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[math-fun] violating transitivity
by Michael Speciner 04 Sep '07

04 Sep '07

My daughter asked me a question based on a homework assignment: Given n objects, and a choice of a comparison (<, =, >) between each of the nChoose2 pairs of them, what is the maximum number of the nChoose3 triples of the objects that can simultaneously violate transitivity (e.g. A > B >= C >= A)? --ms

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Re: [Fwd: Re: [math-fun] violating transitivity]
by Mike Speciner 04 Sep '07

04 Sep '07

Jason, Think of A > B (which is the same as B < A) to mean preferring A to B. And A = B to mean no preference between A and B. You get to pick the preferences between each pair of objects. Some sets of choices will be consistent with a transitive relation, some won't. I'm looking for a choice that results in the maximum amount of inconsistencies with transitivity. For example, with three objects, let's call them rock, paper, scissors, I might have rock < paper < scissors < rock. --ms >> From: Jason <jason(a)lunkwill.org> >> Date: 2007/09/04 Tue PM 02:18:18 CDT >> To: math-fun <math-fun(a)mailman.xmission.com> >> Subject: Re: [math-fun] violating transitivity > >> On Tue, 4 Sep 2007, Michael Speciner wrote: >> >>> My daughter asked me a question based on a homework assignment: >>> >>> Given n objects, and a choice of a comparison (<, =, >) between each of the >>> nChoose2 pairs of them, what is the maximum number of the nChoose3 triples >>> of the objects that can simultaneously violate transitivity (e.g. A > B >= C >>>> = A)? >> You must be using a specialized definition of <,=,> if A>B>=C>=A is true. Can >> you be more specific? >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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[math-fun] Toss Up Komi
by Thane Plambeck 04 Sep '07

04 Sep '07

Target is selling a game called Toss Up. It comes with 10 identical 6-sided dice and a rulebook. Each die has 3 green faces, 2 yellow faces, and 1 red face. http://www.areyougame.com/Interact/item.asp?itemno=PA7367 On a player's turn, he starts by rolling all 10 dice. Dice that come up as green are added to the player's score, one point per die, and are put aside. The player then has the option to roll again with the remaining dice. If at least one is green, all the just-rolled greens are then put aside, he gets that many more points, and he has the option to roll again, etc. If ever the player fails to roll a green and also has at least one red, he is bankrupted. His total accumulated score becomes zero and all the dice are passed to the next player. If alternatively he manages to score ten points, he gets to start over with all ten again. Finally, a player can also decide to stop at any time, passing the dice on to the next player at any point. Whoever is the first to score 100 points wins. In a two person game, it's presumably got to be advantageous to be the first to roll. But suppose the second player is given a "fair" initial positive score X (ie, a komi) to compensate for that. What is X? -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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Re: [math-fun] Toss Up Komi
by Eugene Salamin 04 Sep '07

04 Sep '07

As Neller and Presser say, "There is no known general method for solving equations of the form x = max(Ax+b, A'x+b')." Assuming A, A', b, b' are fixed given numbers, the graph of y = max(Ax+b, A'x+b') is continuous, piecewise linear, and consists of two segments. Its intersection with y = x consists of 0, 1, 2 or points or a continuum, and these are the solutions. Gene ____________________________________________________________________________________ Got a little couch potato? Check out fun summer activities for kids. http://search.yahoo.com/search?fr=oni_on_mail&p=summer+activities+for+kids&…

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[math-fun] Neal Koblitz article on Math vs Crypto
by Richard Schroeppel 03 Sep '07

03 Sep '07

Neal Koblitz's article in the Notices, The Uneasy Relationship Between Mathematics and Cryptography is available, apparently free, at http://www.ams.org/notices/200708/tx070800972p.pdf I disagree with Neal on a number of points, but the article does discuss several interesting issues. Rich

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Re: [math-fun] Constructing a constrained sudoku grid
by Bernie Cosell 02 Sep '07

02 Sep '07

one funster emailed me privately: > Well, I would start with a grid "clued" with your diagonal. > That should get you there pretty fast if one exists. And I can't imagine that I missed that approach! I tried to start with a grid and manipulate it to have the desired diagonal. By just starting with the desired diagonal, I could then add "possible" cells until the solver fills in the rest of the grid. Very neat. I easily banged out several more with little trouble. THANKS! Given this approach, it makes me think that every legal diagonal can be extended to a full grid [where I think the only 'legality' constraint is that none of the three 3x3's on the diagonal contain a repeated #]. Just have the solver show you all the remaining possibilities for each unfilled and un-deduced cell, and fill in the unfilled-ones at random... Thanks all for both the sample-grids for me to use and suggestions for how I could generate more! /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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Re: [math-fun] Constructing a constrained sudoku grid
by Bernie Cosell 02 Sep '07

02 Sep '07

On 2 Sep 2007 at 1:14, Michael Reid wrote: > i didn't do anything special; started with a solved sudoku grid, > then permuted rows and columns and permuted the labels 1-9 to get it. Boy, I don't know how you managed that. I tried that approach on a dozen or more solved-grids and couldn't get it to work out [hence my inquiry]. There are a couple million ways to relabel the grid and after that I'm not sure how many different ways there would be arrange the digits on the diagonal and you can permute the blocks-of-nine, and I guess I just never hit on a combination that would work. I wonder if the way the numbers work out if you were just lucky or if I was unlucky...:o) /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] Constructing a constrained sudoku grid
by Bernie Cosell 02 Sep '07

02 Sep '07

I want to construct an ordinary 9x9 sudoku grid [I'm not worrying [yet] about clueing it to be unique, solvable, proper difficulty, etc, I'm just working on "making the constrained Latin square" part now]. What I want is the main diagonal to be 1, 2, 3, 3, 4, 5, 6, 4, 5. I've been dinking around with generating sudoku squares and then relabeling and rotating rows and columns (within the six "3-wide" row and column blocks, of course) and trying to get that pattern on the diagonal. So far, I haven't been able to manage it, which has left me with three questions: 1) can *any*sudoku square be shifted/relabeled to get that pattern [that is, is it just lack of smarts/tenacity on my part that the few grids I've tried brute-forcing into that pattern don't work? [I'd guess not. I'm guessing that among the equivalence classes of sudoku grids some will admit my desired diagonal pattern and some won't. 2) Is it possible that there is *NO* sudoku square with the desired pattern on the diagonal [that is, is there some constraint I don't understand in setting up a Latin square?]. Again, I'm guessing [hoping actually..:o)] not: Just as I'm guessing that some equiv classes will for-sure not allow my diagonal pattern, I'm also guessing that for ANY diagonal pattern [and the one I want in particular], there'll always be at least ONE equivalence class that'll allow it. 3) and the middle ground: if it IS possible, but just not for any particular sudoku grid I'm left with a quandry: There are a zillion equivalence classes of 9x9 Latin squares (fewer, I guess of legal sudoku grids but probably still "zillions")... and so how might I go about *finding* one that has the desired main-diagonal constraint? /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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RE: [math-fun] Algebra I (RiP)
by Bill Gosper 29 Aug '07

29 Aug '07

I know some kids who go to fancy private schools--maybe they'll let me sneak a peek at their texts. Sorry about the five times less--I felt a twinge, but couldn't find an obvious misinterpretation. Dan> Which is a bit like using the phrase > > "most unsuperlative" > > ; or one could ask which imaginary animal is > > "most nonexistent" > > ; and it could be quite useful to ask what of all things is the > > "most vacuous" > > , given how complicated a vacuum seems to be in modern physics. There's a comedian (Steve Wright?) who perplexes burger-flippers by *insisting* on extra-medium. I concluded: >To sum up, p295, Caption: Wall Drug is a landmark in South Dakota. The store's fame began during the 1930s--the Great Depression--when it offered free ice water to travelers. Seriously, to sum up, at the end of a course taught from this book, I doubt anyone could answer the question "What is algebra?". And he would have approximately zero manipulative skill. This book could serve a course called Compulsory Algebra ("No lout nor ditz left behind"). It doesn't matter that the Snowflake photo on p15 *completely* misses the point of fractals--the sullen sufferers would forget the correct, self-similar picture just as quickly. But there must also be Nerd Algebra for those who damnwell *will* leave the others behind. And the only pictures in its book would be parts of problem descriptions. (E.g., click on Algebra Lesson on my bogus Google page: http://gosper.org/Googebra.htm .) PS, p 73: IMPROVING YOUR REASONING SKILLS A glyph is a symbol that presents information nonverbally. No it isn't. It's a carving, e.g. petroglyph or dermatoglyphics. They probably mean ideogram. Way to go, Accuracy Checker. --------------------------------- Fussy? Opinionated? Impossible to please? Perfect. Join Yahoo!'s user panel and lay it on us.

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Re: [math-fun] reminder - this is a family list
by James Propp 28 Aug '07

28 Aug '07

Thanks for posting the link to that article, Dan! If you step back a bit, it's kind of interesting that there's an article about this at all. None of this Prom Theorem stuff is really news to social scientists (aside from the way it's packaged); it's laypeople and science journalists who need to be reminded of things like this. I suspect a review article on gender differences and "danciness" in the social science literature would mention all the points that we've mentioned in this forum, and more. (I mean, most of us doing the posting are amateurs when it comes to social science --- unless we're convinced that our Ph.D.'s in math make us competent in ALL disciplines, in which case we're not social science amateurs --- we're social science cranks. :-) ) So why does the Prom Theorem show up as news? Maybe there's an Emperor's New Clothes appeal to the story ("Mathematician Refutes Social Scientists!") --- although (a) competent social scientists have always understood the Prom Principle (and the ways it both is and is not relevant to the sorts of questions social scientists ask about gender difference), and (b) David counts as a social scientist at this point given all the work he's done in economics, regardless of what it says on his Ph.D. diploma or what department his office is in. I guess it's good when newspapers offer remedial math in the guise of news. Maybe we could push this further, and have an article in which a physicist-statistician announces that causation is different from correlation. This is something that 99.9999% of educated people (myself included) should be reminded of on a frequent basis. Incidentally, when we talk about whether men are "dancier" than women, we usually don't mean that men actualy DO more "dancing" than women; what we usually mean is that if men had their way, there'd be a lot more men-dancing-women than there'd be if women had THEIR way. And there's evidence for this if we look at the amount of dancing done by same-sex dance-couples. So there's a lot to be said in favor of the popular belief that men have trouble keeping their feet in their shoes (as it were). Jim Propp P.S. I haven't read the New York Times article; only the one that Dan gave the link to. So it's possible that Kolata's Times article is different and that some of what I wrote above doesn't apply to it.

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