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[math-fun] another pizza problem, perhaps discussed here already
by Thane Plambeck 15 Dec '17

15 Dec '17

divide a pizza into n congruent pieces at least one of which does not touch the center -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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Re: [math-fun] Cutting a pie ...
by Don Reble 15 Dec '17

15 Dec '17

> Suppose we want to make cuts in a circular pie (the usual kind of cut, > from the center to a point on the perimeter) in such a way that, for > every k between 1 and n, the pieces can be divided equally among k > people, each of whom gets the same amount of pie. > Searching the OEIS for "cut pizza" ... "cut pie" [didn't help] Alas, it has nothing to do with circularity of the item, so "pie", "pizza" mightn't work. But if it's a cut _cake_: A265286. IIRC, this was discussed on Seqfan. -- Don Reble djr(a)nk.ca

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Re: [math-fun] rabbits, syllables, and Fibonacci numbers
by Dan Asimov 15 Dec '17

15 Dec '17

I was invovled for several years with the Berkeley Math Circle, which was great fun. The founders of that group, Zvezda Stankova and Tom Rike, wrote a couple of books about how to go about organizing a math circle, containing ABOVE ALL lots and lots of interesting material to discuss. I have the first volume: https://www.amazon.com/Decade-Berkeley-Math-Circle-Mathematical/dp/08218468…. If you go to Amazon and search on math circles many books come up, and going by customer reviews there are a lot of good ones among 'em. —Dan -----Original Message----- >From: Cris Moore <moore(a)santafe.edu> >Sent: Dec 15, 2017 12:50 PM >To: math-fun <math-fun(a)mailman.xmission.com> >Subject: [math-fun] rabbits, syllables, and Fibonacci numbers > > >This week I did an activity at my daughters school (7th grade) talking about Fibonacci numbers, including > >. rabbit genealogies >. sequences of short and long syllables (1 and 2 units long) in Sanskrit poetry >. domino tilings of 2xn rectangles > >I challenged them to find a bijection between the rabbits after n generations and patterns n syllables long. > >We then looked at ratios between successive Fibonacci numbers, and noted that they seem to be settling down. I showed them phi=(sqrt(5)+1)/2, derived the quadratic equation that it should satisfy, and showed by computing phi+1 and phi^2 that it does. > >I then used phi = 1+1/phi iteratively to show them the continued fraction expansion, and showed (by example) that cutting it off at various places gives the ratios between adjacent Fibonacci numbers. > >We ended with slanted diagonals in Pascal’s triangle, sunflowers, artichokes, Lucas numbers (and the fact that their asymptotic growth rate is the same), “tribonacci” numbers, and so on. > >I know there are a zillion things out there on the Web of this kind — Vi Hart’s video, Math Circles http://www.mathcircles.org/wp-content/uploads/2017/10/FibAndOtherSpecNumber… >and so on. My question: is there a good, wiki-editable, collection of math activities like this, suitable for middle school and high school, with nice visuals and suggestions for teachers about what questions they should challenge students with? I want an emphasis on “why is this pattern here? is it just a coincidence, or is there a good explanation?" > >(I apologize in advance, since I know I’ve asked this list similar questions before…) > >thanks >Cris > > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] rabbits, syllables, and Fibonacci numbers
by Cris Moore 15 Dec '17

15 Dec '17

This week I did an activity at my daughters school (7th grade) talking about Fibonacci numbers, including . rabbit genealogies . sequences of short and long syllables (1 and 2 units long) in Sanskrit poetry . domino tilings of 2xn rectangles I challenged them to find a bijection between the rabbits after n generations and patterns n syllables long. We then looked at ratios between successive Fibonacci numbers, and noted that they seem to be settling down. I showed them phi=(sqrt(5)+1)/2, derived the quadratic equation that it should satisfy, and showed by computing phi+1 and phi^2 that it does. I then used phi = 1+1/phi iteratively to show them the continued fraction expansion, and showed (by example) that cutting it off at various places gives the ratios between adjacent Fibonacci numbers. We ended with slanted diagonals in Pascal’s triangle, sunflowers, artichokes, Lucas numbers (and the fact that their asymptotic growth rate is the same), “tribonacci” numbers, and so on. I know there are a zillion things out there on the Web of this kind — Vi Hart’s video, Math Circles http://www.mathcircles.org/wp-content/uploads/2017/10/FibAndOtherSpecNumber… and so on. My question: is there a good, wiki-editable, collection of math activities like this, suitable for middle school and high school, with nice visuals and suggestions for teachers about what questions they should challenge students with? I want an emphasis on “why is this pattern here? is it just a coincidence, or is there a good explanation?" (I apologize in advance, since I know I’ve asked this list similar questions before…) thanks Cris

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Re: [math-fun] Cutting a pie ...
by Henry Baker 15 Dec '17

15 Dec '17

What if one of those receiving a slice is the Lion? What most people don't realize is that "The Lion's Share" isn't the majority, or even "the vast majority", but *100%* !! http://www.aesopfables.com/cgi/aesop1.cgi?sel&TheLionsShare So, *zero* cuts suffice (although in the fable, the Lion has the dead stag quartered, but then proceeds to take all 4 quarters). Ãsop. (Sixth century B.C.) Fables. The Harvard Classics. *1909Â14*. [Just prior to the 16th "Income Tax" Amendment.] At 08:46 AM 12/15/2017, Michael Kleber wrote: >Guy: What if after pie-spinning, Alice thinks she got the worst piece and >would prefer either Bob's or Carol's, while Bob and Carol would each prefer >Alice's? Then everyone is unhappy, but Alice gets to trade with someone, >and Bob and Carol fight with each other over which one she trades with. >Seems unsatisfying. > >--Michael > >On Fri, Dec 15, 2017 at 10:23 AM, Guy Haworth <g.haworth(a)reading.ac.uk> wrote: > >> The problem is usually expressed as one of dividing a cake (or pie) and >> distributing it so that no-one is in a position to complain about the size >> of their slice ... >> >> There's this ... https://www.scientificamerican.com/ >> article/the-mathematics-of-cake-cutting/ >> >> ... and then there's this by Ian Stewart https://www.amazon.co.uk/How- >> Cut-Cake-Mathematical-Conundrums/dp/0199205906 >> >> However, the best solution I know of (which Ian Stewart confirmed he had >> not heard of) came from the 13-year-old son of a colleague of mine .... >> >> Everyone gets a knife and makes a cut in a circular cake which is standing >> on a turntable ... with the aim of making equal-size slices. >> >> The cake is then spun, and everyone gets the slice opposite them. >> >> [ In other words, the randomness takes the decision about who gets what >> away from the hungry mathematicians ] >> >> Guy

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Re: [math-fun] Draft of my December 2017 blog post
by Henry Baker 15 Dec '17

15 Dec '17

You might want to defer mention of elliptic curve methods to a future posting. I can't wait to see how you're going to make ECC understandable to high schoolers. :-) BTW, modular arithmetic is taught in U.S. grade schools/junior high schools as "clock arithmetic". Check out the Khan Academy videos for more info. Are you going to mention Chinese Remainders: those Chinese children "left behind", while none of U.S. children have been "left behind". ;-) https://www.khanacademy.org/ Re bookkeepers: Talk about the Department of Redundancy Department: 3 doubled letters in a row! You might mention that IBM worked on schemes to detect arithmetic errors in their computing circuits using similar ideas to casting out nines. I think that there may even be some IBM patents from the 1960's along these lines. I do believe that crypto blockchains are the biggest advance in bookkeeping since the Venetians invented double-entry bookkeeping. Perhaps someone needs to engineer a Babbage-style mechanical mechanism to compute blockchain hashes. ;-) At 09:49 AM 12/15/2017, James Propp wrote: >Good point --- though if the elliptic curve methods are still secure, and >are based on finite fields (which are field-extensions of fields of the >form Z/pZ), then maybe my remark is in a sense true. What do you think? > >(And while we're probing beneath my glib prose, is it really accurate to >say that computers have drastically reduced the ranks of salaried >bookkeepers? It's my sense that this is true, but is it?) > >Jim Propp > >On Fri, Dec 15, 2017 at 12:35 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> "And now that we have those computers, mod p arithmetic (where p is a huge >> prime) is the cornerstone of cryptographic security for internet commerce." >> >> Not so much. There have been so many problems with RSA that the most >> recent recommendations are to disable RSA completely, in favor of elliptic >> curves. >> >> "A number of the most popular websites and services online, including >> Facebook and *PayPal*, are vulnerable to an exploit which has resurfaced >> from 1998." >> >> "We believe RSA encryption modes are so risky that the only safe course of >> action is to disable them." >> >> http://www.zdnet.com/article/robot-exploit-from-1998- >> resurrected-leaves-top-sites-crypto-vulnerable/ >> >> https://eprint.iacr.org/2017/1189.pdf >> >> At 08:42 AM 12/15/2017, James Propp wrote: >> >Hi, >> > >> >I'm almost finished with an essay entitled "The Roots of Unity" and would >> >love to get people's feedback. I'll be publishing it in less than 48 hours, >> >and I don't expect many people to have a chance to read it (let alone offer >> >comments) before I publish it. But I'll continue to tinker with it as >> >comments come in, so if you find mistakes or other opportunities for >> >improvement, please let me know, even if it's after the 17th. >> > >> >I tried (and failed) to find a doable puzzle that makes use of the the >> >multiplicative structure of the complex numbers. Can you think of a good >> >one that'd be accessible to a bright high school student? I was going to >> >include the question "Let p be the probability that the sum of the numbers >> >you get when you roll six fair dice is a multiple of seven. Do you think p >> >is equal to, greater than, or less than 1/7? What if we roll seven dice >> >instead of six?" (There's a nice anaysis that uses 7th roots of unity.) But >> >this strikes me as likely to be too tricky for nearly all of my readers. >> >Can anyone suggest a substitute? >> > >> >I hope you'll sign your comments, even negative ones; sometimes I have a >> >follow-up question (like "Okay, would this way of phrasing it be less >> >dreadful?"), which I can't ask you if you've commented anonymously. I want >> >to make the essay better, and I have a fairly thick skin. >> > >> >Please leave your feedback here: >> > https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af >> > >> >Title: The Roots of Unity >> >Beginning: Two primal pleasures for me from the years of my childhood (and >> >maybe the years of infancy that I can't remember) are feelings that I'll >> >call coziness and spaciousness. The first is the feeling I'd get at night >> >pulling the blankets up over my head; the other is the feeling I'd get >> >entering a landsca... >> >Read more: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af >> >Thanks, >> >jamespropp

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[math-fun] Cutting a pie to share equally among up to n people
by James Propp 15 Dec '17

15 Dec '17

Suppose we want to make cuts in a circular pie (the usual kind of cut, from the center to a point on the perimeter) in such a way that, for every k between 1 and n, the pieces can be divided equally among k people, each of whom gets the same amount of pie. How many cuts are needed? (I'm guessing I'm not the first person to ask this question. Searching the OEIS for the words "cut pizza" yielded several hits, none of which were my sequence. Searching the OEIS for the words "cut pie" yielded no hits at all. Not sure what this says about the eating preferences of mathematicians.) Jim Propp

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Re: [math-fun] Draft of my December 2017 blog post
by Henry Baker 15 Dec '17

15 Dec '17

"And now that we have those computers, mod p arithmetic (where p is a huge prime) is the cornerstone of cryptographic security for internet commerce." Not so much. There have been so many problems with RSA that the most recent recommendations are to disable RSA completely, in favor of elliptic curves. "A number of the most popular websites and services online, including Facebook and *PayPal*, are vulnerable to an exploit which has resurfaced from 1998." "We believe RSA encryption modes are so risky that the only safe course of action is to disable them." http://www.zdnet.com/article/robot-exploit-from-1998-resurrected-leaves-top… https://eprint.iacr.org/2017/1189.pdf At 08:42 AM 12/15/2017, James Propp wrote: >Hi, > >I'm almost finished with an essay entitled "The Roots of Unity" and would >love to get people's feedback. I'll be publishing it in less than 48 hours, >and I don't expect many people to have a chance to read it (let alone offer >comments) before I publish it. But I'll continue to tinker with it as >comments come in, so if you find mistakes or other opportunities for >improvement, please let me know, even if it's after the 17th. > >I tried (and failed) to find a doable puzzle that makes use of the the >multiplicative structure of the complex numbers. Can you think of a good >one that'd be accessible to a bright high school student? I was going to >include the question "Let p be the probability that the sum of the numbers >you get when you roll six fair dice is a multiple of seven. Do you think p >is equal to, greater than, or less than 1/7? What if we roll seven dice >instead of six?" (There's a nice anaysis that uses 7th roots of unity.) But >this strikes me as likely to be too tricky for nearly all of my readers. >Can anyone suggest a substitute? > >I hope you'll sign your comments, even negative ones; sometimes I have a >follow-up question (like "Okay, would this way of phrasing it be less >dreadful?"), which I can't ask you if you've commented anonymously. I want >to make the essay better, and I have a fairly thick skin. > >Please leave your feedback here: > https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af > >Title: The Roots of Unity >Beginning: Two primal pleasures for me from the years of my childhood (and >maybe the years of infancy that I can't remember) are feelings that I'll >call coziness and spaciousness. The first is the feeling I'd get at night >pulling the blankets up over my head; the other is the feeling I'd get >entering a landsca... >Read more: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af >Thanks, >jamespropp

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[math-fun] Draft of my December 2017 blog post
by James Propp 15 Dec '17

15 Dec '17

Hi, I'm almost finished with an essay entitled "The Roots of Unity" and would love to get people's feedback. I'll be publishing it in less than 48 hours, and I don't expect many people to have a chance to read it (let alone offer comments) before I publish it. But I'll continue to tinker with it as comments come in, so if you find mistakes or other opportunities for improvement, please let me know, even if it's after the 17th. I tried (and failed) to find a doable puzzle that makes use of the the multiplicative structure of the complex numbers. Can you think of a good one that'd be accessible to a bright high school student? I was going to include the question "Let p be the probability that the sum of the numbers you get when you roll six fair dice is a multiple of seven. Do you think p is equal to, greater than, or less than 1/7? What if we roll seven dice instead of six?" (There's a nice anaysis that uses 7th roots of unity.) But this strikes me as likely to be too tricky for nearly all of my readers. Can anyone suggest a substitute? I hope you'll sign your comments, even negative ones; sometimes I have a follow-up question (like "Okay, would this way of phrasing it be less dreadful?"), which I can't ask you if you've commented anonymously. I want to make the essay better, and I have a fairly thick skin. Please leave your feedback here: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af Title: The Roots of Unity Beginning: Two primal pleasures for me from the years of my childhood (and maybe the years of infancy that I can't remember) are feelings that I'll call coziness and spaciousness. The first is the feeling I'd get at night pulling the blankets up over my head; the other is the feeling I'd get entering a landsca... Read more: https://mathenchant.wordpress.com?p=2052&shareadraft= 5a33f99f1b1af Thanks, jamespropp

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Re: [math-fun] math-fun Digest, Vol 178, Issue 23
by Don Reble 15 Dec '17

15 Dec '17

> "Given a circle of n lights, exactly one of which is initially on, it > is permitted to change the state of a bulb provided that one also > changes the state of every dth bulb after it (where d is a divisor of > n strictly less than n), provided that all n/d bulbs were originally > in the same state as one another. Is it possible to turn all the bulbs > on by making a sequence of moves of this kind?" > OEIS http://oeis.org/A103314 looks very apropos. A070894 looks even better. The set of bulbs changed by a move has a center-of-light at zero. (Arrange the bulbs evenly around the circle. And then all go on or all go off.) The original center is non-zero, and a move changes it to a weighted combination of zero and the old value: but never to zero. So you can't even turn them all off. Please refute this, if you're the last one to bed. -- Don Reble djr(a)nk.ca

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