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[math-fun] Cubulated three-sphere puzzle
by Dan Asimov 01 May '19

01 May '19

Suppose the sphere S^2 is made of (2D) square pieces of rubber sewn together abstractly along common edges. (I.e., the result is topologically equivalent to a sphere.) According to this rule, there is a trivial case where only 2 squares are sewn together, by identifying corresponding edges — this is indeed a sphere topologically. The fact that [in 3-space any two planar squares with the same boundary are the same] is not our concern. It's seems clear that — except for this trivial example — the smallest number of squares in such a thing is 6. Now suppose the 3-sphere S^3 = {x in R^4 | ||x|| = 1} is, similarly, built abstractly out of (solid) cubes, copies of Q = [0,1]^3, that are allowed to intersect only along entire common square faces, edges, or vertices. Puzzle: ------- What is the smallest number of cubes with which it's possible to build something topologically equivalent to S^3 this way? ------ I think I know, but I haven't tried to prove it yet. —Dan

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[math-fun] Attacking the MIT time capsule cipher?
by Jason Holt 30 Apr '19

30 Apr '19

(Background at the bottom). Reading through the implementation (link at very bottom), they seem to be giving away quite a few bits of the secret, which makes me wonder if they did that on purpose. The challenge is to find the secret value w = 2^(2^t) (mod n) given t and n. But then they also give us w XOR an ASCII congratulatory message sgen, which includes some known appended plaintext: sgen = sgen.append(" (seed value b for p = "+prand.toString()+")"); prand is an ASCII decimal representation, where they find the next prime after 5^prand % (2^1024) to get p, one of the factors of n, which lets you prove that you've solved the puzzle. They also right justify the congratulatory message before XOR, so they're presumably also giving us some number of high order bits of w. So that makes me wonder things like: * Is it easier to compute w = 2^(2^t) (mod n) given known bits of w? * Is there some relationship between 5^prand % (2^1024) and w = 2^(2^t) (mod n) such that, say, the first ASCII digit of prand XOR the appropriate bits of w lets us recover prand a piece at a time? * Seems like w should be hard to predict but isn't expected to be as random as a secure PRNG. Does that give us any further hints about w XOR sgen? Background: https://hackaday.com/2019/04/30/mit-cryptographers-are-no-match-for-a-deter… Twenty years ago, a cryptographic puzzle was included in the construction of a building on the MIT campus. The structure that houses what is now MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) includes a time capsule designed by the building’s architect, [Frank Gehry]. It contains artifacts related to the history of computing, and was meant to be opened whenever someone solved a cryptographic puzzle, or after 35 years had elapsed. The puzzle was not expected to be solved early, but [Bernard Fabrot], a developer in Belgium, has managed it using not a supercomputer but a run-of-the-mill Intel i7 processor. The capsule will be opened later in May. ... https://people.csail.mit.edu/rivest/lcs35-puzzle-description.txt

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Re: [math-fun] Hot tea sucks
by Bill Gosper 30 Apr '19

30 Apr '19

Les, thanks for the headsup. Is any of this stuff still classified? Has any DoD person challenged your claims? Putting together the observations that it was huge, powerful, secretive, and diligently operated despite supposed fraudulence, I have a conjecture: Maybe the radar was so powerful and Russian jamming was so lame that SAGE could simply overwhelm any jamming equipment the Russians could fly across the Pacific? They may have been unable to jam more than a few frequencies at the same time. And their very act of jamming should have lost them the element of surprise. —Bill On Tue, Apr 30, 2019 at 12:22 PM Lester D Earnest <learnest(a)stanford.edu> wrote: > Below is a note that I sent at the beginning of this year to POST, the > nonprofit that owns Mt. Umunhum. > > After our discussion last July, in which you declined to get involved with > my effort to reveal the truth about the Mt. Umunhum radar tower, I did an > interview with a web news service that was carved into five pieces, > then posted online a couple of weeks ago and has drawn national interest, > with several journalists around the country planning to interview me. > > The first segment specifically addresses the SAGE air defense > system, with the Mt. Umunhum radar being a small piece of it. If you are > interested in seeing that 21 minute segment, go to *Cold War Radar System > a Trillion Dollar Fraud > <http://salsa4.salsalabs.com/dia/track.jsp?v=2&c=ziQ4WZg%2B%2F9ZbZ7Z%2BBbb8Y…>*. > I still aim to get that fraudulent system exposed, given that I was partly > responsible for it, and get your organization to tear down that decrepit radar > tower, or at least add a display that tells the truth instead of the lies > that have been posted there. > > Lester Earnest Call land line 650-941-3984 any day from 9am till 11pm > Pacific Time > > Senior Research Computer Scientist Emeritus, Stanford University > > https://web.stanford.edu/~learnest/ > ------------------------------ > *From:* Bill Gosper <billgosper(a)gmail.com> > *Sent:* Sunday, April 28, 2019 11:06 PM > *To:* math-fun(a)mailman.xmission.com > *Subject:* Hot tea sucks > > I *thought* Tom Knight beat Brent Meeker with the top it up solution, but > that > mail seems only intermittently visible in my RoundCube. > > On 2019-04-26 09:05, Veit Elser wrote: > > First hike up Mt. Umunhum (always wanted to use that in a sentence) or > > someplace higher. > > > You've touched on an interesting subject which I encourage you to Google > further. > Les Earnest claims the station up there was part of the SAGE air defense > system, > which was the biggest taxpayer fraud in history. But he also mentions > that, 20 miles > away, it was blitzing the Stanford AILab computer at 13 second intervals. > I.e., > whatever they were doing, the people up there were serious. The actual > radar > building is improbably large: > http://www.redwoodhikes.com/Skyline/MtUmunhum1.jpg > There were people actually living there year-round. They even had a > bowling alley. > (I bet that, summer afternoons, it's *way* too windy for volleyball!) > And for years after it closed, the area was very off-limits to hikers and > motorists. > —rwg > >> On Apr 26, 2019, at 11:12 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> > >> I brew it in a 46oz pickle jar that easily tips over, and leaks if the > lid > >> isn't snug. > >> But if I forget to momentarily loosen the lid within a few minutes, a > >> powerful vacuum > >> develops that makes it virtually impossible to unscrew. What is the > >> trivial solution? > >> —BillI >

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[math-fun] An Unusual Integral?
by Brad Klee 29 Apr '19

29 Apr '19

3==Integrate[Hypergeometric2F1[1/3,2/3,1,x-(1/4)*x^2],{x,0,2}] ( Cf. https://oeis.org/A006480 and thereafter ) Gosper said that this calculation was "weird". I think it would be rare to get an integer value from an integral of 2F1(a,b,c,z), with non-cancelling (a,b,c), z=P(x) in Q[[x]], and z over 0...1, between D.E. singular points. Does anyone else have an opinion? --Brad

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Re: [math-fun] A few questions about prime generating functions
by Dan Asimov 29 Apr '19

29 Apr '19

That Wikipedia article Formulas for primes contains this interesting tidbit: ----- It is not even known whether there exists a univariate polynomial of degree at least 2, that assumes an infinite number of values that are prime; ----- The article appears to be referring to *integer* polynomials. What about a *real* polynomial P(x), deg(P) >= 2 ??? Can some such P(x) be proved to represent infinitely many primes? —Dan ----- II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? -----

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[math-fun] A few questions about prime generating functions
by Dan Asimov 29 Apr '19

29 Apr '19

I. It's widely believed that there's no nice function f : Z+ —> Z+ whose values are all prime: f(Z+) ⊂ Primes , where Primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,...}. Are there any actual theorems that say as much — necessarily defining "nice" ? Or at least restricting the classes of nice functions for which this might be possible? How about heuristics indicating why this should be hard? * * * II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? Dirichlet's famous 1837 theorem states that every arithmetic sequence of form {X_n = A + B n}, where A, B are relatively prime integers, contains infinitely many primes.* But I don't know of any other cases. Is that because they aren't known? —Dan ————— * I once read the proof (as a very readable chapter in Elliptic Curves by Anthony Knapp) and thought it astonishing innovative, but especially for that long ago!

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[math-fun] Hot tea sucks
by Bill Gosper 29 Apr '19

29 Apr '19

I *thought* Tom Knight beat Brent Meeker with the top it up solution, but that mail seems only intermittently visible in my RoundCube. On 2019-04-26 09:05, Veit Elser wrote: > First hike up Mt. Umunhum (always wanted to use that in a sentence) or > someplace higher. > You've touched on an interesting subject which I encourage you to Google further. Les Earnest claims the station up there was part of the SAGE air defense system, which was the biggest taxpayer fraud in history. But he also mentions that, 20 miles away, it was blitzing the Stanford AILab computer at 13 second intervals. I.e., whatever they were doing, the people up there were serious. The actual radar building is improbably large: http://www.redwoodhikes.com/Skyline/MtUmunhum1.jpg There were people actually living there year-round. They even had a bowling alley. (I bet that, summer afternoons, it's *way* too windy for volleyball!) And for years after it closed, the area was very off-limits to hikers and motorists. —rwg >> On Apr 26, 2019, at 11:12 AM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> I brew it in a 46oz pickle jar that easily tips over, and leaks if the lid >> isn't snug. >> But if I forget to momentarily loosen the lid within a few minutes, a >> powerful vacuum >> develops that makes it virtually impossible to unscrew. What is the >> trivial solution? >> —BillI

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Re: [math-fun] Hot tea sucks.
by Bill Gosper 28 Apr '19

28 Apr '19

On 2019-04-26 21:17, Andres Valloud wrote: > Does it involve one of those lids that snaps? No, just a wide-mouth twist off Mezzetta's jar. > You could put in the lid > with the snapper up, and when it snaps down then you know you have to > undo the lid. > > I suppose you could also forget you were brewing tea, so the snapper > sound won't be enough to remind you. But perhaps the lid snapper is > enough to keep the strong vacuum from forming in the first place. > > Filling the jar completely [Brent Meeker's suggestion] has the problem >that the jar will very likely > spill while closing as well as opening, and it sounds like that's > something you want to avoid. Exactly. > So, almost fill it, and use a snapping lid. Recall that my (satisfactory) solution of merely blowing cool air under the lid as you replace it the first time. How robust is Brent's top-up suggestion? I purposely underfilled by about .25", and closed it quickly without blowing away the "steam", waiting extra long before trying the lid. It came off! Grudgingly. Believing the experiment to be over, a chugged several swallows, since it wasn't very hot. Then I reclosed it and forgot about it for maybe an hour. Next time, it was hard as heck to get off! The dT was modest enough, but the dV wasn't! Double verification for Brent's volume emphasis. —rwg > > On 4/26/19 15:54 , Bill Gosper wrote: >> I BCCed a lot of people and got a lot of nontrivial suggestions. >> (Restaurateur) Mike Wang's rubber band suggestion >> was probably best. Just encircle the lid with a big one from a broccoli >> bunch. It's amazing if you've never tried it. >> But that's not trivial enough! (I forget where I stashed my rubber band.) >> Someone suggested that if I have a mountain >> handy, I could drive up it. Nobody suggested the dicey solution of >> sticking the jar back in the 𝜇wave. Really hot tapwater >> should work, but our water heater is set way low. C'mon people—*trivial*! >> >> SPOILER: >> In[532]:= StringReverse@"!no ti pals uoy sa dil eht rednu ria )looc >> ylevitaler( fo llufgnul kciuq a wolb tsuJ" >> —rwg >> >> On Fri, Apr 26, 2019 at 8:12 AM Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> I brew it in a 46oz pickle jar that easily tips over, and leaks if the lid >>> isn't snug. >>> But if I forget to momentarily loosen the lid within a few minutes, a >>> powerful vacuum >>> develops that makes it virtually impossible to unscrew. What is the >>> trivial solution? >>> —Bill

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Re: [math-fun] math of carnival games
by Dave Dyer 27 Apr '19

27 Apr '19

This is a very entertaining video illustrating the power of statistics used as a carnival scam. https://www.youtube.com/watch?v=527F51qTcTg&feature=youtu.be&fbclid=IwAR0R3…

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Re: [math-fun] math of carnival games
by Dave Dyer 27 Apr '19

27 Apr '19

This is a very entertaining video illustrating the power of statistics used as a carnival scam. https://www.youtube.com/watch?v=527F51qTcTg&feature=youtu.be&fbclid=IwAR0R3…

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