May 2019 - mathfun@mailman.xmission.com

[math-fun] Smullyan paradox about belief
by James Propp 02 May '19

02 May '19

I recall encountering in one of Raymond Smullyan’s books a thought experiment that convinced me that it is possible to be mistaken about what one believes. That is, one can say “I believe X” and be speaking falsely even though one is not intentionally lying. Does anyone know what thought experiment I’m dimly recalling, and which of his books it appears in? Thanks, Jim Propp

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

01 May '19

Now that I've thought about it for a moment, it's clear that even the condition (*) (that each k-face is determined uniquely by its set of vertices) is too weak. Two squares could each share 3 of their vertices with the other. For a change I will attempt to hold my peace while thinking. —Dan

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

01 May '19

Good points, Tom and James! Let X (like the S^3 in question) be the abstract union of unit n-dimensional cubes Q = [0, 1]^n, where any two intersect only along entire common k-faces, for some k. Let Vert(X) denote its set of vertices (i.e., coming from the n-cubes Q). By definition, a "subcube" of X is any k-dimensional cubical face of one of its component n-cubes Q, for any k. I've been trying to think of what additional constraint(s) will fix this problem of too many examples. A natural condition is to require that: (*) Any subcube is uniquely determined by its vertices. (See below for another related condition.) Assuming this condition: ------------------------ This rules out the examples of Tom and James, and leads to the 6 square faces of a cube as the unique smallest possible cubulated S^2. (Sorry for having overlooked the need for this, in my headlong, willy-nilly rush to post my half-baked idea. I hope this time it's at least 3/4-baked.) ----- Question: --------- Under the additional constraint of I) above, what is a (the?) cubulated 3-sphere S^3 with the fewest cubes Q = [0, 1]^3 ??? ----- —Dan ————— *PS Also consider this condition: (**) X embeds in some Euclidean space R^n with Vert(X) = X ∩ Z^n. I.e., there is a mapping f : X —> R^n that is a topological equivalence with its image f(X), such that the vertices of X are precisely X ∩ Z^n. It seems clear to me that (**) implies (*). --------- QUESTION: Does (*) imply (**) ??? --------- —D. --------------------------------------------------------------------------- James Buddenhagen wrote: ----- More generally for 2D squares, let n be any positive integer greater than 2. Form a (say regular) n-gon as an equator of S^2 and join all vertices to both the north and south pole. Now erase the equatorial polygon and you are left with a topological S^2 composed of n topological squares. On Tue, Apr 30, 2019 at 10:38 PM Tom Karzes <karzes(a)sonic.net> wrote: > For the case with 2D squares, can't you join three squares at a common > vertex, as in the corner of a cube, and then do the same thing with the > opposite corners to obtain a sphere from 3 squares? > > Also, can't you join two squares at an edge to obtain a rectangle, then > join two rectangles (as in the two square case) to obtain a topological > sphere from 4 squares? Or is there a constraint that I missed that > these examples don't satisfy? > > Tom > > Dan Asimov writes: > > 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. -----

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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 01 May '19

01 May '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 01 May '19

01 May '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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