April 2008 - mathfun@mailman.xmission.com

[math-fun] Smallest magic squares of powers
by Christian Boyer 01 Sep '08

01 Sep '08

Oct 7th, 2007: the smallest known magic square of 4th powers is now a 44x44 square constructed by Jaroslaw Wroblewski (Poland) and Hugo Pfoertner (Germany). Jaroslaw built the semi-magic square in June, then Hugo needed a very impressive computing job of 3 months, on 57 processors of a SGI computer http://www.sgi.com/products/remarketed/origin3000/, in order to find two magic diagonals in this square! A big progress, because the previous smallest known was a 243x243 square, coming from the 243x243 tetramagic square of Pan Fengchu (China). The best known results, and their first authors, are now: Magic square Smallest known ------------ --------------- Of squares 4x4 (Euler, 1770) Of cubes 9x9 (Boyer, 2006) Of 4th powers 44x44 (Wroblewski-Pfoertner, 2007) Of 5th powers 729x729 (from the pentamagic square of Li Wen, 2003) ON ANY POWER, WE DO NOT KNOW WHAT THE SMALLEST POSSIBLE MAGIC SQUARE IS! For example, as you know, we still do not know if a 3x3 magic square of squares is possible... 4x4, 5x5, 6x6, 7x7, and 8x8 magic squares of cubes are unknown... And the problem of the diagonals is extremely difficult, as proved by the impressive computing search of Hugo. It remains a big gap between the smallest known magic and semi-magic squares (=with or without magic diagonals) of 4th powers: 44x44 versus 4x4... Here is the 4x4 semi-magic square of 4th powers (Morgenstern-Boyer, 2006): 14101^4 938^4 931^4 37762^4 35866^4 20881^4 21038^4 13393^4 24806^4 30191^4 30418^4 9263^4 413^4 32026^4 31787^4 1106^4 Christian.

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[math-fun] Morpion Solitaire
by Christian Boyer 06 Jun '08

06 Jun '08

An easy game: Only paper and pencil needed. A game for one player. And an open game: Who will improve the current world records??? Here is my new website dedicated to "Morpion Solitaire": www.morpionsolitaire.com Fully bilingual, English/French. Feel free to report any error, or any missing information. As you can read in the "Origin" page, any new information on the origin of this game will be welcomed. Some other questions. Is it possible to do best than the two grids given in "Other 5"? Or worst than my very bad grids at the end of the "Limits" page? Christian.

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[math-fun] The value of PI
by Bernie Cosell 01 May '08

01 May '08

Pi shows up a lot in geometry. I was wondering when "we" (ahem) realized that it was just one underlying constant. For example, the area of a circle is 3.14... times the square of the radius and the volume of a sphere is 4.18... times the cube of the radius. It is already profound discovering that those ratios are fixed [and to be able to calculate a bunch of digits of them]; I don't know if he did or not, but I can' easily see Archimedes managing to calculate a bunch of digits of the 4.18... constant even as he did for the 3.14. one]. But when did mathematicians realize that those weren't two separate gnarly constants, but actually "reflections" [if you will] of a single underlying constant? I don't think the Greeks had enough math machinery to figure all that out, did they? [that is, that the ratio of those two constants is exactly 3:4. Or that the ratio of the radius of a circle to its curcumferenace is really just exactly twice the ratio of the square of the radius of the circle to its area; 3.14... and 6.28... will certainly *look* like 1:3 [not clear the 3:4 ratio is as obvious..:o)], but could they actually 'know' that enough to say that that was really "proven", in some sense? If not the greeks, then when did we figure that out? /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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Re: [math-fun] The value of PI
by Dan Asimov 30 Apr '08

30 Apr '08

Bernie asks: << . . . the area of a circle is 3.14... times the square of the radius and the volume of a sphere is 4.18... times the cube of the radius. . . . But when did mathematicians realize that those weren't two separate gnarly constants, but actually "reflections" [if you will] of a single underlying constant? I don't think the Greeks had enough math machinery to figure all that out, did they? . . . >> Yes, the fact that the ratio of the volume of a sphere to the volume of the circumscribed cylinder is 2/3 was one of Archimedes' proudest achievements: This is said to be inscribed on his gravestone. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] And I suppose heavy objects fall faster than light ones?
by Eugene Salamin 30 Apr '08

30 Apr '08

----- Original Message ---- From: Steve Witham <sw(a)tiac.net> To: math-fun(a)mailman.xmission.com Sent: Wednesday, April 30, 2008 12:18:23 PM Subject: Re: [math-fun] And I suppose heavy objects fall faster than light ones? ... If everything stopped in its orbit, and was only held up by bouncing against other things (and against the Sun??) it would be like Earth's atmosphere or oceans, or a centrifuge, and then dense stuff would settle toward the center. Then, you've got the fluid pressure pushing up in proportion to size and the gravity pulling down in proportion to mass. I think if the force of gravity dropped off more slowly or more quickly, you'd have the effect of a wall pushing inward or outward, respectively, so this raises a more metaphysical question, why is gravity just so? I wonder whether slight changes in gravity's exponent either way would mean chaotic orbits that collide with each other much more often. --Steve ______________________________________________ This description, where buoyancy plays a role and denser objects settle lower, appplies to fluids. For ideal gases, each type of molecule settles to its own Boltzman distribution, independently of the other types of molecules. In this case, the stratification is according to molecular weight, not molecular density. The 1/r^2 force (and the harmonic oscillator r force) are special in that, for any given orbit, the radial oscillation period equals the angular oscillation period. If T denotes this common period, we have r(t + T) = r(t) and theta(t + T) = theta(t), so that orbits are closed. Other force laws produce nonclosed orbits, but slight deviations from Newtonian (e.g. relativistic corrections) produce only slightly nonclosed orbits. In fact, orbits in the r^1 and r^-2 force fields are related through a z --> z^2 transformation of the complex plane. This is explained in the very nice book: V. I Arnol'd, "Huygens and Barrow, Newton and Hooke". Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ

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Re: [math-fun] And I suppose heavy objects fall faster than light ones?
by Steve Witham 30 Apr '08

30 Apr '08

>From: Henry Baker <hbaker1(a)pipeline.com> > >what makes a centrifuge different from the solar system? I.e., at >what densities of planetary matter would things start working >differently? >From: "Paul R. Pudaite" <paul.r.pudaite(a)gmail.com> > >The solar system doesn't have a wall. (?) Gravity, to things in orbits, is not like a wall because the force of gravity and the momentum are both proportional to the mass. With a gas of particles of different densities, in orbit, the gas pressure will be equal from all directions and particles of different densities are happy to share similar orbits. (The pressure is related to "temperature" of the particles moving randomly relative to a lockstep orbit, and some of the energy of collision is dissipated by radiated heat... and yet they can't settle to a lockstep because rings at different distances have to go at different speeds... Does that mean everything has to become separated solid rings?) If everything stopped in its orbit, and was only held up by bouncing against other things (and against the Sun??) it would be like Earth's atmosphere or oceans, or a centrifuge, and then dense stuff would settle toward the center. Then, you've got the fluid pressure pushing up in proportion to size and the gravity pulling down in proportion to mass. I think if the force of gravity dropped off more slowly or more quickly, you'd have the effect of a wall pushing inward or outward, respectively, so this raises a more metaphysical question, why is gravity just so? I wonder whether slight changes in gravity's exponent either way would mean chaotic orbits that collide with each other much more often. --Steve

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Re: [math-fun] And I suppose heavy objects fall faster than light ones?
by Henry Baker 30 Apr '08

30 Apr '08

Lemesee... How come our solar system isn't a centrifuge? Does the solar system order the planets in terms of density? If not, what makes a centrifuge different from the solar system? I.e., at what densities of planetary matter would things start working differently? At 07:24 PM 4/29/2008, Mike Stay wrote: >It separates things by density, not mass. > >On Tue, Apr 29, 2008 at 6:50 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> Mike Speciner wrote: >> >> << >> [quote of passage from NY Times article that Mike evidently considers erroneous] >> >> >> >> Mike, I don't recall if I ever understood just how centrifuges work. Would you be so kind as to explain? >> >> --Dan

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Re: [math-fun] And I suppose heavy objects fall faster than light ones?
by Eugene Salamin 30 Apr '08

30 Apr '08

----- Original Message ---- From: Henry Baker <hbaker1(a)pipeline.com> To: math-fun <math-fun(a)mailman.xmission.com> Cc: Dan Asimov <dasimov(a)earthlink.net> Sent: Wednesday, April 30, 2008 8:26:20 AM Subject: Re: [math-fun] And I suppose heavy objects fall faster than light ones? Lemesee... How come our solar system isn't a centrifuge? Does the solar system order the planets in terms of density? If not, what makes a centrifuge different from the solar system? I.e., at what densities of planetary matter would things start working differently? ------------------------------- In a centrifuge, the gas is in an equilibrium established by collisions between molecules. The gas is everywhere all rotating at the same angular velocity as the rotor, since otherwise there would be shear between the rotor and gas, and between different regions of gas, and this shear would be dissipated through viscosity. The planets in the solar system have not yet had time to equilibrate through their gravitational interactions. Gravitationally bound systems do not possess a state of thermodynamic equilibrium; some parts will acquire escape velocity and go off to infinity, and the rest will collapse into the center. Gene ____________________________________________________________________________________ Be a better friend, newshound, and know-it-all with Yahoo! Mobile. Try it now. http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ

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Re: [math-fun] And I suppose heavy objects fall faster than light ones?
by Dan Asimov 30 Apr '08

30 Apr '08

Mike Speciner wrote: << [quote of passage from NY Times article that Mike evidently considers erroneous] >> Mike, I don't recall if I ever understood just how centrifuges work. Would you be so kind as to explain? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] And I suppose heavy objects fall faster than light ones?
by Dan Asimov 30 Apr '08

30 Apr '08

Mike Stay wrote: << It separates things by density, not mass. On Tue, Apr 29, 2008 at 6:50 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > Mike Speciner wrote: > > << > [quote of passage from NY Times article that Mike evidently considers erroneous] > >> > > Mike, I don't recall if I ever understood just how centrifuges work. Would you be so kind as to explain? >> Hmm, guess I did know that, at least intuitively. SO -- since the article discusses using centrifuges to separate isotopes of uranium in gaseous form: How does one define the density of a single molecule? (Especially considering that it has no boundary, but rather a gradually attenuating force field.) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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