October 2014 - mathfun@mailman.xmission.com

[math-fun] Self-referential counterexamples to modus tollens
by James Propp 22 Oct '14

22 Oct '14

Spurred by a chance encounter with Allan Wechsler, I just came up with a pair of sentences that have opposite truth values even though they are (formally) contrapositives: 1. If the premise of this implication does not contain the letter "p", then two is not even. 2. If two is even, then the premise of this implication contains the letter "p". Note that 1 is (vacuously) true whereas 2 is false. Of course there is no contradiction here, since "this implication" has different meanings in the two sentences. (Thanks, Allan, for teaching me the word "deictic" today!) Can anyone find a neater pair of sentences along these lines? My esthetic preference would be for eschewing tricks like referring to the letters that occur in different parts of the sentence. The sorts of sentences I would prefer would be things along the lines of "If the contrapositive of this sentence implies the inverse of this sentence, then either the premise of this sentence is true or the conclusion of this sentence is false." I've made unsuccessful attempts in the past to construct a true sentence of this kind whose formal contrapositive is false. Jim Propp

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[math-fun] Someone needs to take a photo of the Riemann hypothesis
by Thane Plambeck 22 Oct '14

22 Oct '14

https://photomath.net/ -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] "My" solution of Wilson's ordering problem in all dimensions... scooped.
by Warren D Smith 22 Oct '14

22 Oct '14

I proudly wrote up & posted it here http://rangevoting.org/WilsonOrder.html then made mistake of telling hyperplane-arrangements expert Richard P. Stanley about it. He soon found I.J. Good & T.N. Tideman: Stirling Numbers and a Geometric Structure from Voting Theory, J. Combin. Theory, Ser. A 23,1 (1977) 34-45 http://ac.els-cdn.com/0097316577900772/1-s2.0-0097316577900772-main.pdf?_ti…' which solved my R(N,D) problem. They give a closed form in terms of sum over Stirling numbers, which I did not know. On other hand I actually think I prefer my answer to their (presumably equivalent) one, and I'm not sure their proof really is rigorous, there are a couple fine points they blissfully ignored. But basically, they scooped me 37 years ago. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] square into acute, isosceles ▲s
by Warren D Smith 21 Oct '14

21 Oct '14

Many (all?) of those dissections "cheat" in the sense they are not "face to face" triangulations. Can it be done under this demand? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] cylinder intersection
by Warren D Smith 21 Oct '14

21 Oct '14

QUESTION (Dan Asimov): Suppose given N bi-infinite cylinders of radius 1 in 3-space, whose axes all contain the origin. What is the maximum number of (curved) faces that their common intersection can have? --I think the answer should be O(N) [so Grossman guess of a quadratic(N) is wrong]. Am I confused? More generally I think, if you intersect N convex 3-dimensional objects (to get a convex result of course) and the original objects' surfaces are defined by piecewise algebraic equations with bounded degree and bounded #pieces, then the final object should enjoy O(N) bounds on its face, edge, and vertex count, and can be computed via an O(NlogN)-time "randomized incremental" algorithm. Or at least this ought to be true of objects like spheres, halfspaces, or cylinders which have the property that when you intersect two, you get 1 convex object with at most two pieces discarded (if intersection nontrivial). The general idea to prove that is in: K.L.Clarkson & P.W.Shor: Applications of Random Sampling in Computational Geometry, II, Discrete and Computational Geometry 4,5 (1989) 387-421. and probably is also discussed in e.g. K.Mulmuley book.

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[math-fun] Lockheed Martin Claims Fusion Breakthrough That Could Change World Forever
by Ray Tayek 21 Oct '14

21 Oct '14

http://www.forbes.com/sites/williampentland/2014/10/15/lockheed-martin-clai… <http://www.lockheedmartin.com/us/products/compact-fusion.html>Lockheed Martin, the aerospace and defense conglomerate based in Bethesda, Md., is claiming to have made a major breakthrough in <http://www.dvice.com/2013-2-22/lockheeds-skunk-works-promises-fusion-power-…>nuclear fusion, which could lead to development of reactors small enough to fit on the back of a truck within a decade. ... --- co-chair http://ocjug.com/

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[math-fun] Marble problem
by David Wilson 21 Oct '14

21 Oct '14

A clear marble has N tiny bubbles in it, numbered 1 through N. Roll the marble on the floor, then list the bubbles in order of their distance from the floor (ignore situations where two or more bubbles are at the same height). Given an optimal distribution of bubbles, what is the largest possible number of bubble orders you could record? What if you measure distance from the contact point of the marble and floor?

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[math-fun] square into acute, isosceles ▲s
by Bill Gosper 21 Oct '14

21 Oct '14

Yesterday, without telling him http://www.mathpuzzle.com/21oct02.html, I challenged Nico Brown, http://mathcircle.berkeley.edu/images/nico-vincent-bmmt.JPG (left, here winning the Berkeley Mini-Math Tournament) to find such a dissection. I think there were only three known. He just found the fourth: https://mail.google.com/mail/?ui=2&ik=97b2a014c7&view=att&th=149301580b6c08… --rwg

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[math-fun] Petersen's booboo
by Bill Gosper 21 Oct '14

21 Oct '14

At Saturday's Stanford Celebration of Mind, one of the speakers (DEK?), keeping with the Centennial theme, exhibited Ernie Parker's 10x10 orthogonal Latin square, as featured in what was probably Gardner's first Scientific American cover story. Most of us have heard that the nonexistence of a 4n+2 square was one of Euler's few unlucky conjectures. But the speaker added that Julius Petersen, the father of Graph Theory, had actually published a (bogus) proof of Euler's incorrect dismissal of 4n+2. This is impressive scholarship--not in Wikipedia? --rwg

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[math-fun] puzzle: honest rank-orderings in D dimensions...now (almost) completely solved.
by Warren D Smith 20 Oct '14

20 Oct '14

My alleged solution to this puzzle is now posted here: http://rangevoting.org/WilsonOrder.html (Warnings: PRELIMINARY, may contain bugs and there are 4 annoying questions at the end I have not quite solved, though all are probably doable.) Comments? It seems to be a problem of fundamental importance in voting theory. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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