August 2007 - mathfun@mailman.xmission.com

Re: [math-fun] How we "do mathematics"
by Eugene Salamin 02 Feb '08

02 Feb '08

----- Original Message ---- From: David Gale <gale(a)math.berkeley.edu> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Thursday, August 9, 2007 3:17:48 AM Subject: RE: [math-fun] How we "do mathematics" ... >A (very) old one I like : > >Two Death Valley marathonians friends complete their run >together and seek shadow in their tent nearby the finish >line. > >The first one had prepared a fresh 5-liter tank of water >in a cooling box and his friend a similar 3-liter one. > >They were about to start drinking when a journalist enters >the tent desperately asking for water. > >OK, the water is divided into three equal parts -- and drunk. > >The thankful journalist insists to pay his share and leaves >8 dollars to be split between the two friends. > >How? > >---- Why is this problem so fascinating? The two friends are paid $8. The guy who contributed 5 l gets $5, and the guy who contributed 3 l gets $3. Or maybe the $8 goes towards paying for their campsite. I must be missing something. Gene ____________________________________________________________________________________ Pinpoint customers who are looking for what you sell. http://searchmarketing.yahoo.com/

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[math-fun] Atle Selberg
by Schroeppel, Richard 25 Sep '07

25 Sep '07

I noticed a long obit in the Los Angeles Times last week for Atle Selberg, who provided the first(*) elementary proof of the Prime Number Theorem. http://www.latimes.com/news/printedition/california/la-me-selberg22aug22 ,1,4873005.story?coll=la-headlines-pe-california free, short: http://www.norwaypost.no/cgi-bin/norwaypost/imaker?id=96351 Rich ----- * Modulo a priority dispute with Erdos.

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[math-fun] Algebra I (RiP)
by Bill Gosper 07 Sep '07

07 Sep '07

I said >PPS, my former student aced his tests but skipped his homework, and is now repeating Algebra I. As soon as I can photocopy a few pages, I'll share some of the stupendous howlers I found on a quick perusal of his new text. Amazingly, it's *not* Prentice-Hall ... Key Curriculum Press, Discovering Algebra, An Ivestigative Approach, by Murdock, Kamischke, & Kamischke. Along with a committee filling two columns on page ii, including five Multicultural and Equity Reviewers, two Social Sciences and Humanities reviewers, and, get this, three Scientific Content Reviewers, and one Accuracy Checkers (sic!). Seven hundred pages of irrelevant color pictures (e.g., Elvis impersonators) with inane captions (e.g., "Dead skin cells are one of the components of dust."). The people I know who are good at algebra had books with *zero* pictures, and few, if any, black and white graphs. Those books weighed maybe five times less, and contained way more than twice the algebra. (I learned it from my grandmother's.) If I were an aspiring algebra student, I would be deeply offended by this book for diluting the subject beyond recognition, as if algebra were so difficult and distasteful as to require huge doses of eye and brain candy, and for forcing me to lug to school hundreds of stupid pictures far less interesting than an equally heavy stack of magazines of my own choosing. And I came here to learn algebra, not to be cured of racism and sexism. And as a teacher faced with this book, I would invite any "student" who wished he was elsewhere to place his favorite magazine inside his open book. The pictures would be at least as relevant. I'll bet that even those who get an A in the course will be completely unable to apply algebra to any problem that appears to differ significantly from those "investigated" in the book. Howlers: In a section devoted to wind chill, p227, photo: snow blowing across highway across a barren plain. Caption: High wind speeds in Saskatchewan, Canada, drop temperatures below freezing. Just to make sure you understand, p228, photo: orange tents and heavily clad people on an expanse of snow. Caption: The wind chill factor drops temperatures well below zero in Antarctica. Wow, I had no idea how many of those poor Katrina victims must have frozen to death. According to this table, a temperature of 5F with a 20mph wind gives a chill of -31. So the temperature of an object at 0F should decrease rapidly. Think of the starvation we could avert by giving each victim a cheap refrigerator--a fan in a box! And those stupid citrus growers actually think they can avert frost damage with giant blowers! And imagine the wind chill at 40 below and 1000mph. It must be dang near minus infinity. And yet those idiot Europeans built the Concorde out of titanium because they thought aluminum would melt! Geez I'm glad I stayed in school. Oops, here in the main text it says wind chill is only how cold it *feels*. How do you measure how cold it feels? Does snow in Saskatchewan stay frozen because the wind makes it feel cold? In a section on radioactive decay, photo: human skull. Caption: Archaeologists can approximate the age of artifacts with carbon dating. This process uses the rate of radioactive decay of cabon-14. Carbon is found in all living things, so the amount left in a bone, for example, is an indicator of the bone's age. This is a plastic casting of a skull found in 1997 in Richland Washington. Carbon dating has dated this skull as 9200 years old. It's less frightening now that I know it's plastic. I wonder if the Scientific Content Reviewers and the Accuracy Checker used their real names. This skull, by the way, is of the infamous Kennewick Man, http://www.harbornet.com/folks/theedrich/hive/Kennewic.htm . Apparently, someone slipped it past the Multicultural and Equity Reviewers. In the fractals section, p19(!): As the Koch curve develops [For some reason, the book studiously avoids the name Snowflake], segment length decreases as the number of segments increases. As you draw higher stages, the length of individual segments approaches, or gets closer and closer to, what number? What number does the number of line segments approach? ZAP! Without even knowing it, the student has been infected with the poisonous notion that infinity is a number. And the poisonous notion that this line segment recursion actually defines which points are on the fractal. There is, of course, no mention of dimension. As a student, I would wonder how Mandelbrot and Sierpinski got so much credit for "discovering" designs that people have doubtless doodled since the invention of writing. The authors could have promised a discussion of dimension in a later chapter, although it might have required a few thousand more intervening pages of photographs. E.g., antique photo of seated woman. Caption: Maria Mitchell (1818-1889) was the first professional woman astronomer in the United States. Relevance: nil. But the Multicultural and Equity Reviewers must have detected a patch of inequity. I would think that even female students would begin to doubt the competence of women, if they require such heavy-handed propaganda. Side by side on p15, Caption 1: Because a coastline, like a fractal curve, is winding and irregular, it is not possible to measure its length accurately. The Koch curve can be used as a model of a coastline. [...] Photo: Aerial shot of a pair of beautifully smooth, crescent beaches! Then, to perfect the confusion, Caption 2: At later stages, the Koch curve looks smoother and smoother.[!] But if you magnify a section at a later stage, it is just as jagged as at Stage 1. So much for self-similarity. Photo: a minority adolescent holding a magnifying glass over a Snowflake approximation to reveal a part of the initial hexagram (Heaven forbid that I say Star of David), but with HUGE, FAT line segments! Line segments are the bane to understanding fractals. A properly drawn line segment is invisible! (Except as a color boundary.) p72, problem 9: The graph at right is a hexagon [True] whose vertices are seven[!] ordered pairs. Two of the points are (3,0) and (1.5,2.6). The hexagon is centered at the origin. a. What are the coordinates of the other points? HUH?? Assuming regular, even though it isn't? There are three kinds of mathematicians: ... 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. --rwg (Confidential to you-know-who-you-are: There is only one b in Fibonacci.) --------------------------------- Take the Internet to Go: Yahoo!Go puts the Internet in your pocket: mail, news, photos & more.

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[math-fun] Constructivist logic
by Jason Holt 07 Sep '07

07 Sep '07

I recently went through my approximately 3-year cycle of wondering what the significance of the Halting problem is all about and wrote up my thoughts here: http://credentiality2.blogspot.com/index.html (I include the link for completeness, but it's not the point of this message, and I can't claim that it's in any way worth the time it takes to read it.) My old CS theory prof tells me I take offense to the notion of uncomputability because I prefer the constructivist school of logic: http://en.wikipedia.org/wiki/Constructivism_(mathematics) I poked around wikipedia a bit and have to agree that it does appeal to me. In particular, I like the idea that the reals are computable and perhaps thus not 200% more infinite than the naturals as I had been lead to believe, but rather 100% less different in cardinality, there being no more fewer reals than naturals than there are naturals and integers. Excuse me, I think I need to sit down for a moment. Anyway, it seems that Constructionism has not yet converted all the heathen masses, and I'm curious what this mass of heathens thinks of it. In particular, I'm curious what it means for diagonalization proofs, which have always seemed suspicious to me, doing magical things that only work out at infinity.

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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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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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[math-fun] RE: Base-10 Skolem-Langford numbers
by Eric Angelini 27 Aug '07

27 Aug '07

... full lists available here (french foreword): http://www.cetteadressecomportecinquantesignes.com/SkoLang.htm (where there is a link to the "Strong-SLs" (= "SL-forts") Best, É. -------------------------------------------------------------- ... are now here, thanks to David Wilson: http://www.research.att.com/~njas/sequences/A108116 ... and "Base-10 strong Skolem-Langford numbers" are there: http://www.research.att.com/~njas/sequences/A132291 Grazie David! Best, É.

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[math-fun] Base-10 Skolem-Langford numbers
by Eric Angelini 27 Aug '07

27 Aug '07

... are now here, thanks to David Wilson: http://www.research.att.com/~njas/sequences/A108116 ... and "Base-10 strong Skolem-Langford numbers" are there: http://www.research.att.com/~njas/sequences/A132291 Grazie David! Best, É.

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