January 2016 - mathfun@mailman.xmission.com

[math-fun] Triangle question
by Dan Asimov 13 Jan '16

13 Jan '16

Ghosts of a departing triangle ------------------------------ Given a triangle whose angles have sizes u, v, w, draw its 3 angle bisectors. Now draw the 6 rays, 2 emanating from each vertex, that make angles of epsilon > 0 with the bisectors at the vertices. Call them "epsectors". Much like the trisectors in Morley's theorem, adjacent epsectors from different vertices will intersect pairwise at 3 points. Call the triangle formed by those 3 points T(epsilon). Question: --------- As epsilon approaches 0, what are the limiting angles of T(epsilon) (as functions of u, v, w) ??? —Dan

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[math-fun] GCHQ puzzle
by rcs＠xmission.com 10 Jan '16

10 Jan '16

Has anyone done the GCHQ puzzle yet? The grid & constraints are at http://www.bbc.com/news/uk-35058761 I noticed one row was completely specified by the information supplied. Rich

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[math-fun] 2016 , a simple expression
by Simon Plouffe 10 Jan '16

10 Jan '16

Hello, here is one for 2016. 2016 = 4/a and a is infinity ----- 5 \ n ) --------------- / exp(2 Pi n) - 1 ----- n = 1 nice. Best regards, Simon Plouffe

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[math-fun] lottery thought problem
by Thane Plambeck 10 Jan '16

10 Jan '16

design a simple-looking powerball-like lottery variant that has three properties: (0) it's easy to determine that a given bet has won after the drawing is made (1) it's nevertheless quite difficult, even for a mathematician, to work out the exact chances that a random bet wins [ this is the hard one. i want to do this via some type of hysteresis related to the ensemble of numbers played in the previous lottery draw, maybe ] (2) it has the property that with probability at most 1/10,000, no one will win even if hundreds of millions of people play every week for one thousand years. -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] Superexponential haystacks
by James Propp 08 Jan '16

08 Jan '16

What are people's favorite examples of problems in which one is looking for needles in a superexponential haystack that we suspect contains exponentially many needles but can't prove contains any at all? Here I am adopting Richard Guy's nice way of describing such problems, which he used last summer when he taught me about a beautiful example of this phenomenon (which I hope I am remembering correctly): It is almost certainly true that for all n sufficiently large, you can arrange the integers from 1 to n in a circle in such a fashion that any two consecutive numbers add up to a perfect square. But no one can prove it. I just learned about another nice example from Barry Cipra. His Bricklayer's Challenge ( http://www.pavelspuzzles.com/2012/11/the_bricklayers_challenge.html) appears to be solvable for every n, but nobody knows how to prove it for infinitely many n. I hereby rule out answers to my question in which the "needle-ness" of the needle takes a long time to verify. This is the case for some Ramsey theory problems that are shown to be solvable by the probabilistic method. It is also the case for codes that come close to the Shannon bound (which again are proved to exist by the probabilistic method). Jim Propp

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[math-fun] Guess my numbers
by James Propp 08 Jan '16

08 Jan '16

This may be an old chestnut, but I only encountered it once (and that was almost forty years ago), so I'd appreciate references. Suppose I've picked n positive integers x_1, x_2, ..., x_n, which it's your job to guess. Each of your questions must be of the form "What is the value of a_1 x_1 + a_2 x_2 + ... + a_n x_n?", where a_1, a_2, a_3, ... are positive integers you get to choose. How many questions must you ask me to be assured of being able to deduce my n numbers? I got this wrong when I first heard it. Hint: The answer isn't n. Please, no spoilers till Thursday; but any leads that don't include spoilers are welcome right away! Thanks, Jim

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Re: [math-fun] Draft of my next blog post
by Warren D Smith 07 Jan '16

07 Jan '16

I found it extremely annoying to read it. The author kept saying over and over he was not going to tell us what named initials were, and what named problems were. What the hell was he going to tell us? Who the hell cared -- I quit reading it before I found out. He still could not give me a single clue even after a whole page! I mean, there is such a thing as writing in a style guaranteed to annoy readers and make them say "this author is an asshole." Well: You found it. Your job as an author is to make it EASY for readers to learn. Pre-digest. Pre-think organization for maximum efficiency. Your job as an author is NOT to make is maximally hard for the reader because you enjoy masturbating in public.

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[math-fun] Draft of my next blog post
by James Propp 07 Jan '16

07 Jan '16

I've posted a draft of my essay "How to Be Wrong" and would appreciate comments: https://mathenchant.wordpress.com?p=482&shareadraft=568d547f0934f If you send comments via WordPress, keep in mind that they're anonymous by default, which may suit you or may not. I plan to publish, as always, on the 17th. Jim

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[math-fun] Asimov triangle problem
by Warren D Smith 07 Jan '16

07 Jan '16

Dan Asimov: For those of you who found this too easy, here is a problem I don't know the answer to that is inspired by the last one: Let N be a positive integer. Suppose we are given 3N points in 3-space such that no 4 of them are coplanar. Say the points are divided into 3 mutually exclusive groups of N points each: the reds, the greens, and the blues. True or false: There exists a partition of the 3N points into sets of size 3 that each contains one red, one green, and one blue point, such that there are no intersections among the resulting N triangles. (The triangle of a triple is the convex hull of its 3 points.) --WDS: The answer is true. More generally, given N*D points in D-dimensional space, N of each color (D colors), no D+1 lying on a hyperplane, there is a way to draw N disjoint all-color (D-1)-simplices using them as vertices. Regard each point as a normal distribution unit-mass "blob" with very small variance (limit as variance-->0). Use the "ham sandwich theorem" from topology to see each of the D color-measures can be simultaneously cut exactly in half by a hyperplane. https://en.wikipedia.org/wiki/Ham_sandwich_theorem Do so. If N is odd then hyperplane will necessarily pass thru exactly 1 point of each color and use that simplex and "discard" those D points. Subdivide into the points on each side of the cutplane and recursively solve the two subproblems. QED

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[math-fun] chameleon tongues
by Warren D Smith 05 Jan '16

05 Jan '16

http://www.newsweek.com/chameleon-tongues-are-amongst-worlds-fastest-most-p… says they accelerate their tongues at 264 gees, i.e. 2.6 km/sec^2, a record. Claims muscles are not believed capable of producing accelerations above about 18 gees. The chameleon does it with pre-stored elastic energy in the fashion of a catapult. I believe some jumping insects use the same kind of trick. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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