September 2005 - mathfun@mailman.xmission.com

[math-fun] Additive & Multiplicative Magic Squares
by Schroeppel, Richard 23 Sep '05

23 Sep '05

There's an interesting oblique way to look at these. If a square is mul(tiplicatively ma)gic, then taking logs gives you a regular magic square (albeit with transcendental entries). Equally, looking at the exponents of particular primes also gives a magic square. So the log-square falls in the class of squares that are (additively) magic, and also magic if a function F() is applied to the cells. Simpler examples have F() = squaring; there may be examples with both squaring & cubing. --- The 3x3 magic squares are parameterized as K+a K-a-b K+b K-a+b K K+a-b K-b K+a+b K-a I don't know of a pretty parameterization for the 4x4s. You can pretty much pick any 7 cells (with no lines or other forced totals), and the sum S. a b c - d e f - g - - - - - - - Just fill in the other values to make the total right. Remember the sum of the 4 corners is also S. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of David Wilson Sent: Friday, September 23, 2005 8:33 AM To: Michael Kleber; math-fun Subject: Re: [math-fun] Multiplicative Magic Squares Nice argument. Can we satisfy a+b+c = abc two different ways as a first step towards a 4x4? ----- Original Message ----- From: "Michael Kleber" <michael.kleber(a)gmail.com> To: "math-fun" <math-fun(a)mailman.xmission.com> Sent: Friday, September 23, 2005 10:02 AM Subject: Re: [math-fun] Multiplicative Magic Squares David Wilson wrote: > Can a 3x3 magic square be both additively and multiplicatively magic? Not even close, even if you don't try for diagonals. There's at most one solution to {x+y=S, xy=P} for any S and P -- as we were taught when we learned to factor polynomials -- and any z in a both-ways-magic square would force its row-mates and column-mates to be two solutions to such a system. Nice work on the possible 4x4 magic products, by the way... --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] 177 million tons of ice???
by Michael D Beeler 23 Sep '05

23 Sep '05

Henry Baker> What makes D2O toxic? I wondered too! Excerpts from the top 2 Google results for "heavy water toxicity" are below. -- Mike ------------------------------------------------ >If I recall correctly, there was an experiment done some years ago with >mice and deuterium. The mice were fed normal food, but given pure deuterium >instead of, "water". As I remember, the mice died of dehydration. >The cells were unable to use the deuterium. Not the kind of dehydration you're probably thinking of, though. When body deuterium reaches about 50%, it inhibits mitosis because spindle microtubules won't form (some hydrogen bond effect inhibiting self-polymerization, I think). So all eucaryotic cells are poisoned at about these concentrations, or a little higher (bacteria can survive full deuteration -- they just grow half as fast). The consequences of failure of cell division for an intact animal like a rodent, are somewhat like those of radiation or chemo -- the bone marrow and gut lining cells suffer. Animals die of infection or diarrhea. That might make a good cerebral mystery for a student text: a person with full control of an elderly relative deuterates them to death over a couple of weeks to a month (which would take about 50 kg of heavy water at $300 a kg, but nevermind). Forensics thinks the person died of some strange disease or poison, but can't identify one (since they don't routinely do any tests to identify D!). The detective only recognizes something's amiss when the bad guy offers him ice in his lemonade, and has forgotten to dump the ice in the icebox, which is made from heavy water. Detective notices the ice SINKS in his glass. Hmmmm. Steve Harris, M.D. ------------------------------------------------ Despite the fact the light water and heavy water are chemically identical, heavy water is mildly toxic. How can this be? Since heavy water is heavier than normal water, the speed of chemical reactions involving it is altered somewhat, as is the strength of some types of bonds it forms. This affects certain cellular processes, notably mitosis, or cell division, due to the difference in binding energy in the hydrogen bonds needed to make certain proteins. Mouse studies have shown that drinking only heavy water along with normal feed eventually causes degeneration of tissues that need to replenish themselves frequently, and leads to cumulative damage from injuries that don't heal as quickly. One study likens the effects to those suffered by chemotherapy patients. Heavy water toxicity manifests itself when about 50% of the water in the body has been replaced by D2O. Prolonged heavy water consumption can cause death. Don't get any funny ideas about using heavy water as a virtually untraceable and undetectable murder weapon, though. Given its role in breeder reactors for producing weapons-grade plutonium, production and distribution of heavy water is closely monitored and controlled. Obtaining a significant amount is damn near impossible for the average Joe, and you'd need a LOT of it to kill anyone. It's also expensive--one estimate puts the price at about $300 per kilogram. Hit 'em over the head with a bottle of Poland Spring and save yourself some grief. --SDSTAFF Q.E.D. Straight Dope Science Advisory Board

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[math-fun] Sum-Product Magic Square
by David Wilson 23 Sep '05

23 Sep '05

I wrote a brute force program to generate 3x3 magic product squares. The findings lead me to conjecture that these squares, up to rotation and reflection, are of the form ac bbc aabc aabbc abc c bc aac abbc where a, b, c are chosen so as to make all cells distinct. If this is true, then a 3x3 magic square with all cells distinct cannot be at once additively and multiplicatively magic. I imagine the additive 3x3 magic squares have been fully parameterized, I suppose a proof should have been attempted from that tack. -------------------------------- - David Wilson

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RE: [math-fun] 177 million tons of ice???
by Schroeppel, Richard 23 Sep '05

23 Sep '05

Remember that the density of water is maximal at 4C, which, by coincidence (?) is the melting point of D2O. Depending on various temperatures, a D2O ice cube could pick up a coating of H2O or HDO ice, which might refloat it. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com]On Behalf Of Michael Kleber Sent: Wednesday, September 21, 2005 12:04 PM To: math-fun Subject: Re: [math-fun] 177 million tons of ice??? Tom Knight meant to write: > The molecular size and charge distribution is almost identical, so the > crystals form without distinguishing between H2O and DHO. > It would be rare to find any significant number of molecules of D2O. [I corrected a DH2O to DHO, per later conversation.] Let's be clear here: heavy water is indeed D2O. You can't have a sample of pure, um, welterweight water, DHO: the hydrogen atoms are exchanged between water molecules all the time, so you'd really have a sample with half DHO and a quarter each H2O and D2O. Per Steve Gray's density calculation, you would need about (1/.914 - 1)/(20/18 - 1) = "85% heavy" water to make ice the density of normal liquid water. -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] 177 million tons of ice???
by Schroeppel, Richard 22 Sep '05

22 Sep '05

What's D2O cost? I imagined some off-the-wall applications (safer swimming pool, extending the ski season, rocket fuel). Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of Tom Knight Sent: Thursday, September 22, 2005 2:26 PM To: math-fun Subject: Re: [math-fun] 177 million tons of ice??? I can report experimentally that indeed D2O ice sinks. In addition, I made ice from a 17% H2O + 83% D2O solution, which was almost neutrally buoyant (I was off a bit, and it sank, but clearly was very close to buoyant). On Sep 21, 2005, at 2:11 PM, Schroeppel, Richard wrote: > Remember that the density of water is maximal at 4C, which, by > coincidence (?) is the melting point of D2O. > Depending on various temperatures, a D2O ice cube could pick up a > coating of H2O or HDO ice, which might refloat it. > > Rich > > -----Original Message----- > From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com > [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com]On > Behalf Of Michael Kleber > Sent: Wednesday, September 21, 2005 12:04 PM > To: math-fun > Subject: Re: [math-fun] 177 million tons of ice??? > > > Tom Knight meant to write: > >> The molecular size and charge distribution is almost identical, so >> the crystals form without distinguishing between H2O and DHO. >> It would be rare to find any significant number of molecules of D2O. > > [I corrected a DH2O to DHO, per later conversation.] > > Let's be clear here: heavy water is indeed D2O. You can't have a > sample of pure, um, welterweight water, DHO: the hydrogen atoms are > exchanged between water molecules all the time, so you'd really have a > sample with half DHO and a quarter each H2O and D2O. > > Per Steve Gray's density calculation, you would need about > (1/.914 - 1)/(20/18 - 1) = "85% heavy" water to make ice the density > of normal liquid water. > > -- > It is very dark and after 2000. If you continue you are likely to be > eaten by a bleen. > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Mulgic Squares
by Schroeppel, Richard 22 Sep '05

22 Sep '05

Minimality of K = 216 for 3x3 mulgic square: Proof sketch: K must be a cube, with at least 9 divisors. 1,8,27,64,125 have 1,4,4,7,4 divisors. Done. -----Original Message----- From: Schroeppel, Richard Sent: Thursday, September 22, 2005 10:07 AM To: 'math-fun' Cc: Schroeppel, Richard; 'rcs(a)cs.arizona.edu' Subject: RE: [math-fun] Multiplicative Magic Squares There's a standard proof that the center of a 3x3 magic square is K/3, where K is the row sum. (Add up the four lines through the center, subtract the whole square.) Of course this works for multiplication too, so the magic product is always the cube of the center. There's a collection of similar results for a 4x4 square, such as "the sum of the four corners is K" etc. Although the arguments offered so far are plausible, I think a bit more proof is in order before claiming minimality of the various squares. Maybe a program? There aren't that many potential products < 5040 with 16 divisors, and we can also require the exponents in the prime factorization form a descending staircase. Allowing 0 as an entry doesn't seem helpful. I suspect that allowing negatives will turn out to be equivalent to the positive case with duplications allowed (but not triplications & up). But I don't see the proof that minus signs can always be attached to a positive square in a consistent way. Rich rcs(a)cs.arizona.edu -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com] On Behalf Of Christian Boyer Sent: Thursday, September 22, 2005 9:02 AM To: 'math-fun' Subject: RE: [math-fun] Multiplicative Magic Squares Of course replace "magic sum" by "magic product", twice in my previous message... -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr(a)mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : jeudi 22 septembre 2005 16:30 À : 'math-fun' Objet : RE: [math-fun] Multiplicative Magic Squares In his old book on recreative mathematics, Boris Kordiemski described exactly the Michael's method (using two latin squares) for 4x4 multiplicative squares. He also described a method for 3x3 multiplicative squares: a b² a²b a²b² ab 1 b a² ab² With a=2 and b=3, you get 2 9 12 36 6 1 3 4 18 Magic sum = 216 (*). It is the smallest possible multiplicative square. In 1667 (a long time ago...), Arnauld gave a 3x3 multiplicative square. Perhaps the oldest published multiplicative square? Using powers of 2, its magic sum was bigger : 4096. Christian. (*) 216 is also the smallest cube being also sum of three cubes. 216 = 6^3 = 3^3 + 4^3 + 5^3 _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] Multiplicative Magic Squares
by Torgerson, Mark D 22 Sep '05

22 Sep '05

Replacement goes a long way. You may take all the 3s in the square below and turn them into 29s. Or some other square having large primes may be replaced with a square with smaller corresponding primes. Even primes within a particular square may be switched. 55440=2^4*3^2*5*7*11 so you can make a corresponding square with 2*3^2*5*7*11^4. No matter the starting square, this reduction leads to some sort of minimal representation, where the factorization of the product gives 2 with the largest exponent, 3 the next and so on. -----Original Message----- From: math-fun-bounces+mdtorge=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com] On Behalf Of Michael Kleber Sent: Wednesday, September 21, 2005 2:32 PM To: math-fun Subject: Re: [math-fun] Multiplicative Magic Squares Just blathering off the top of my head: Ed pegg wrote: > Are multiplicative magic squares well studied? For example, using > dots merely as spacers, the following has a multiplicative constant of > 55440. > > 231 .40 ..3 ..2 > ..6 ..1 132 .70 > ..4 .21 .20 .33 > .10 .66 ..7 .12 For any prime p, this must still be multiplicatively magic if you replace each entry with the largest power of p dividing it. And that must be additively magic if you take its log_p. For instance, for p=2, the above becomes 0 3 0 1 1 0 2 1 2 0 2 0 1 1 0 2 with additive constant 4, the power of 2 in 55440. Of course, this projection destroys distinctness of entries. For that you just need to find another one like the above such that the ordered pairs of corresponding entries (a,b) are all distinct, and then make a multiplicative one out of 2^a 3^b. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] Multiplicative Magic Squares
by Schroeppel, Richard 21 Sep '05

21 Sep '05

Start with the old 4x4 magic square construction 1 - - 4 - 15 14 - - 6 7 - merged with 12 - - 9 - 10 11 - 8 - - 5 13 - - 16 - 3 2 - Subtract 1 (to make the range 0-15) and then take binary bit planes. This gives the patterns --xx -x-x -xx- -xx- xx-- x-x- -xx- x--x xx-- x-x- x--x -xx- --xx -x-x x--x x--x and each row, column, and diagonal has two xs. Stacking the four patterns gives all 16 possible Z combinations. Assign the primes 2,3,5,7 to the four patterns. Multiply the pattern entries, getting all the divisors of 210 as the square entries. The magic product is 210^2 which is 44100. We can lower the product a little by using 4 in place of 7; the product is then 14400. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com]On Behalf Of ed pegg Sent: Wednesday, September 21, 2005 3:50 PM To: math-fun Subject: RE: [math-fun] Multiplicative Magic Squares If you just use powers of 2, the minimal square with distinct entries has multiplicative constant 1073741824 (2^30). 55440 is much smaller. Is it the smallest? --Ed Pegg Jr --- "Torgerson, Mark D" <mdtorge(a)sandia.gov> wrote: > Replacement goes a long way. You may take all the 3s in the square below > and turn them into 29s. Or some other square having large primes may be > replaced with a square with smaller corresponding primes. Even primes > within a particular square may be switched. 55440=2^4*3^2*5*7*11 so you > can make a corresponding square with 2*3^2*5*7*11^4. No matter the > starting square, this reduction leads to some sort of minimal > representation, where the factorization of the product gives 2 with the > largest exponent, 3 the next and so on. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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RE: [math-fun] Multiplicative Magic Squares
by Torgerson, Mark D 21 Sep '05

21 Sep '05

Just looking and hoping my finger arithmetic is correct. In the 55440 square, I think you can turn 11s into 3s and the 7s into 5s giving 2^4*3^3*5^2=10800. -----Original Message----- From: math-fun-bounces+mdtorge=sandia.gov(a)mailman.xmission.com [mailto:math-fun-bounces+mdtorge=sandia.gov@mailman.xmission.com] On Behalf Of ed pegg Sent: Wednesday, September 21, 2005 3:50 PM To: math-fun Subject: RE: [math-fun] Multiplicative Magic Squares If you just use powers of 2, the minimal square with distinct entries has multiplicative constant 1073741824 (2^30). 55440 is much smaller. Is it the smallest? --Ed Pegg Jr --- "Torgerson, Mark D" <mdtorge(a)sandia.gov> wrote: > Replacement goes a long way. You may take all the 3s in the square > below and turn them into 29s. Or some other square having large primes > may be replaced with a square with smaller corresponding primes. Even > primes within a particular square may be switched. > 55440=2^4*3^2*5*7*11 so you can make a corresponding square with > 2*3^2*5*7*11^4. No matter the starting square, this reduction leads to > some sort of minimal representation, where the factorization of the > product gives 2 with the largest exponent, 3 the next and so on. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] 177 million tons of ice???
by Steve Rowley 21 Sep '05

21 Sep '05

>Date: Wed, 21 Sep 2005 13:46:23 -0700 (PDT) >From: Eugene Salamin <gene_salamin(a)yahoo.com> > >--- Steve Rowley <sgr(a)alum.mit.edu> wrote: >> Just put a magnet in your water bottle and you're ready to sell >> "ortho-water bottles", since the magnetic field will make the ortho >> polarization energetically favored. Slap a little sermon on the side >> about statistical thermodynamics and the magical incantation >> "exp(-uH/kT)", and it might even become trendy. "Refrigerate for >> higher polarization", that sort of thing. > >This is good marketing. It combines magnets with "structured water". >Now throw in homeopathy too. Homeopathy? Wow, that's even more twisted than I could think of... :-) >> You could even sell "after-market ortho-water bottle adapters" (aka >> magnets), to tape onto existing water bottles. > >No, no, no, disposibles are where the money is. Right, disposable magnets! Even better. (See? Don't ever let anybody tell you peer review doesn't work.) >> Strangely enough, spin-polarized hydrogen has occasionally been >> mentioned as a very dense energy storage mechanism. Its instability >> makes it more of a bomb than a gas tank, alas. > >This is electron spin, not nuclear spin. Hydrogen atoms with opposite >electron spin can bind to form a molecule, whereas atoms with the same >spin cannot. So by polarizing a gas of atoms, giving them all the same >spin, you keep them in a high energy state. I know; I was joking. A bit of Googling ressurects the fact that it was Tom Greytak and Dan Kleppner at MIT who were doing this back in the 80's: http://web.mit.edu/physics/greytak-kleppner/ -- Steve Rowley <sgr(a)alum.mit.edu> http://alum.mit.edu/www/sgr/ ICQ: 52-377-390

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