September 2016 - mathfun@mailman.xmission.com

[math-fun] An enigma solved on magic squares
by Christian Boyer 16 Sep '16

16 Sep '16

In 2010, twelve enigmas were launched: www.multimagie.com/English/MagicSquaresEnigmasE.pdf <http://www.multimagie.com/English/MagicSquaresEnigmasE.pdf> Three of them were solved from 2010 to 2015. I am pleased to announce that a fourth enigma (#6b) is now solved! Here is the first known 7x7 additive-multiplicative magic square, constructed by Sébastien Miquel. The 7 rows, 7 columns and 2 diagonals have the same sum. The 7 rows, 7 columns and 2 diagonals have the same product. 126 66 50 90 48 1 84 20 70 16 54 189 110 6 100 2 22 98 36 72 135 96 60 81 4 10 49 165 3 63 30 176 120 45 28 99 180 14 25 7 108 32 21 24 252 18 55 80 15 More details will be given at www.multimagie.com <http://www.multimagie.com> in the coming weeks. Before this square, 8x8 was the size of smallest known add-mult magic squares. 5x5 and 6x6 are still unknown (enigmas #6 and #6a), but 3x3 and 4x4 are proved impossible. Now, there remain eight unsolved enigmas for winning €6,500 (~$7,300) and eight bottles of champagne! Christian Boyer.

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Re: [math-fun] Moore curve drawn with epicycles
by Bill Gosper 15 Sep '16

15 Sep '16

Jörg nailed it. This is similar to http://gosper.org/TDrag3c.mp4 (plays in Safari but not Firefox). Gawd, I can't believe E. H. Moore would perpetrate such an inelegant kludge! Gluing four Hilbert arcs onto the sides of a degenerate rhombus apparently for the mere purpose of creating a closed squarefilling curve. Whose quadrants are not Moore curves! Better would have been filling a domino with two Hilberts (back to back) on the sides of a digon. gosper.org/fst.pdf explains how to get the Fourier series of a curve to repeat on the sides of a regular n-gon, indented for negative n. http://gosper.org/TDrag3c.mp4 is three triadic dragons on the sides of an equilateral triangle. Actually, for Fourier purposes, the curve need not even close. It can just snap back after each period, interpolating just one point halfway through the snap. Better yet, why didn't "algoritmic" draw a https://en.wikipedia.org/wiki/Sierpiński_curve, which is closed and legitimately four copies of itself. Apparently the "disks" are to emphasize that the rotors (harmonics) retain fixed amplitudes. But they don't assure the skeptical viewer that their relative frequencies and phases remain fixed. I wonder if "algoritmic" had an analytic formula for the nth Hilbert harmonic. Given some such formula describing an arc, I'll bet there's a neat formula for arranging them on the sides of an arbitrary rhombus. I'm not surprised there seem to be excess rotors in the video. Spacefilling Fourier series converge very slowly, You typically need to double the number of harmonics for each additional level of detail of the spacefiller. And, it would help if "algoritmic" started and ended with zero angular velocity, like http://gosper.org/TDrag3c.mp4 . --rwg On 2016-09-08 16:30, Joerg Arndt wrote: > * James Propp <jamespropp(a)gmail.com> [Sep 08. 2016 17:59]: >> Can Bill Gosper or anyone else explain what's going on with >> https://twitter.com/algoritmic/status/772699702064254976 >> ("Moore curve drawn with epicycles")? >> >> Jim Propp >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > Unless I am very mistaken this is more and more terms of the Fourier > series (in polar coordinates). The disks seem to indicate the > magnitude of the terms of the series (but there seem to be too many > disks for my feeling after the first go-around is finished). > > One of the many many cases where people really should indicate > what they are doing when putting pictures/movies online. > > Best regards, jj >

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[math-fun] A powerful Zero Torque Square
by Ed Pegg Jr 15 Sep '16

15 Sep '16

Consider one of the 880 order 4 magic squares. .6, 12, .9, .7, 15, .1, .4, 14, .3, 13, 16, .2, 10, .8, .5, 11 If these weights are placed on the odd-odd vertices from (-3,-3) to (3,3), then this will balance perfectly on the center point (0,0). Square each number, and it will still remain perfectly balanced. There are 48 4x4 magic squares with this property. .36, 144, .81, .49, 225, ..1, .16, 196, ..9, 169, 256, ..4, 100, .64, .25, 121 Cube each number, and this magic square is still perfectly balanced. And the same when the numbers are raised to the 4th powers. (But not the 5th or 6th powers) This is the only magic square that remains balanced at powers 3 and 4. .216, 1728, .729, .343, 3375, ...1, ..64, 2744, ..27, 2197, 4096, ...8, 1000, .512, .125, 1331 --Ed Pegg Jr

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[math-fun] thermodynamic question
by Henry Baker 15 Sep '16

15 Sep '16

[This is a thermo question, not a climate troll.] California just passed energy efficiency requirements for computers. 1. If I have an inefficient computer that requires twice as much power to run a particular computation as another, "efficient" computer, then almost certainly my inefficient computer is "worse for the environment" than an efficient computer. 2. What if, on the other hand, I power my "inefficient" computer using solar power ? To avoid grid issues, I'll power my inefficient computer directly from a solar panel so that it isn't connected to the grid at all. The sunlight is going to be eventually converted to heat whether my computer is there or not; so what difference does it make?

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[math-fun] Draft of my September 2016 blog post
by James Propp 15 Sep '16

15 Sep '16

I started writing a new draft titled "Going Negative" and would love to get your feedback. I plan on publishing it on the 17th and will continue to tinker with it between now and the weekend. Is it too long? If so, what passages would you suggest I compress, or cut? I'm not an historian. Do I get my math history wrong, either blatantly (with mis-statements of fact) or subtly (by misplacement of emphasis or other forms of distortion)? I'm still confused about the differences between Frend's position and De Morgan's (and I'm sure the difference is an important one). I'm not a pre-college teacher. Do I misunderstand the pedagogical issues posed by negative numbers at the pre-college level? Suggestions for references, and comments of all kinds, are welcome. I appreciate candid criticism from people who are sympathetic to my aims but think I've fallen short of them. My prose style is designed to make the material accessible, but my breeziness should not be interpreted as indicating indifference to scholarly correctness. Please leave your feedback at https://mathenchant.wordpress.com?p=1196&shareadraft=57d777dd7094f keeping in mind that the comments of EVERYONE on math-fun will get funneled through the same email conduit; if you want me to know who you are, please sign your comments! If you sign your comments I'll assume (unless you indicate otherwise) that you don't mind my acknowledging your contribution. Title: Going Negative Beginning: Minus times minus equals plus / The reason for this we will not discuss. --- W. H. Auden, recalling a popular verse from his school days Ever tried mixing together your two least favorite foods? I suspect you haven't. Nobody mixes two noxious ingredients and expects the results to be tasty... Read more: https://mathenchant.wordpress.com?p=1196&shareadraft=57d777d d7094f Thanks, Jim Propp

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[math-fun] gyroelongated bifastigium
by Bill Gosper 14 Sep '16

14 Sep '16

is nonconvex, but the name is irresistible. gosper.org/gyroelongatedbifastigium.png Resistince is futile. --rwg

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[math-fun] Packings of Regular Pentagons in the Plane
by Hans Havermann 14 Sep '16

14 Sep '16

The pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice (formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons) has now been proved. http://arxiv.org/abs/1602.07220

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[math-fun] Although there's nothing wrong with my car's 3rd gear,
by Bill Gosper 14 Sep '16

14 Sep '16

I never use it when driving home along my street. I'm always in either 2nd or 4th until I reach my driveway. The nearest traffic light is miles away. Explain. --rwg

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Re: [math-fun] Although there's nothing wrong with my car's 3rd gear,
by Bill Gosper 13 Sep '16

13 Sep '16

On 2016-09-13 16:02, Tom Karzes wrote: > Ok, second attempt: Could you be at the bottom of a hill, using 2nd > for braking when there are cars ahead of you? > > Tom > Partial credit. (I should have added that there's almost never a car in front of me. That could get me into 3rd, but I don't think it's ever happened.) Hilarie's been on my street and knows the answer. And NeilB for sure--from his driving lesson there! My street branches steeply off another steep street (with a stop sign at the top). Approaching from the top, there is a rollercoaster down, slightly left onto my street, then suddenly up. I shift straight from 1st to 4th on the down, shoot onto my street, and roll over the hill, which is shallow near my house. Approaching my street from the downhill side entails a sharp, steep right--a bit exciting even in 2nd gear--leaving inadequate rpm to upshift afterward. Near the inside curb there are several feet of nearly 20% grade. I'm trying to perfect a slinky that will function there. --rwg > Bill Gosper writes: > > I never use it when driving home along my street. I'm always in either 2nd > > or 4th until I > > reach my driveway. The nearest traffic light is miles away. Explain. > > --rwg

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[math-fun] Eight prime square theorem?
by David Wilson 12 Sep '16

12 Sep '16

Lagrange's four square theorem states that every nonnegative integer is a sum of four squares. Empirically, it looks as if every nonnegative integer is a sum of 8 squares of primes (counting 1 as prime). It looks as if arbitrarily large numbers require 8 terms.

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