December 2017 - mathfun@mailman.xmission.com

[math-fun] Deep Mind crushes chess
by Hans Havermann 08 Dec '17

08 Dec '17

https://chess24.com/en/read/news/deepmind-s-alphazero-crushes-chess "Stockfish is the reigning TCEC computer chess champion, and while it failed to make the final this year it went unbeaten in 51 games. In a match with the chess-trained AlphaZero, though, it lost 28 games and won none, with the remaining 72 drawn. With White AlphaZero scored a phenomenal 25 wins and 25 draws, while with Black it 'merely' scored 3 wins and 47 draws. It turns out the starting move is really important after all!"

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[math-fun] How do you ask a question in mathematical notation?
by Scott Kim 06 Dec '17

06 Dec '17

I'm thinking about teaching the problem solving process in mathematics, and have run into a curious question: can one ask a mathematical question purely in mathematical notation? I believe the answer is no — mathematical questions always require human language in addition to mathematical notation. Problem statements are therefore always extramathematical in nature. In practice, school kids frequently see problems stated in forms like: 13+78 = ___, which means "what is the sum of thirteen and seventy eight?" But that involves a nonmathematical symbol (the blank), AND ends up misleading kids to think that the equals sign means "the answer is". Very sloppy. We can do better. This is an extension of something that has always bothered me: if mathematics is so rigorous, then how come it is conducted in a mish mosh of English and formal notation? The practical reason for mixing in human language is clear…like a computer program, a formal mathematical proof is more readable if it is annotated with comments. But too much reliance on human language means that formal proofs are not checkable by computer — a weird situation at best. I suppose that makes me a formalist, or a computer scientist…I've certainly got the latter bias because I don't trust anything I can't program. — Scott

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[math-fun] Introduction
by Andres Valloud 06 Dec '17

06 Dec '17

Hello, my name is Andres, I just joined the list. Generally my favorites tend toward the integers (hashing, factoring, quadratic residues, modular arithmetic, combinatorics). I also enjoy algorithm analysis. Over the past few years I've been studying the Collatz problem, and reproduced some known results on the subject. Professionally, I've been writing dynamic language virtual machines for about 11 years. Cheers, Andres.

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Re: [math-fun] Air traffic control
by Henry Baker 06 Dec '17

06 Dec '17

With modern "Graphics Processing Units" ("GPU's") having vast parallel floating point processing capabilities, it's easy to forget how hard it was to do real-time graphics programming -- even of the minimal line-drawing variety -- back in the early 1980's. I can well imagine that the simple "(x'-x)^2+(y'-y)^2+(z'-z)^2<1 mile" calculation would be too much for C(N,2) distances to compute for N airplanes. Many graphics operations first did *bounding box checks* prior to doing more expensive double precision multiplications. I can easily imagine doing such rectangular box checks in the early 1980's. Bounding using octagonal boxes (probably using 45 degree angles) instead of rectangular boxes (at least in the latitude/longitude dimensions, but probably not in the altitude dimension, where airplanes are *never* simultaneously allowed unless they're in a proper direction) could be done, and might still be slightly less expensive than a true ball radius calculation, but the advantage would seem to be minimal. At 06:44 PM 12/5/2017, James Propp wrote: >An air traffic controller in the 1980s told me that the antiquated software >that was then in use nationally approximated circles by polygons >(dodecagons I think, though maybe octagons). There was a rule saying that >two planes shouldn't be within a mile of each other, and the rule was >supposed to enforced by software (so that some sort of alarm would go off >if one plane was in the circular one-mile neighborhood of the other), but >the software used a polygonal approximation to those circles, more >precisely INSCRIBED circles, so it was possible for two planes to be >slightly less than a mile apart without setting off the alarm. He told me >that the controllers sometimes would arrange for this to happen for sport. > >Anyone know anything about this? > >My memory is unreliable, so I may have major details of the story wrong.

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[math-fun] Air traffic control
by James Propp 06 Dec '17

06 Dec '17

An air traffic controller in the 1980s told me that the antiquated software that was then in use nationally approximated circles by polygons (dodecagons I think, though maybe octagons). There was a rule saying that two planes shouldn't be within a mile of each other, and the rule was supposed to enforced by software (so that some sort of alarm would go off if one plane was in the circular one-mile neighborhood of the other), but the software used a polygonal approximation to those circles, more precisely INSCRIBED circles, so it was possible for two planes to be slightly less than a mile apart without setting off the alarm. He told me that the controllers sometimes would arrange for this to happen for sport. Anyone know anything about this? My memory is unreliable, so I may have major details of the story wrong. Jim Propp

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[math-fun] Rubik's cube: the base case
by Cris Moore 05 Dec '17

05 Dec '17

https://www.youtube.com/watch?v=LyjGhtba_zs <https://www.youtube.com/watch?v=LyjGhtba_zs> - Cris

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[math-fun] SET / Rubik mashup
by James Propp 05 Dec '17

05 Dec '17

Is this sort of hybrid between the games SET and the Rubik's cube a new thing? https://twitter.com/nomad_penguin/status/937810975025086464 Jim Propp

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Re: [math-fun] How do you ask a question in mathematical notation?
by Dan Asimov 03 Dec '17

03 Dec '17

When scribbling a math question to myself on a pad of paper, I'll often use notation like ∃? x in X ∋— [condition] [* In case it's illegible in your mail reader: i) the first symbol is a backwards capital E, a "there exists" symbol] ii) the symbol ∋— after the X looks like a pitchfork with the handle to the right. This means "such that". to mean "Does there exist a member of the set X such that [condition] holds?" where the such-that symbol ∋— is something I picked up somewhere from my professors, but haven't seen many other people using it. (To simulate it here I've used the "contains as a member" backwards epsilon special character followed by an em-dash (option-shift-plus on a Mac keyboard).) —Dan

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Re: [math-fun] Parenthesizing
by Dan Asimov 02 Dec '17

02 Dec '17

Suppose ^ means exponentiation and we want to count the *different integers* that 2^2^...^2 is equal to (N 2's in all) using various parenthesizations. E.g. #(2^2) = #{4} = 1 #(2^2^2) = #{16} = 1 #(2^2^2^2) = #{64, 256, 65536} = 3 #(2^2^2^2^2) = ??? Of course, any 3 twos grouped together make 4^2 = 2^4 = 16. Is there an asymptotic formula? Or better, an exact one? —Dan

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Re: [math-fun] Parenthesizing
by Dan Asimov 02 Dec '17

02 Dec '17

If the arrows are taken as implication, as in the predicate calculus, then Jim's question seems to make good sense to me. (But the question can of course be asked in general with other meanings assigned to the arrows.) —Dan ----- > Do all the Catalan-many parenthesizations of p_1 --> p_2 --> ... --> p_n > yield distinct Boolean functions? That doesn't quite parse. ... -----

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