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[math-fun] Edwards book on Integration
by Guy Haworth 02 Jun '17

02 Jun '17

In the UK, 'Professor' Jimmy Edwards was a prominent comedian and actor through to 1970. His father was a Professor of Mathematics (hence the 'Professor' above) and wrote a book on Integration solutions - my school had a copy (with hard, orange covers). I can't find any trace of this book on the web but some may know of it. Clues welcome. Guy

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[math-fun] Calculus (?) problem ?
by Henry Baker 01 Jun '17

01 Jun '17

[A modern Zeno problem.] Achilles runs a total distance D=2^k meters in a total time T seconds. For example, Achilles might run 2048 meters in 491.52 seconds (= 8:11.52). Achilles wasn't terribly fast by today's world record standards. :-) Achilles's average *pace* (=1/speed) for the first half (=D/2) is 5 seconds/500m (= 0.01 seconds/meter) *faster* than his average pace for the second half. Achilles's average pace for the first quarter (=D/4) is 0.01 seconds/meter faster than his average pace for the second quarter. and so forth. Achilles's motion is a smooth analytic function of time and/or distance, excepting perhaps the origin. What is the functional form of his motion -- e.g., distance as a function of time, time as a function of distance, speed as a function of time, speed as a function of distance, pace as a function of time, pace as a function of distance -- any one of these forms should be useful. If I haven't made a mistake, Achilles's absolute *average pace* over the entire run shouldn't matter; clearly if the average pace for the first half differs by 5 seconds/500m from the average pace for the second half, then we could simply set the average pace for the entire run to zero (i.e., Newton relativity), so he may end up going backwards part of the time (i.e., relative to a tortoise moving at constant pace = Achilles's average pace). But this still leaves the problem of the functional form of his speed.

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Re: [math-fun] The bathroom square dance
by Fred Lunnon 01 Jun '17

01 Jun '17

At second attempt, I did detect a slight twisting effect. I imagine this is the same mechanism that causes the room apparently to rotate in the reverse direction after one suddenly ceases spinning about a vertical axis (how do ballerinas cope, I wonder?); or the floor to roll and pitch after one disembarks from a small boat. I doubt whether retinal cell configuration has any part to play in such effects, which are manifestations of a general mental process, occurring at many levels, which seeks to filter out constant background from our perception, leaving us conscious only of any unpredictable variation. I once encountered a spectacular example of this phenomenon, involving hearing rather than sight. Standing beside the Victorian wall clock in the hall of my home late one quiet night, I failed to hear its customary loud tick, and concluded that it must have broken down. Visual inspection however revealed that the pendulum continued to swing; and as I stared at it in some perplexity, over several seconds its tick gradually faded in to become once more clearly perceptible. Similarly as Sarah Silverman recently pointed out, it is impossible to register one's own body odour, whatever olfactory violence one may inadvertantly be perpetrating upon one's neighbours. The distinctive feature of sight in this context is that relative motion possesses sign, making its (sudden) absence more directly contrasted with presence. Fred Lunnon On 6/1/17, Bill Gosper <billgosper(a)gmail.com> wrote: > What's supposed to happen after 20 sec? > > You're supposed to quickly redirect your stare to top left of your screen. > > > (That's my sixties cred shot to pieces.) WFL > > > Or just your retinal cone cells. If you do not see disturbing, startling > twisting, > > consider an ophthalmological consult. --Bill > > > > On 5/31/17, Bill Gosper <billgosper(a)gmail.com> wrote: > > On 2017-05-29 10:55, Stuart Anderson wrote: > > To make the tiles really dance ... > > Danny Caligari > > ; > > > https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-… > > > Bad URL there. Use > > http://www.math.brown.edu/~rkenyon/gallery/squarespiral2.gif > > > While you're there, stare at > > http://www.math.brown.edu/~rkenyon/gallery/squarespiral.gif > > for ~20 secs, then the top corner of your screen. --rwg >

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[math-fun] The bathroom square dance?
by Bill Gosper 31 May '17

31 May '17

On 2017-05-29 05:48, James Propp wrote: > Can a square-tiling of the plane, like a square-tiling of a square, be > converted into an electrical network (a la Brooks, Smith, Stone, and > Tutte)? That might give a picture that would be suggestive of a different > name than "bathroom floor tiling". > > Incidentally, I've always thought that the tiling of the plane by squares > and octagons was the "bathroom floor tiling": The tiles of the MIT AI Lab bathroom were just squares, randomly dark and light. I remember because, as I looked at them, they'd spontaneously run a couple of Life generations, so long had I spent watching the CRT. Strangely, the dark ones were 1 and the light ones 0. Maybe the dark ones were sparser? --rwg > > https://www.houzz.com/photos/48092404/Victorian-Octagon-With-White-Dot-Porc… > > Jim Propp > > On Sun, May 28, 2017 at 7:24 PM, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > >> As on page 494 of the handy 900-page >> http://paulbourke.net/dataformats/postscript/psref.pdf >> for instance. >> >> I cannot honestly pretend that "PostScript type 10 halftone dictionary" >> constitutes a great improvement on "bathroom floor tiling". >> >> Where are the ancient Greeks, the Moors (Moroccan, not Yorkshire), >> the Alhambra when we need them? >> >> WFL >> >> >> On 5/28/17, Mike Speciner <ms(a)alum.mit.edu> wrote: >> > This reminds me of PostScript's type 10 halftone dictionaries. In >> > ancient history, when I was implementing halftones in firmware, I >> > realized that any rational-tangent halftone could be represented by a >> > single orthogonal rectangle of values that repeated horizontally but was >> > offset vertically. Later, PostScript 3 appeared with type 10 halftone >> > dictionaries, defined by two orthogonal squares tiling the plane. It >> > took me a little while to realize that these approaches are equivalent. >> > (Halftones are normally defined by a single square of values with some >> > arbitrary angle to the coordinate axes. When the angle has a rational >> > tangent, we can get either of the above techniques.) >> > >> > I guess I didn't answer your question. >> > >> > --ms >> > >> > On 28-May-17 11:13, Fred Lunnon wrote: >> >> A well-known tiling of the plane employs offset square tiles of two >> >> sizes. >> >> See the example below using tiles of 2 and 3 units, somewhat perturbed >> >> on this occasion by proportional spacing rather than incompetent home >> >> improvement skills. >> >> >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> [ * * * * * * * * * >> >> ] >> >> >> >> I can't resist remarking that such diagrams provide a particularly >> >> elegant >> >> demonstration of Pythagoras' theorem, apparent when the period along >> >> a horizontal axis is calculated. >> >> >> >> A particularly humiliating programming exercise involves generating a >> >> matrix in which unoccupied cells above are represented by 0 and >> >> asterisked cells by +1 or -1 , chosen to satisfy the local >> constraints >> >> on >> >> an integer number wall (frame theorems). >> >> >> >> But all that's completely irrelevant to my purpose, which is simply >> to >> >> enquire if these tilings have an established name --- preferably one >> >> rather >> >> more euphonious than my current nomenclature: The bathroom floor tiling! >> >> >> >> Fred Lunnon >> >>

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Re: [math-fun] Random binary experiment
by Henry Baker 31 May '17

31 May '17

I'd make one minor modification to Rich's procedure: result is the 13th bit of a SHA-256 hash of everyone's "votes", which "vote" could be any number they want to contribute, but it must be fixed & written down (in some fixed sequence). The problem with "odd" instead of "hash" is that every vote *counts*, but in a way such that an additional vote is trivially guaranteed to change the parity and hence the result; it is too easy to hide a person or vote twice. With a hash -- even a hash of subsequences of the list -- there is no simple relationship between the intermediate result and the final result. Yes, checking the SHA-256 hash of a long list can't be done in anyone's head, but everyone's cellphone can do SHA-256. At 06:26 PM 5/31/2017, rcs(a)xmission.com wrote: >Announce in advance "an odd number of Heads is outcome A". >Everyone in the crowd flips his own coin in private, and reports the >answer in a sealed envelope, the "ballot". >The ballots are opened in full view, and Heads are counted. >No particular problems with a stuffed ballot box, or multiple voting, >or individuals choosing to used biased coins or just plain cheat. >As long as observers are sure that every ballot is counted, anyone >who cares about the outcome has contributed his own version of a >random bit. > >Rich > >------------ >Quoting Henry Baker <hbaker1(a)pipeline.com>: >>In general, I don't think that any amount of *passive* observation >>will be convincing. >> >>Science is only believable when *independent* experiments are >>supplied with *independent* random streams of bits; >>i.e., if I am allowed to perform the experiment myself using my own >>source of random bits. ("Supply a wiggle in, and see what wiggles >>out") >> >>Of course, this raises the following possibility (sci-fi writers, >>listen up!): >> >>In the future, the NSA will be tasked as a monopoly providing all >>necessary random bits. It does this by producing quantum entangled >>pairs of particles, supplying the market with one of each pair, and >>keeping the other particle in labeled storage. >> >>If/when the NSA customer "measures" one of the "market" particles, >>the entangled particle will also assume the same state, so the >>customer will have his random bit, and the NSA will also be happy. >> >>At 02:31 PM 5/31/2017, Dan Asimov wrote: >>>You have to perform a random binary experiment in front of a crowd >>>of people Â so no fooling is allowed Â in in such a way that >>>everyone is convinced that the experiment was fair. >>> >>>The people include some technical experts but many who are not. >>> >>>What is the simplest / easiest / cheapest way to ensure that the >>>crowd will be convinced that the experiment was fair (the two >>>outcomes had an equal chance of occurring) ??? >>> >>>ÂDan >>> >>>P.S. I do not have ann answer to this, but maybe there is a "best" answer.

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[math-fun] The bathroom square dance
by Bill Gosper 31 May '17

31 May '17

On 2017-05-29 10:55, Stuart Anderson wrote: > To make the tiles really dance ... > Danny Caligari > ; > https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-… > Bad URL there. Use http://www.math.brown.edu/~rkenyon/gallery/squarespiral2.gif While you're there, stare at http://www.math.brown.edu/~rkenyon/gallery/squarespiral.gif for ~20 secs, then the top corner of your screen. --rwg

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[math-fun] Mechanical computer for kids: Turing Tumble
by Henry Baker 30 May '17

30 May '17

FYI -- https://www.kickstarter.com/projects/871405126/turing-tumble-gaming-on-a-me… Turing Tumble: Gaming on a Mechanical Computer Build mechanical computers powered by marbles to solve logic puzzles. Escape planet Eniac and discover how computers work. Turing Tumble is a new type of game where players (ages 8+) build mechanical computers powered by marbles to solve logic puzzles. It's fun, addicting, easy-to-learn, and while you're playing, you discover how computers work. I'm all about teaching kids to code. When I was a professor at the University of Minnesota, I saw how valuable it is for all students to be coders. I have three young kids and I've tried all sorts of games to build their interest in coding. The problem is that they all treat computers like abstract, black boxes. They overlook the fundamental, most amazing concept: how simple switches, connected together in clever ways, can do incredibly smart things. Kids learn best when they use their senses to explore concepts. Turing Tumble is the only game that lets kids see and feel how computers work. The logic isnÂt hidden inside a computer chip Â it's all right there in front of them. It builds logic and critical thinking skills, fundamental coding concepts, and grounds their understanding of computers. And adults, this game is not just for kids! It's a game unlike any you've ever played. It takes creative thinking to come up with a solution to each puzzle - you can't just stumble across the solution by trial and error - and it's extremely gratifying to build it and watch it work. Even if you're an expert programmer, I guarantee you'll find the latter puzzles a challenge. ... Also, this page has some animated gifs showing some of the mechanical logic at work: https://boingboing.net/2017/05/30/cool-marble-powered-mechanical.html

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[math-fun] The bathroom square dance
by Stuart Anderson 29 May '17

29 May '17

To make the tiles really dance ... Danny Caligari ; https://www.google.com.au/amp/s/lamington.wordpress.com/2013/01/13/kenyons-…

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[math-fun] The bathroom square dance?
by Fred Lunnon 29 May '17

29 May '17

A well-known tiling of the plane employs offset square tiles of two sizes. See the example below using tiles of 2 and 3 units, somewhat perturbed on this occasion by proportional spacing rather than incompetent home improvement skills. [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] [ * * * * * * * * * ] I can't resist remarking that such diagrams provide a particularly elegant demonstration of Pythagoras' theorem, apparent when the period along a horizontal axis is calculated. A particularly humiliating programming exercise involves generating a matrix in which unoccupied cells above are represented by 0 and asterisked cells by +1 or -1 , chosen to satisfy the local constraints on an integer number wall (frame theorems). But all that's completely irrelevant to my purpose, which is simply to enquire if these tilings have an established name --- preferably one rather more euphonious than my current nomenclature: The bathroom floor tiling! Fred Lunnon

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Re: [math-fun] Chess puzzle
by Guy Haworth 29 May '17

29 May '17

Andy Latto asks the key questions. Is there an en passant capture? Is castling allowed? The rules of chess composition at http://www.wfcc.ch/1999-2012/codex/ esp. article 16, solve this: 1) Castling is permitted unless it can be proved that it is not 2) En passant capture is not permitted unless it can be proved that it is available. Because the last move could have been Kd7-e8, we cannot prove that that the last move was g7-g5 (and, agreed, it was certainly not g6-g5). Therefore we must assume that an en-passant capture is not permitted, and that the last move was by K or R. Therefore castling is not permitted, and so Kf5-e6 is followed by Rd8#. ... but in fact, the assumption is not necessary. We do not even have to rely on the Codex of Composition to resolve the ambiguities of the position inherent in the diagram: If g7-g5 had been played, as stated, h5-g6 followed by Rd8# suffices. G

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