May 2017 - mathfun@mailman.xmission.com

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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Re: [math-fun] Chess puzzle
by Guy Haworth 28 May '17

28 May '17

I don't see mate in 2 but 1. h7 stops the castling, and it's then mate in 3. Guy Haworth

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Re: [math-fun] Chess puzzle from Eric Angelini; Partial spoiler
by Andy Latto 28 May '17

28 May '17

I think that the intent of the puzzle is that if black's last move was g7-g5, white mates by ph5xg6 (en passant), threatening Rd8 mate, and if black castles, white mates with h6-h7, while if black's last move was with the king or rook, white plays Ke6, with an unstoppable threat of Rd8 mate, since black, having moved the king or rook, cannot castle. So white has a move that mates in two, but doesn't know what the move is without knowing the history of the game to this point. But I'm missing a step; If black's last move was g6-g5, I can't see a proof that black can't castle, so I can't find a mate. White's move before that might have been, for example, RxQd4, and black's move before that was Qd4. The white pawn on h6 could have gotten there from g5 long ago, capturing a knight, before black moved g7-g6. What am I missing? Andy Latto On Sat, May 27, 2017 at 2:03 PM, <rcs(a)xmission.com> wrote: > [from Eric Angelini -- Rich] > > ------------ > Date: Tue, 23 May 2017 11:36:01 +0200 > Subject: Chess weirdness > From: Eric Angelini <Eric.Angelini(a)kntv.be> > > Hello math-fun, > Here is the chess problem I was mentioning in my last post: > > +---+---+---+---+---+---+---+---+ > | | | | | k | | | r | 8 > +---+---+---+---+---+---+---+---+ > | | | | | | | | | 7 > +---+---+---+---+---+---+---+---+ > | | | | | | B | | P | 6 > +---+---+---+---+---+---+---+---+ > | | | | R | | K | p | P | 5 > +---+---+---+---+---+---+---+---+ > | | | | | | | | | 4 > +---+---+---+---+---+---+---+---+ > | | | | | | | | | 3 > +---+---+---+---+---+---+---+---+ > | | | | | | | | | 2 > +---+---+---+---+---+---+---+---+ > | | | | | | | | | 1 > +---+---+---+---+---+---+---+---+ > a b c d e f g h > > White to move; mate in 2 > > The "problem" is that there is -- and there is no > mate in 2! Kind of a 2-states quantum superposition! > If you had the White pieces, what would your first > move be? > > Best, > É. > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -- Andy.Latto(a)pobox.com

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[math-fun] Chess puzzle from Eric Angelini
by rcs＠xmission.com 28 May '17

28 May '17

[from Eric Angelini -- Rich] ------------ Date: Tue, 23 May 2017 11:36:01 +0200 Subject: Chess weirdness From: Eric Angelini <Eric.Angelini(a)kntv.be> Hello math-fun, Here is the chess problem I was mentioning in my last post: +---+---+---+---+---+---+---+---+ | | | | | k | | | r | 8 +---+---+---+---+---+---+---+---+ | | | | | | | | | 7 +---+---+---+---+---+---+---+---+ | | | | | | B | | P | 6 +---+---+---+---+---+---+---+---+ | | | | R | | K | p | P | 5 +---+---+---+---+---+---+---+---+ | | | | | | | | | 4 +---+---+---+---+---+---+---+---+ | | | | | | | | | 3 +---+---+---+---+---+---+---+---+ | | | | | | | | | 2 +---+---+---+---+---+---+---+---+ | | | | | | | | | 1 +---+---+---+---+---+---+---+---+ a b c d e f g h White to move; mate in 2 The "problem" is that there is -- and there is no mate in 2! Kind of a 2-states quantum superposition! If you had the White pieces, what would your first move be? Best, É.

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[math-fun] Microsoft Has a Plan to Add DNA Data Storage to Its Cloud
by Henry Baker 27 May '17

27 May '17

FYI -- Provides a whole new meaning for 'cloud seeding'... BTW, I seem to recall previously suggesting that HAKMEM be published in DNA, but got no interest. Should I try again? 'Think of those old floppy disks you can't read anymore or clay tablets with indecipherable hieroglyphs. Unlike such media, DNA probably won't ever go out of style.' https://www.technologyreview.com/s/607880/microsoft-has-a-plan-to-add-dna-d… 'The aim is a "proto-commercial system in three years storing some amount of data on DNA in one of our data centers, for at least a boutique application," says Doug Carmean, a partner architect at Microsoft Research. He describes the eventual device as the size of a large, 1970s-era Xerox copier.' 'Internally, Microsoft harbors the even more ambitious goal of replacing tape drives, a common format used for archiving information. "We hope to get it branded as 'Your Storage with DNA,'" says Carmean.' 'Formatted in DNA, every movie ever made would fit inside a volume smaller than a sugar cube.' 'Last July, Microsoft publicly announced it had stored 200 megabytes of data in DNA strands, including a music video, setting a record. The work, described in a paper published in March on the pre-print server Biorxiv, has been led by Carmean and Karin Strauss, both of Microsoft Research, and the University of Washington laboratory of computer scientist Luis Ceze.' http://biorxiv.org/content/early/2017/03/07/114553 'Based on the several weeks it took to carry out their experiment, Carmean estimates that the rate of moving data into DNA was only 400 bytes per second. Microsoft says that needs to increase to 100 megabytes per second.' 'Think of those old floppy disks you can't read anymore or clay tablets with indecipherable hieroglyphs. Unlike such media, DNA probably won't ever go out of style.' '"We'll always be reading DNA as long as we are human," says Carmean.'

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[math-fun] Turing machine code golf for producing Fibonacci numbers.
by Allan Wechsler 22 May '17

22 May '17

I have come up with a protocol for realizing certain integer sequences in the behavior of a two-symbol Turing machine. My machines always start with a tape populated with all zeros. First, I dispense with the halt state; then, I designate some state, symbol, and direction as the reporting criteria of the machine. Whenever the machine performs a transition, and finds itself in the "report" state and looking at the "report" symbol, it looks in the specified direction and counts the number of 1s in that direction, and reports that count as the next element of the sequence. In general, I want to come up with a machine that realizes a given integer sequence in the fewest number of states. I don't care about efficiency. I decided to test the concept by creating a machine that realizes the Fibonacci sequence 0, 1, 1, 2, 3, 5 ... I succeeded, but I was surprised that it took 10 states; my intuition was that it would be about 5. I'm not going to reveal my machine right away, but I want to hear if any funsters can realize the Fibonacci sequence with a Turing machine (using my protocol, please) with fewer than 10 states.

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Re: [math-fun] 45º pessimal?
by Henry Baker 21 May '17

21 May '17

Didn't some young kid recently find a closed-form solution for the cannonball-in-the-atmosphere problem? At 05:45 PM 5/17/2017, Bill Gosper wrote: >Everybody knows that, in vacuo, 45Âº elevation maximizes projectile range. Trick question: Given our atmosphere, is the range maximized by > or < 45Âº? SPOILER. Baseball: <. Cannonball: >. Well, long long distance artillery anyway. The idea is to fly above the denser, low level atmosphere. This raises the interesting question: Are there projectile densities and energies for which the range improves for elevations both > and < 45Âº? --rwg

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[math-fun] Finding other people
by Keith F. Lynch 20 May '17

20 May '17

The recent discussion of giving directions to people who don't know where they are reminded me of a problem that I think is original with me. Suppose a large number of people are placed randomly in an immense forest. There are no landmarks, the ground is level, and it's always heavily overcast so there's no way to tell which way is north. You can leave marks. People and marks can be seen iff you get within 10 meters of them. If everyone knows the above before they're transported to the forest, so they can discuss the problem with the others, what would be a good strategy? There isn't necessarily one best answer. But some answers are better than others.

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Re: [math-fun] Finding other people
by Keith F. Lynch 18 May '17

18 May '17

Dave Dyer <ddyer(a)real-me.net> wrote: > You and your partner are located at separate locations somewhere in > the milky way galaxy. You have a ship capable of acceleration to > light speed in negligible subjective time, and a beacon that can be > detected over any distance its light reaches. > Rendezvous for a victory drink before either of you dies of old age. I'm assuming you mean that both parties have ships that can accelerate and decelerate an unlimited number of times and can travel at any speed less than that of light. (If, as your statement seems to imply, your partner has neither a ship nor a beacon, your partner has a negligible chance of finding you or of being found by you. (This is math, so I won't say zero or infinitesimal. You could just stumble into each other by chance. But that's very very very very very unlikely.)) Set your beacon where you are, at rest relative to the center of the galaxy. Fly in tight circles around it, maybe one light-day out, at close to c. (If you can survive a near-instananeous acceleration to close to c, you can certainly survive flying in a tight circle.) Your partner travels to the center of the galaxy, or as close as is safe, then flies in a tight circle at close to c until they detect your beacon. Your partner travels to your beacon, then switches it off to signal their arrival, then switches it back on so you can find it again. As soon as you detect that it's been switched off and on, you fly directly to it. Total subjective time, well under a week. Was that your intended solution?

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[math-fun] Mathematica 11.1 solves 7-11 problem in one hour,
by Bill Gosper 18 May '17

18 May '17

finds 5 "antimony" solutions. In[405]:= Reduce[a > b > c > d && a b c d/10^6 == a + b + c + d == 711, {a, b, c, d}, Integers] // tim During evaluation of In[405]:= 3369.569497,6 Out[405]= (a == 316 && b == 150 && c == 125 && d == 120) || (a == 600 && b == 250 && c == -60 && d == -79) || (a == 900 && b == 316 && c == -5 && d == -500) || (a == 1185 && b == 50 && c == -24 && d == -500) || (a == 1250 && b == 15 && c == -80 && d == -474) || (a == 1500 && b == 25 && c == -24 && d == -790) In[406]:= ToRules@% Out[407]= Sequence[ {a -> 316, b -> 150, c -> 125, d -> 120}, {a -> 600, b -> 250, c -> -60, d -> -79}, {a -> 900, b -> 316, c -> -5, d -> -500}, {a -> 1185, b -> 50, c -> -24, d -> -500}, {a -> 1250, b -> 15, c -> -80, d -> -474}, {a -> 1500, b -> 25, c -> -24, d -> -790}] In[410]:= {a b c d, a + b + c + d} /. List@%407 Out[410]= {{711000000, 711}, {711000000, 711}, {711000000, 711}, {711000000, 711}, {711000000, 711}, {711000000, 711}} (Dividing these "pennies solutions" by 100. will give $7.11.) Does anyone know the origin of this problem? --rwg I once found a (messy) addition formula yielding an infinitude of rational solutions, including some with nothing smaller than half cents.

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