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[math-fun] MacOS Mail
by Henry Baker 09 Jun '20

09 Jun '20

I realize that this is off-topic, so feel free to delete this if you wish. I personally use Linux & Windows, but my wife uses MacOS. Over the past 2 years, the MacOS "Mail" program has become impossibly slow -- all the while use the same hardware! I can't believe that Mac users would put up with this. Note that everything else works fine; it's just the Mail program. Do any of you use Macs? Do you find the Mac Mail program intolerably slow?

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[math-fun] Euphoria sepulcralis
by Brad Klee 07 Jun '20

07 Jun '20

Also Known As the Dark Flower Scarab, but this week I am calling them the field looter flower beetles, especially now that they've teamed up with the hyper-destructive Japanese beetles. The coloration of E. Sepulcralis is mostly black, with a green, sometimes red undersheen. Like the Japanese beetles, and many others, their Iridescent light reflections are circularly polarized: https://www.inaturalist.org/observations/48771137 https://www.inaturalist.org/lists/3394445-Chiral-Scarabs-of-Arkansas The North Shore disc golf course cuts one of their habitats into narrow little swaths, where the Queen Anne's lace grows wild around the manicured lawn. Disc golfers think they have really accomplished something when they get a birdie or an eagle, but it is nothing compared to what the flower beetles can now accomplish after millions of years of evolution. Although they seem to be doing fine statistically, habitat destruction and degradation could introduce them to unnecessary suffering. Other insects face similar problems, and possibly extinction, see also: https://www.wildlifearkansas.com/materials/2006_updates/03SGCN.pdf The sky is blue, the fields and the forests are calling, --Brad

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[math-fun] Spherical geometry puzzle
by Dan Asimov 06 Jun '20

06 Jun '20

I recently searched in vain for an expression for the area of a spherical triangle in terms of its *sides* having lengths a, b, c. (Assume that the triangle is drawn on the surface of a sphere of radius = 1, with sides that are arcs of great circles, and whose angles are all < π radians.) So I figured out my own formula, which definitely took a few minutes. Puzzle: Find such a formula without consulting any references. —Dan

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[math-fun] A generalization of an old Conway Problem
by Victor Miller 06 Jun '20

06 Jun '20

John Conway once posed the following problem: The Ten Divisibilities I have a ten digit number, abcdefghij. Each of the digits is different, and a is divisible by 1 ab is divisible by 2 abc is divisible by 3 abcd is divisible by 4 abcde is divisible by 5 abcdef is divisible by 6 abcdefg is divisible by 7 abcdefgh is divisible by 8 abcdefghi is divisible by 9 abcdefghij is divisible by 10 Find the number. One can do this computationally in a number of ways, such as exhausting over all permutations of [1,10], or branch and bound. I decided to do this via integer programming to investigate the generalization to other bases than 10. Interestingly enough, I found that there were only solutions for bases 2,4,6,8,10,14 (tested through base 27). I don't know if they've petered out completely. Any ideas? Victor

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[math-fun] Strange matrix products
by Bill Gosper 05 Jun '20

05 Jun '20

Simon Plouffe's old http://wayback.cecm.sfu.ca/cgi-bin/isc/lookup?number=1.64163256065515386629… told me that In[202]:= Dot @@ Table[{{1/2 + 2^-n, -1/2}, {-1/2, 1/2}}, {n, ∞}] = {{1,-1},{1-𝜗, 𝜗-1}] (if only Mathematica supported infinite Tables) where 𝜗 = In[186]:= Sum[2^-Binomial[n, 2], {n, ∞]}] Out[186]= EllipticTheta[2, 0, 1/√2]/2^(7/8) I.e., this product converges to something interesting even though the determinant approaches zero. I can't wait to try other things, but agonizingly, Simon's website seems up only a few seconds every few minutes! —rwg

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[math-fun] multimagie.com update
by Christian Boyer 05 Jun '20

05 Jun '20

www.multimagie.com updated this week, but: « TEN YEARS after their opening (*), there still remain eight unsolved enigmas for winning €6,500 and eight bottles of champagne! The remaining enigmas are on small squares, from 3x3 to 6x6. » Christian. (*) April 2010: www.multimagie.com/English/MagicSquaresEnigmasE.pdf

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Re: [math-fun] Sages' hats and prime-richness
by Steve Witham 04 Jun '20

04 Jun '20

Dan, thanks for the MSRI math problems. The sages' hats (quoted below) was about my speed. Spoilers. I liked finding that solving a little problem was one way to solve the whole problem, then realizing that sometimes the big problem *was* the little problem, so it didn't do any good to look for another method, I had to solve *that* little problem. Involutions were involved (pun). OEIS A000085 counts the number of involutions of n things. I printed out entries 0..40 and their factorizations... They seem pretty "primey"... |Basically, although the recurrence for A000085 is fairly simple, it has the talent of *ruling out* a sizeable number of prime factors (for instance 3, 7, 11, 17...). |||OEIS A264737 lists primes that do appear as factors of A000085 entries. So (except for the powers of two in every entry) you get numbers with fewer, larger prime factors than... my intuition expects anyway. | What a woild. --Steve |> Subject: [math-fun] Some math puzzles > From: Dan Asimov <dasimov(a)earthlink.net> > Date: 5/29/20, 4:58 PM > > Here are four problems from the puzzle column in the latest > newsletter from MSRI, faintly edited. > ... > 3. Suppose there are n hats, each with a different color. > These hats are placed on the heads of n sages. All of the > sages know all the colors: their own hat and everyone elseÂ’s > hats. A referee then announces the "correct" hat color that > should be on the head of each sage. The sages are then allowed > "swap" sessions: in one session disjoint pairs of sages are > allowed to interchange their hats. Can the sages fully correct > their hat colors in two swap sessions? -fin-

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Re: [math-fun] Some math puzzles
by Dan Asimov 03 Jun '20

03 Jun '20

MSRI is the Mathematical Sciences Research Institute in Berkeley, California. —Dan Bernie Cosell wrote ----- >> Here are four problems from the puzzle column in the latest newsletter >> from MSRI, faintly edited. > >What's MSRL? The puzzles are fun. > /b\ -----

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[math-fun] A chain of mean terms
by Éric Angelini 03 Jun '20

03 Jun '20

Hello Math-Fun, 109,87,65,43,21,1,23,45,67,89. Is this the longest chain where the mean of two adjacent terms is given by the concatenation of the two digits framing the comma between them? Best, É.

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[math-fun] 1,3,11,13,17,23,73,...
by James Propp 03 Jun '20

03 Jun '20

I was surprised to see that the integer sequence 1,3,11,13,17,23,73,... consisting of all the record-breakers in sequence A005589 (i.e., all positive integers whose representation as an English word or phrase is longer than the English representations of all earlier positive integers) isn’t a sequence of its own. Did I make a mistake in computing the list of record-breakers? I actually don’t find such sequences very interesting, but I know others do. In particular, my daughter has asked me to write a blog essay about the fact that repeatedly applying the map k->A005589 <http://oeis.org/A005589>(k) to any starting value n always leads to 4 (cf. A016037 <http://oeis.org/A016037>, A133418 <http://oeis.org/A133418>). Has anyone (perhaps Diane Karloff, who is credited with this observation) written about this? Then maybe I could give the article to my daughter instead of having to write it myself. :-) My daughter and I learned about this fact from my son, who came home from camp last summer challenging us to make sense of baffling conversations like this: Son: Pick another number. Me: Okay, seventy-seven. Son: ... Seventy-seven is twelve, twelve is six, six is three, three is five, five is four, four is the magic number. Have any of you heard this “game” before? Jim Propp

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