com ----- Date: Sat, 7 May 2016 17:52:34 -0500 From: "Dave Greene david.m.greene(a)gmail.com [LifeCA]" <LifeCA(a)yahoogroups.com> Subject: [LifeCA] No Grandparents! To: LifeCA <LifeCA(a)yahoogroups.com> On May 5th, 'mtve' on the conwaylife.com forums solved the grandfather problem, with a 298-cell pattern: x = 24, y = 23, rule = B3/S23 8ob5ob4ob3o$obo2b2o2b3o2b4ob2o2bo$2obob4ob4ob2obo2b3o$3obobobobo2bob3o b3o$ob2o6bo2b5o3b2o$bobob9obobo$obob2o4bo2bob4obo$obobo2b3ob3ob2o$4o2b 2ob3ob4o4bo$4ob6o2bo6bo$obob3ob2ob2ob3ob2o3bo$ob3obob6ob4obo$2b6obob2o 2b3o$2o2bobobob4o$b4ob4o2b6o4b2o$obobobo2b2ob2o2b2o2b3o$o3b2ob2o2b3ob 2o3bo$2ob2o4b2o4b3o$3obob5o3b4obobobo$ob4o2b2o3bobo4bo2bo$2ob2o4b2o3bo 5b2obo$b4obo2bobo4bob2o2b2o$3o3b2ob3o2bobo6bo! The pattern is rumored to have 17,920 parents that all contain the same 622-cell orphan. Matthias Merzenich ran a search in JLS to verify: http://www.conwaylife.com/forums/viewtopic.php?f=2&t=576&p=30713#p30694 Some details on the algorithm used to make the discovery are here: http://www.conwaylife.com/forums/viewtopic.php?f=2&t=576&p=30713#p30667 Is the $50 prize mentioned in LifeLine #6 still available, or was there a statute of limitations that expired sometime in the intervening four decades? Keep the cheer, Dave Posted by: Dave Greene <david.m.greene(a)gmail.com>

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[math-fun] Question about a set
by David Wilson 07 May '16

07 May '16

Consider the set S of numbers that can be gotten from elements of Q by addition, multiplication, and taking integer roots? That is, a in Q => a in S a, b in S => a + b in S a, b in S => ab in S a in R, k in Z, a^k in S => a in S I understand S would be a proper subset of the algebraic numbers. Does S have a name? Is S a field?

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[math-fun] regular polytopes in 3D and 4D, file of coordinates
by Warren D Smith 07 May '16

07 May '16

In case anybody wants them, this file https://dl.dropboxusercontent.com/u/3507527/RegPolytopes.c contains the coordinates of the vertices of all regular polytopes in 3D and 4D in a form ready to use inside C programs. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] differential equations -- cause flight delays and interrogations
by Henry Baker 07 May '16

07 May '16

FYI -- I've always been suspicious of economists, too, but only the *macro* economists like Bernanke who destroyed the financial system in 2008; whatever equations they were smoking back then didn't work! Ivy League economist ethnically profiled, interrogated for doing math on American Airlines flight <https://www.washingtonpost.com/news/rampage/wp/2016/05/07/ivy-league-econom…>https://www.washingtonpost.com/news/rampage/wp/2016/05/07/ivy-league-econom… By Catherine Rampell May 7 at 9:19 AM 11:20 a.m.: This post has been updated to include additional details from American Airlines. On Thursday evening, a 40-year-old man -- with dark, curly hair, olive skin and an exotic foreign accent -- boarded a plane. It was a regional jet making a short, uneventful hop from Philadelphia to nearby Syracuse. Or so dozens of unsuspecting passengers thought. The curly-haired man tried to keep to himself, intently if inscrutably scribbling on a notepad he'd brought aboard. His seatmate, a blond-haired, 30-something woman sporting flip-flops and a red tote bag, looked him over. He was wearing navy Diesel jeans and a red Lacoste sweater Â a look he would later describe as "simple elegance" Â but something about him didn't seem right to her. She decided to try out some small talk. Is Syracuse home? She asked. No, he replied curtly. He similarly deflected further questions. He appeared laser-focused Â perhaps too laser-focused Â on the task at hand, those strange scribblings. Rebuffed, the woman began reading her book. Or pretending to read, anyway. Shortly after boarding had finished, she flagged down a flight attendant and handed that crew-member a note of her own. Then the passengers waited, and waited, and waited for the flight to take off. After they'd sat on the tarmac for about half an hour, the flight attendant approached the female passenger again and asked if she now felt okay to fly, or if she was "too sick." I'm OK to fly, the woman responded. She must not have sounded convincing, though; American Airlines flight 3950 remained grounded. Then, for unknown reasons, the plane turned around and headed back to the gate. The woman was soon escorted off the plane. On the intercom a crew member announced that there was paperwork to fill out, or fuel to refill, or some other flimsy excuse; the curly-haired passenger could not later recall exactly what it was. The wait continued. Finally the pilot came by, and approached the real culprit behind the delay: that darkly-complected foreign man. He was now escorted off the plane, too, and taken to meet some sort of agent, though he wasn't entirely sure of the agent's affiliation, he would later say. What do know about your seatmate? The agent asked the foreign-sounding man. Well, she acted a bit funny, he replied, but she didn't seem visibly ill. Maybe, he thought, they wanted his help in piecing together what was wrong with her. And then the big reveal: The woman wasn't really sick at all! Instead this quick-thinking traveler had Seen Something, and so she had Said Something. That Something she'd seen had been her seatmate's cryptic notes, scrawled in a script she didn't recognize. Maybe it was code, or some foreign lettering, possibly the details of a plot to destroy the dozens of innocent lives aboard American Airlines Flight 3950. She may have felt it her duty to alert the authorities just to be safe. The curly-haired man was, the agent informed him politely, suspected of terrorism. The curly-haired man laughed. He laughed because those scribbles weren't Arabic, or some other terrorist code. They were math. Yes, math. A differential equation, to be exact. Had the crew or security members perhaps quickly googled this good-natured, bespectacled passenger before waylaying everyone for several hours, they might have learned that he -- Guido Menzio Â is a young but decorated Ivy League economist. And that he's best known for his relatively technical work on search theory, which helped earn him a tenured associate professorship at the University of Pennsylvania as well as stints at Princeton and Stanford's Hoover Institution. They might even have discovered that last year he was awarded the prestigious Carlo Alberto Medal, given to the best Italian economist under 40. That's right: He's Italian, not Middle Eastern, or whatever heritage usually gets ethnically profiled on flights these days. Menzio had been on the first leg of a connecting flight to Ontario, where he would give a talk at Queen's University on a working paper he co-authored about menu costs and price dispersion. His nosy neighbor had spied him trying to work out some properties of the model of price-setting he was about to present. Perhaps she couldn't differentiate between differential equations and Arabic. Menzio showed the authorities his calculations and was allowed to return to his seat, he told me by email. He said the pilot seemed embarrassed. Soon after, the flight finally took off, more than two hours after its scheduled departure time for what would be just a 41-minute trip in the air, according to flight-tracking data. The woman never reboarded to the flight. Casey Norton, a spokesman for American Airlines (whose regional partner Air Wisconsin operated the flight), said the woman had indeed initially told the crew she was sick, but when she deplaned she disclosed that the reason she was feeling ill was her concern about the behavior of her seatmate. At that time, she requested to be rebooked on another flight. The crew then called for security personnel, who interviewed Menzio and determined him not to be a "credible threat." Norton did not know whether the woman was ever notified that Menzio had been cleared. (He said he was not allowed to give out her name for privacy reasons, and since Menzio did not know it either, I have not been able to contact the woman for comment.) Whenever there are conflicts between passengers, Norton said, "we try to work with them peacefully to resolve it," whether that means changing seat assignments or switching someone to take a different flight. When asked how often customers raise similar suspicions about fellow passengers that turn out to be unfounded, he said it happens "from time to time" but declined to provide details about frequency. Menzio for his part says he was "treated respectfully throughout," though he remains baffled and frustrated by a "broken system that does not collect information efficiently." He is troubled by the ignorance of his fellow passenger, as well as "A security protocol that is too rigidÂin the sense that once the whistle is blown everything stops without checksÂand relies on the input of people who may be completely clueless." Rising xenophobia stoked by the presidential campaign, he suggested, may soon make things worse for people who happen to look a little other-ish. "What might prevent an epidemic of paranoia? It is hard not to recognize in this incident, the ethos of [Donald] Trump's voting base," he wrote. In this true parable of 2016 I see another worrisome lesson, albeit one also possibly relevant to Trump's appeal: That in America today, the only thing more terrifying than foreigners isÂmath.

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[math-fun] Propp/Asimov sequence problem
by Warren D Smith 07 May '16

07 May '16

--NJA Sloane tells me "Dan Asimov's misunderstanding of Propp's question, which of course is https://oeis.org/A165911, ... Does it go into a cycle? WDS gave an inconclusive analysis. The graph of the first 1000 terms (go to the OEIS A165911 entry and click "graph") was also inconclusive. So in the night I added more terms to the b-file, and now the graph definitely is heading off to infinity. Roughly like e^(n/10)." --And I also think my analysis can be made more definitive (although still nonrigorous) as follows. problem: can Propp/Asimov's sequences f[n] = CorePart( f[n-1]+f[n-2] ) ever (with suitable starting integers) diverge to infinity without instead falling into a finite length cycle? Here CorePart(X) means the product of all the primes inside X's factorization, e.g. CorePart( 2*3*3*5*7*7*7*7 ) = 2*3*5*7. (Is that what Asimov meant?) If we model the X as "random" and ask about factors of P inside them (P=prime) then the chance X has exactly K factors of P inside it (K>=0) is exactly (P-1) * P^(-K-1) and then max(K-1, 0) of those factors need to be removed from X. So the expected additive reduction in ln(X) when we remove those factors is DELTA = DOUBLESUM( (K-1)*ln(P)*(P-1)*P^(-K-1), summed over primes P and integers K>=2 ). We can evaluate the sum over K in closed form, then the remaining sum over prime P is done numerically: DELTA = SUM( ln(P)/ [(P-1)*P], primes P ) =0.7554 meanwhile the increase in ln(X) due to the Fibonacci recurrence should be upper bounded by ln(2) = 0.6931 so it seems that the expected decrease in ln(X) ought to outweigh the expected increase due to summing, and therefore with "probability=1" this analysis would predict NON-growth to infinity and hence ultimate cycling. Exactly the opposite of Sloane's empirical computer finding. So evidently something is wrong with my cheesy probability model. And sure enough, examining Sloane's computer output https://oeis.org/A165911/b165911.txt we see that his numbers are NOT random uniform mod 2. Instead they are odd,odd,even,odd,odd,even,odd,odd,even... hence even only 1/3 of the time, not 1/2 as would naively be expected. Can other modular patterns be found (e.g. mod 3, etc)? Probably just a few such patterns will suffice to make the probability argument then reach the correct prediction. Maybe. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Rot13 and other substitution ciphers
by Jim Gillogly 06 May '16

06 May '16

> rwg > wrote: > > On 2016-04-15 09:25, Jim Gillogly wrote: > > Keith F. Lynch writes: > >> The longest words that remain words after being rot13'd are > >> Chechen <-> Purpura. > > > > Checking my unabridged, I see another cromulent pair at 7: > > nowhere <-> abjurer > > > > They also have the advantage of being usable for Scrabble. > > What's Scrabbologically wrong with purpura? (Except it being the > name of Edward Teller's secretary at LLNL.) > Nothing's wrong with "purpura", which is in the Official Scrabble dictionary. It's the *pair* that you won't find in a game, since Chechen is a proper noun. I grant that it's not a very serious advantage, but hey... this is about fun, right?? -- Jim Gillogly

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[math-fun] 5 and 7 outcome fair dice
by Bill Gosper 06 May '16

06 May '16

Odd barrel dice have the problem that two faces are always uppermost, so how do you label them? Alternative to face-transitivity, there are fair dice which cannot rest on any face, and yet have one face uppermost. Spoiler: gosper.org/5&7dice.png --rwg I assume these are old ideas. Check out Wikipedia if you haven't seen spherical, 6 outcome dice.

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Re: [math-fun] Lasker
by Bill Gosper 06 May '16

06 May '16

On 2016-04-08 14:39, Warren D Smith wrote: > Macaulay's book gave a star role to the amazing theorem of Lasker, > also mentioned here > > https://en.wikipedia.org/wiki/Primary_decomposition > > It is due to Emanuel Lasker, world chess champion 1894-1921, > which I think made him the longest-reigning world champ. > So, I had not realized this, but Lasker also made a lasting contribution > to mathematics. > >From http://exeterchessclub.org.uk/content/chess-stories Lasker incognitoLasker often used to play without letting on who he was: he had fun at the expense of one poor chap who managed always to win against his mystery opponent when he gave knight odds, but lost when he had the extra knight! Probably apocryphal (Hans?), but the story I heard was that, en route to his 1921 Havana defeat, a fellow passenger found him analyzing a position and, unsuspecting, introduced himself as president of his local chess club. And then beat Lasker in game. "This piece here, how do you call it?" "The queen." "Do you mind if I play without it? It is impairing my mobility." [You can guess the rest.] --rwg

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Re: [math-fun] Rot13 and other substitution ciphers (was Re: Quickies)
by Jim Gillogly 06 May '16

06 May '16

Keith F. Lynch writes: > The longest words that remain words after being rot13'd are > Chechen <-> Purpura. Checking my unabridged, I see another cromulent pair at 7: nowhere <-> abjurer They also have the advantage of being usable for Scrabble. -- Jim Gillogly

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[math-fun] Propp's "squarefree-fibo" sequence
by Warren D Smith 05 May '16

05 May '16

Well, can Propp's sequences f[n] = SquareFreePart( f[n-1]+f[n-2] ) ever (with suitable starting integers) diverge to infinity without instead falling into a finite length cycle? Here SquareFreePart(X) means the product of all the primes that had odd powers inside X's factorization, e.g. SquareFreePart( 2*3*3*5*7*7*7*7 ) = 2*5. If we model the X as "random" then with chance 1/4 they contain a factor of 4 that will be removed, with chance 1/9 they contain a factor 9, etc. So ln(X) is going to decrease due to these removals, by an expected additive amount DELTA with DELTA <= SUM_{k>1} k^(-2) * ln(k^2) = 1.875096508631687507405148... The expected decrease amount also is lower bounded by the same sum but summed over prime k only: DELTA > 0.9861. Also note, the "probability distribution" of DELTA has finite variance. Meanwhile, ln(X) is going to grow due to the Fibonacci-like recurrence, by some amount presumably less than ln(2) = 0.693147180559945 in "expectation." A lower bound on the expected decrease amount is clearly greater than an upper bound on the expected increase amount. Therefore by the strong law of large numbers" of probability theory, it "follows" that increase of ln(f[n]) to infinity is impossible. This of course would only be a proof if the f[n] really behaved "randomly," which is bogus, but it is enough to make the "it always cycles" conjecture plausible. I tend to doubt Propp's stronger conjecture that only one particular limit cycle is possible; and to disprove that, it would suffice to find a single counterexample starting 2-tuple. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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