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[math-fun] Legendary Chalk
by Hilarie Orman 25 Aug '20

25 Aug '20

Does anyone really use chalk anymore? If so, is Hagoromo really this good? https://www.cnn.com/2020/08/22/us/hagoromo-chalk-great-big-story-trnd/index… Hilarie

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[math-fun] A variant of the muffin problem
by James Propp 25 Aug '20

25 Aug '20

>From https://mathenchant.wordpress.com/2020/08/16/the-muffin-curse/ (slightly adapted): Imagine an infinite supply of 1-ounce chocolates rolling off an assembly line that's staffed by two immortal and indefatigable employees (call them Ethel and Lucy) who have to feed an infinite line of students. When a new chocolate arrives, Ethel cuts it up and puts pieces into an infinitely large holding area; meanwhile, Lucy takes pieces from the holding area and hands them out to students. All the chocolate pieces must be handed out to students, and each student must get exactly *x* ounces of chocolate. If Lucy and Ethel want the smallest piece any student gets to be as large as possible, what goal should they shoot for, as a function of *x*? I'm pretty sure that when *x* is rational, this is just the muffin problem discussed in this forum back in 2008. But what happens when *x* is irrational? If, say, *x* is the golden ratio ϕ = (1 + sqrt(5))/2 = 1.618..., is there a way for Ethel to divide the 1-ounce chocolates into smaller morsels, and for Lucy to distribute the morsels, so that every morsel gets eaten, and every student gets exactly ϕ ounces of chocolate, and no morsel is less than 0.4 ounces? It may be relevant to mention that for the original muffin problem, f(m,s) (defined as the maximum possible size of the smallest piece among all ways of distributing m muffins among s students) has the property that f(m,s) = g(m/s) for a certain function g that is NOT continuous as a function from the rationals to the rationals (here I am using the epsilon-delta definition of continuity, which makes sense even when the domain of a function is the rationals). So g(.) does not admit an extension to an everywhere-continuous function from the reals to itself (in case you were hoping, as I was, that this would be the answer). There may be some sort of semicontinuous extension at work here. I chose 0.4 because if we let s and m be consecutive Fibonacci numbers, then g(m/s) seems to be converging to something slightly larger than 0.4. Here's Richard Chatwin's article: Richard Chatwin, An optimal solution for the muffin problem, https://arxiv.org/pdf/1907.08726.pdf. Jim Propp

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[math-fun] Another variant of the muffin problem
by Dan Asimov 25 Aug '20

25 Aug '20

Define N(m,s) as *the minimum number of pieces* necessary to distribute m/s muffins to each of s students, starting from m whole muffins. Is N(m,s) easy to determine? —Dan Jim Propp wrote: ----- ... the original muffin problem, f(m,s) (defined as the maximum possible size of the smallest piece among all ways of distributing m muffins among s students) ... -----

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[math-fun] A fractal spiral?
by Éric Angelini 25 Aug '20

25 Aug '20

Hello Math-Fun, (a few friends in copy) this is in French — but easy to understand with the illustrations. In short: 1. seq D 2. digits of D form seq S 3. seq S is turned (on a 2D square grid) into a single-digits spiral 4. This spiral is fractal: its odd digits reshape the spiral itself (the digits have a peculiar design) ... This idea came to me after a discussion on SeqFan about the precise (and seemingly impossible) definition of the term « fractal ». This was submitted this morning to the French magazine Tangente: https://cinquantesignes.blogspot.com/2020/08/partir-en-recursion.html?m=1 Best, É. Catapulté de mon aPhone

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[math-fun] Meeting time
by Kelly DeRango 24 Aug '20

24 Aug '20

Hi Rich: I hope your conference went well last week. We would love to schedule a time for you to speak to our math club. School is beginning this week in Michigan and so I am either teaching or in meetings until 3:30 pm eastern time M-Th and until 2:45 pm on Friday. I am available after those times. Right now we have a Math Club virtual meeting set up for 3 pm Friday. Let me know if you can make a meeting sometime this week, it wouldn't have to be the Friday meeting we have set up but could be if that is what you prefer. Stay safe! Kelly -- Kelly DeRango Battle Creek Area Math and Science Center -- ------------------------------------------------------------------------------------------------------ This email is intended only for the use of the addressee(s) named herein. It may contain legally privileged and confidential information. If you are not the intended recipient, or an authorized representative of the intended recipient, you are hereby notified that any review, copying or distribution of this email and its attachments, if any, is strictly prohibited. If you have received this email in error, please immediately notify the sender by return email and delete this email from your system. Thank you.

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Re: [math-fun] Mike, on the PDP-6, when you were chasing Mersenne primes (e.g. 11213, 23209, ...) vs Brillhart etc @ TJWatson
by Landon Curt Noll 23 Aug '20

23 Aug '20

Hello, I was asked to supply some background on FFTs primality testing history. I hope you find this useful. =-= As part of the Amdahl 6 group back in the late 1980’s: http://www.isthe.com/chongo/tech/math/prime/amdahl6.html where in 1990, we dethroned Mersenne primes for a few years, we used: http://www.isthe.com/chongo/tech/math/prime/p391581_216193.html http://www.isthe.com/chongo/src/calc/lucas-calc We wrote a 'ultra-high speed multi-precision' package on an Amdahl 1200 (think IBM 370-195 with a long trace vector processor add-on) and implemented a Riesel test. The heart of the package was a multiplication and square routine that was based on the PFA Fast Fourier Transform and on Winograd's radix FFTs. The IBM Floating point used a radix 16 mantissa. But we also had access to Amdahl/IBM micro-code so we could play with the bits "below the end of the double". Our vectors were represented in base 2^23. We used Macsyma using LISP ported to the Amdahl / IBM to build floating point operations to model the Floating-point "math" ops. On top of that we loaded the PFA Fast Fourier Transform and a Winograd's radix FFTs model. It was a HUGE Macsyma operation than ran on the Amdahl 5990 for a few weeks. In the end we were able to prove that using the IBM floating point architecture and base 2^23, we had NO systematic errors. (unlike GIMPS where they just hope it is good enough). While Macsyma was also used to verify the correctness of our code, it played a very important role in our code testing. While the Amdahl 6 was not the first primality testing FFT by any means, it was the first well tuned FFT for primality testing. Our cross-over point for when the FFT was more efficient for Riesel primes, primes of the form: h * 2^n - 1 was just under 1000 decimal digits. Combined with an amazingly efficient sieve on the Amdahl 1200 vector processor, we were cranking out 1000+ digit prime proofs at the rate of 1 every 6 seconds. Our intent was to flood the "Titanic prime list". Until Amdahl 6’s work, a titanic prime was a proven prime that was 1000 or more decimal digits in length. After our flood, where at one point we had 99.95% of the Titanic list, they changed the "Titanic prime" definition: which was fine since that was our goal. :-) =-= I was involved in a Primality test using FFTs on the Cyber 205 in the mid 1980’s. Due to the co-developer, Steve McGrogan wanting to only to start testing when the program was "fully optimized", we missed discovering the prime M86243. While the "fully optimized" routine would have been 42% faster, had we started the first code that passed all of our regression tests, we would have discovered M86243 in only 47 hours. This is why I advise people who are engaged in what Dr. Lehmer called: The great indoor sport of prime number hunting to start with the code that passes your extensive test suite: launch your search modify your source verify by measurement, that your new code rans FASTER verify that your new code passes your QA / regression test suite replace your production code with the new code rinse and repeat Do that until your mods are no longer running faster: likely when the clock time you might save is less than the time it takes to code and test. =-= I wrote a Mersenne LL-test using FFT code that ran on the Cray 1 back in 1980. It was a floating point FFT written in Cray Assembler that was based on the Cray FFT math library. The mods were mostly to deal with the fact that I was doing Mersenne Prime testing that an inline mod 2^n-1 instead of the standard normalization that the FFT required. However that code discovered that there was a reciprocal approximation hardware bug that caused the Mersenne tests for M9689 and M19937 to fail. The fix involved a solder patch to a circuit board in the reciprocal approximation unit. Only the Cray 1’s that were manufactured post Serial number 13 had the fix build in. Unfortunately the 16 machines that were pre SN-13 did not have that fix. There was resistance to fixing the older Cray’s because some numerical code broke when the bug was fixed. Indeed the Cray 1 I used had a jumper that could be changed to turn off the switch. Yes, there were 16 machines prior to #13 .. some Cray machines were not numbered because they were "made to not exist". Indeed the first Cray that could run my FFT code correctly was loaded up into a truck and left in a trailer behind the Safeway supermarket in Livermore California: it was driven away by authorized people. The repair was tested in the field. My code was used to verify the fix. There was a jumper that could enable/disable the fix. The fix did involve in part, an issue of capacitance. In honor of the PDP-10 MIT hack: https://www.cs.utah.edu/~elb/folklore/magic.html the board that was first field repaired contains the inscription: EVEN MORE MAGIC It is my hope that this machine will be moved to a museum someday. Anyway my FFT Mersenne search was halted for reasons you can read between the lines in the above. :-) FYI: We used the MIT-MC Macsyma machine to verify that the Cray 1 ALU was generating wrong values. The Cray was at an ARPANET link site, and so tipping over to MIT-MC to run Lisp code to test was a logical thing to do at that time. =-= When I helped found the EFF Cooperative Computing Awards: http://www.eff.org/awards/coop I put into the rules: https://www.eff.org/awards/coop/rules guidelines that I am NOT eligible for an award. This is because I am chair of the Cooperative Computing Awards advisory panel who helps the EFF determine when a claim is legit, it would be a conflict of interest if I obtained an award. In order to avoid even the appearance of a conflict of interest: I only advise others on primality testing these days. Much as improved from the Amdahl 6 days! See My tutorial: A Grand Coding Challenge! Finding a new Largest Known Prime http://www.isthe.com/chongo/tech/math/prime/prime-tutorial.pdf incorporates the best practices of finding a "new Largest Known Prime". While maybe not quite at "waist height" as Selfridge liked to call it, there is a plum waiting to be picked, that someone with a modest amount of effort could again de-thrown the Mersenne primes and beat GIMPS to a new prime number record. The tutorial in the URL above describes a few approaches to picking a mew "prime plum". I would also be happy ti give video talk on these details to anyone interested. =-= Since I saw a few Q’s about what I did for Moffett Federal Airfield, Lick, Pluto and Voyager: Yes I had a hand in helping save the Blimp Hanger at Moffett Field although there were others who had much bigger hands in saving the hanger. https://en.wikipedia.org/wiki/Moffett_Federal_Airfield Yes, I had a hand in helping prevent Lick Observatory from closing: http://mthamilton.ucolick.org Yes, I played a direct personal role in talking a US Senator into changing their vote to save the New Horizons mission to the dwarf planet Pluto (yes Pluto is a planet, a Dwarf planet): Current New Horizons status: http://pluto.jhuapl.edu And yes, I found the funds to prevent the Voyager spacecraft from being turned off in the mid 1990’s. Current Voyager status: https://voyager.jpl.nasa.gov/mission/status/ =-= I hope this helps. chongo (Landon Curt Noll) /\oo/\

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[math-fun] Exploiting uniqueness
by James Propp 22 Aug '20

22 Aug '20

Does anyone know of a name attached to this problem-solving tactic? Jim Propp ---------- Forwarded message --------- From: David Will <daw5de(a)virginia.edu> Date: Tue, Jun 30, 2020 at 8:17 PM Subject: Question about the self-referential aptitude test To: <jamespropp(a)gmail.com> Dr. Propp: Good evening. I know I'm a bit late to the party, but I just wanted to say I'm a big fan of your self-referential aptitude test, which I just discovered today. In the instructions, you mentioned this idea that the foreknowledge of the puzzle having a unique solution can, in and of itself, be used as a clue when solving the puzzle. I enjoy Japanese-style puzzles, and I've previously encountered this same phenomenon in games such as slitherlink and masyu. I'm also a graduate student in mathematics at the University of Virginia, so I've been curious about this idea from a mathematical perspective. I was hoping you'd be able to answer a quick question. Do you know of a name for this phenomenon? I'm lacking in knowledge about game theory (is that even the right subject?), so I was unsuccessful in my searches online. I'd like to see if this phenomenon has been studied with any sort of rigor. In particular, I want to know if it would be possible to construct a puzzle that requires this logic in order to be solved (in some loose sense, ignoring brute-force guessing). Thank you for your time. I'd greatly appreciate it if you'd be willing to entertain my curiosity. Best, Andrew Will

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[math-fun] Keller's conjecture now settled in all dimensions
by Dan Asimov 21 Aug '20

21 Aug '20

Keller's conjecture <https://en.wikipedia.org/wiki/Keller's_conjecture> proposed that if Euclidean space R^n were tiled by n-dimensional cubes any which way, then there must be some pair of n-cubes that share a common (n-1)-dimensional face. This is intuitively true in dimensions 2 and 3, but was proved true in dimensions up through 6 by Oskar Perron in 1940. To the surprise of many people, Lagarias & Shor (1994) found a counterexample in dimension 10. This immediately leads to counterexamples in all higher dimensions as well. Soon after a counterexample was found in dimension 8, showing it was also false in dimension 9 and leaving the only dimension where it was open being dimension 7. A team of four people — Joshua Brakensiek, Marijn Heule, John Mackey, and David Narváez — used a computer program to search for counterexamples in dimension 7, and in October 2019 they found one: <https://arxiv.org/abs/1910.03740>. There a nice article about it in Quanta magazine: <https://www.quantamagazine.org/computer-search-settles-90-year-old-math-pro…>. —Dan

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Re: [math-fun] Correction re: Keller's conjecture now settled in all dimensions
by Dan Asimov 21 Aug '20

21 Aug '20

The Wikipedia article just hasn't been updated yet. —Dan Mike Beeler wrote: ----- The Wikipedia article says, "However, the translation from cube tilings to graph theory can change the dimension of the problem, so this result doesn't settle the geometric version of the conjecture in seven dimensions.” -----

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[math-fun] Correction re: Keller's conjecture now settled in all dimensions
by Dan Asimov 21 Aug '20

21 Aug '20

Oops! I said the exact opposite of what happened. Sorry about that. In 7 dimensions it was proved ***true***, not false, contrary to what I wrote. —Dan ----- Keller's conjecture <https://en.wikipedia.org/wiki/Keller's_conjecture> proposed that if Euclidean space R^n were tiled by n-dimensional cubes any which way, then there must be some pair of n-cubes that share a common (n-1)-dimensional face. This is intuitively true in dimensions 2 and 3, but was proved true in dimensions up through 6 by Oskar Perron in 1940. To the surprise of many people, Lagarias & Shor (1994) found a counterexample in dimension 10. This immediately leads to counterexamples in all higher dimensions as well. Soon after a counterexample was found in dimension 8, showing it was also false in dimension 9 and leaving the only dimension where it was open being dimension 7. A team of four people — Joshua Brakensiek, Marijn Heule, John Mackey, and David Narváez — used a computer program to search for counterexamples in dimension 7, and in October 2019 they found one: <https://arxiv.org/abs/1910.03740>. ^^^^^^^^^^^^^^ ***FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE*** There a nice article about it in Quanta magazine: <https://www.quantamagazine.org/computer-search-settles-90-year-old-math-pro…>. -----

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