July 2019 - mathfun@mailman.xmission.com

[math-fun] Power means (was Re: optimal advance tournament scheduling)
by Keith F. Lynch 27 Jul '19

27 Jul '19

Inspired by Marilyn vos Savant's using an arithmetic mean where a geometric mean was clearly called for last weekend (did anyone else catch that?) I decided to look into the optimum solution of the advance tournament scheduling problem for various power means. Power means of n elements include: -1: Harmonic mean, the reciprocal of the arithmetic mean of the reciprocals 0: Geometric mean, the nth root of the product 1: Arithmetic mean, the sum divided by n 2: Quadratic mean (RMS), the square root of the arithmetic mean of the squares The power doesn't have to be an integer. It can be any real number. (For 0, it's taken as the limit when the power approaches 0.) (The power can presumably be a complex number too, but I'll save that for another day.) Here's the result: 2E6BB72 44444444 n:2520 min:18 max:28 0,0,27,175,339,496,370,308,267,243,162,82,51 22.4476 -16.023502 20.896856 0775CF5 55444433 n:10080 min:18 max:28 0,0,182,788,1032,1248,1678,1144,1406,1144,836,440,182 22.7560 -14.684336 21.008307 05F7D78 54444443 n:20160 min:18 max:28 0,0,668,1134,1786,2078,2856,3100,2806,2452,1866,1008,406 22.9565 -9.398537 21.600937 077BE78 44444444 n:2520 min:17 max:28 0,45,98,37,166,264,243,406,360,404,263,176,58 23.3071 -8.666145 21.729642 03FDE78 44444444 n:35 min:16 max:28 1,0,0,2,1,0,4,4,7,3,9,3,1 24.0000 There are five different best solutions, depending on the power. I show the powers at which best solution changed, and what the power mean was at that power. Of course at that power it was a tie between the above and the below graph. For instance 2E6BB72 is the best solution for powers below (more negative than) -16.023502, and 0775CF5 is the best solution for powers above it but below -14.684336. At -16.023502 itself, 20.896856 is the mean for both 2E6BB72 and 0775CF5. (See my July 21 message for how to turn those hexadecimal numbers into the graphs they represent, and what the meaning of the stuff after those hexadecimal numbers is.) It's interesting that 03FDE78, K4,4, is the best for all powers above -8.666145, including the harmonic, geometric, arithmetic, and quadratic means. It's also 7-way-tied for the highest median, 24, and has the unique highest mode, 26. I checked all powers from -200 to +200. Beyond that, I was plagued by underflow, overflow, and precision-loss errors. (Try raising 28 to the 200th power on your scientific calculator, see how far you get.) But I'm convinced that the bottom solution goes all the way down and the top solution goes all the way up. In the limit of high powers, the mean is the maximum value, which in this case is a 1281-way tie for 28. In the case of a tie, it's whichever set has the highest proportion of 28s. And K4,4 does. It may not look like it, since only 1 of the 35 K4,4 graphs has 28, and other graphs have as many as 768 28s. But that's 768 out of 40320, which is less than 1/35. In the limit of low (very negative) powers, the mean is the minimum value, which in this case is a 76-way tie for 18. In the case of a tie, it's whichever set has the lowest proportion of 18s. And, sure enough, 2E6BB72 does. As for the *worst* graph, for any power mean it's 2DDEFC0 55555511 n:28 min:16 max:16 28,0,0,0,0,0,0,0,0,0,0,0,0 16.0000 Since all values are 16, all power means are 16 (as are the median and the mode). It's interesting that the worst graph and the best (for most purposes) graph have the two lowest values of n. Now I'm wondering about the general case of some finite quantity of finite sets of positive integers. As the power is gradually ramped up, can which set has the highest power mean occur more than once? Obviously yes, since, given the two sets {2,9} and {1,6,7,8} the former excels for very low and very high powers, and the latter is best for powers in between. (They're tied at a power of 1. At what other power are they tied?) It there any limit to how many times two sets can trade places as best? How about three or more sets?

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[math-fun] Multiply my digits
by Éric Angelini 26 Jul '19

26 Jul '19

5 terms (with my smartphone)! 6,16,144,1944,3899,... Best, É.

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[math-fun] Sparse Rulers to length 1515
by ed pegg 25 Jul '19

25 Jul '19

At http://demonstrations.wolfram.com/SparseRulers/ are sparse rulers to length 1515. For all of these lengths, with n=length and m=minimal marks, m - round(sqrt(3*n+9/4)) is either 0 or 1. The 0/1 pattern to length 1515 is nice, you can see it at http://demonstrations.wolfram.com/SparseRulers/HTMLImages/index.en/popup_3.… . More discussion is at https://math.stackexchange.com/questions/3279766/sparse-ruler-conjecture and https://math.stackexchange.com/questions/3246000/graceful-graphs-with-valen… . In my larger list, bound is true to at least 1790. --Ed Pegg Jr

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[math-fun] Random nonempty intersection of cube and plane, revisited
by James Propp 25 Jul '19

25 Jul '19

You may remember from a couple of years ago the math-fun discussion of what one gets when one intersects a cube with a random plane that intersects it nontrivially (Subject: Random slice of a cube). Thanks to communal effort, we learned that the expected number of sides is exactly 4, and that this remains true even if one preselects the orientation of the plane. In preparing to give a public talk about this result (see https://www.cityguideny.com/event/MoMath--National-Museum-of-Mathematics--2…) I realized that some members of the audience may ask "What about the expected area of the intersection? What about the expected perimeter?" The answer to the first question is nice (2/3 times the square of the side-length; ask me if you want to know why). I don't know the answer to the second question, but maybe some of you can help me figure it out (e.g., maybe Keith Lynch can dust off his code and use it to estimate expected perimeter). And while we're at it, numerical corroboration of the answer to the first question would be reassuring. For these two questions, it does NOT suffice to focus on one particular orientation of the cutting plane; you have to average over all of them. The measure I'm using is the unique rotationally-invariant and translationally-invariant continuous measure on the affine Grassmannian (the set of all planes in R^3); it's not a probability measure, but it assigns finite measure to the set of all planes passing through a fixed cube, so by renormalizing you can get a probability measure. By the way, one of the themes of my talk is the collaborative process by which math is done (in contrast to the myth of the isolated genius), so the sequence of events that brought these results to light is going to be integral to my talk. If you have any concerns that I might acknowledge your contribution and for some reason you don't want me to, please let me know! Jim Propp

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[math-fun] Geometric probability a la Archimedes
by James Propp 25 Jul '19

25 Jul '19

One can prove that the expected distance from a random point on the surface of a sphere to the equatorial plane is half the radius. Assuming we could rephrase this claim in a form that Archimedes would recognize, how would he have proved it? As an example of the kind of proof I would like to see, consider the proposition that the expected distance from a random point in a disk to the boundary of the disk is 1/3 of the radius. One can prove this using the formula for the volume of a cone. (I came up with this myself but I’m sure others have too.) Further examples of the kind of proof I have in mind are Archimedes’ determination of the surface area and volume of the sphere. Jim Propp

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[math-fun] Multiply my digits
by Éric Angelini 25 Jul '19

25 Jul '19

Hello Math-Fun, say a(n) is the product of the digits of a(n+1); and this a(n+1) is the product of the digits of a(n+2); and this a(n+2) etc. I could not go further than a(n+3) with 8, 42, 1372, 22777,... Of course a way to expand this seq. would be to test all integers written with the digits 1,2,2,7,7,7 or with the digits 1,1,2,2,7,7,7 or with the digits 1,1,1,2,2,7,7,7 etc. (as their product would still be the same = 1372), then check if one of those integers might be the product of single digits only. I got stuck as this task is quickly impossible to execute on a smart- phone! Perhaps starting with another a(1) than 8 is more promising? And if there are many infinite such seqs, which one would be the lexico- graphically earliest? [pardon me of this is old hat] Best, É.

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[math-fun] Threshold thm for noisy GoL?
by Mike Stay 23 Jul '19

23 Jul '19

Does there exist some p < 1/2 and metacircular interpreter pattern such that when the pattern is run on a noisy grid where a cell has probability p of being toggled each generation, the resulting computation runs life cells with error < p? -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com

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[math-fun] Happy approximate pi day
by Lucas, Stephen K - lucassk 23 Jul '19

23 Jul '19

I see from various friends that it is approximate pi day, 22/7 if you live outside the USA. I’m sure many of us know Dalzell’s classic result that int_0^1 x^4(1-x)^4/(1+x^2) dx = 22/7 - pi which proves that pi is strictly less than 22/7, since the integrand is nonnegative. But for fun, here are a couple of infinite series with positive terms that also equal 22/7 - pi: sum_{n=2}^\infty 60/[(4*n^2-1)(16*n^2-1)(16*n^2-9)] sum_{n=1}^\infty 640n/[(4n+1)(4n+3)(4n+5)(4n+7)(4n+9)(4n+11)] sum_{n=6}^infty 2^{n-3}(n+242)/[15(2n+1)(2n+3)C(2n,n)] Stege

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[math-fun] Hey kids, bathroom stuff!
by Bill Gosper 22 Jul '19

22 Jul '19

I'm pretty sure this is the pattern in Oren's Hummus bathrooms on Castro St; Can you find a short proof or disproof that the dark tiles form a square grid? —Bill (During the 70s, I stared at (slowly, in those days) evolving Life patterns so habitually that the AI Lab bathroom tiles would run a couple of steps when I looked down. But those were square tiles, randomly dark and light.)

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Re: [math-fun] Hey kids, bathroom stuff!
by Bill Gosper 22 Jul '19

22 Jul '19

On Mon, Jul 22, 2019 at 7:21 AM James Buddenhagen <jbuddenh(a)gmail.com> wrote: > Your bathroom floor link gives me 404 not found. > I've never seen anything like this. That's the url in my Chrome tab, but indeed it 404s. Prefixing http:// seems to work. Clicking on Save image as insists on trying to save it as html! Fortunately, I still have the original browser tab, so try wtf <http://gosper.org/bathroom-floor.jpg> . —rwg > > On Sun, Jul 21, 2019 at 10:48 PM Bill Gosper <billgosper(a)gmail.com> wrote: > >> Arrgghh, my bad. >> >> http://otterrun.info/wp-content/uploads/2018/02/black-and-white-bathroom-fl… >> —rwg >> >> On Sun, Jul 21, 2019 at 8:40 PM Brad Klee <bradklee(a)gmail.com> wrote: >> >> > I did not see an image. Math fun may scrub those automatically. >> > I would suggest pasting a link to some other host site, or else >> > others will have similar confusion. --Brad >> > >> > On Sun, Jul 21, 2019 at 10:34 PM Bill Gosper <billgosper(a)gmail.com> >> wrote: >> > > >> > > >> > > >> > > On Sun, Jul 21, 2019 at 7:53 PM <bradklee(a)gmail.com> wrote: >> > >> >> > >> This is not making sense to me, did you forget something?? >> > > >> > > Quite possibly. Did you you get an image? Did you zoom it enough to >> see >> > > the hexagonal tiles? I don't think I needed to say "geometrically >> > square grid". >> > > Can you propose a something that would make things sensible? —Bill >> > >> >> > >> >> > >> > On Jul 21, 2019, at 8:03 PM, Bill Gosper <billgosper(a)gmail.com> >> > wrote: >> > >> > >> > >> > I'm pretty sure this is the pattern in Oren's Hummus bathrooms on >> > Castro St; >> > >> > >> > >> > Can you find a short proof or disproof that the dark tiles form a >> > square >> > >> > grid? —Bill >> > >> > (During the 70s, I stared at (slowly, in those days) evolving Life >> > patterns >> > >> > so habitually >> > >> > that the AI Lab bathroom tiles would run a couple of steps when I >> > looked >> > >> > down. But >> > >> > those were square tiles, randomly dark and light.) >> > >> > _______________________________________________ >> > >> > math-fun mailing list >> > >> > math-fun(a)mailman.xmission.com >> > >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >

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