Re: [math-fun] 6x6 pandiag. multiplic. squares ARE NOT impossible!
Michael Kleber's latest brilliant analysis of Christian's new multiplicative magic square -- in terms of bit planes -- is quite reminiscent of (what I think it) Conway & Coxeter's analysis of additive magic squares as it appears in (some of) the most recent edition(s) of W.W. Rouse Ball's "Mathematical Recreations and Essays" (updated by Coxeter). Let N := the set of nonnegative integers. Then {additive magic squares of elements of N^k for some k} is in 1-1 correspondence with {multiplicative magic squares} -- just letting the components of v in N^k correspond to prime exponents. I was wondering if there could be a multiplicative magic square that was somehow not a consequence of additive ones . . . but now I see that's impossible. --Dan
Dan Asimov wrote:
Then {additive magic squares of elements of N^k for some k} is in 1-1 correspondence with {multiplicative magic squares} -- just letting the components of v in N^k correspond to prime exponents.
I specifically thought this was *not* the case. In additive magic squares, for example, 1+1 = 2+0 -- well, that's a bad example since there's a duplicated number, but eg 5+1 = 2+4. But if you're trying to use the bits as bases for prime exponents, this is invalid: in binary you have 101 + 001 = 010 + 100, but if you count the bits on each side you get 102 versus 110. This is was I was fumblinly addressing with my comments about things being magic even if you forget how to carry. I'm not familiar with the Conway/Coxeter analysis Dan refers to, so perhaps there's a deep reason why normal magic squares -- well, some restricted subset of them -- automatically works on a bit-by-bit level. (Clearly not, say, for the 3x3 magic square on 0-8 -- there's only one entry with the 8s bit turned on!) I would guess that understanding when this happens, if it's not already known, is probably quite interesting. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
Dan, you are right, there is a great relationship between additive and multiplicative squares (and cubes), but not always a direct relationship. For example the beauty of the multiplicative squares and cubes is to find the smallest possible entries and the smallest possible products, and -as far as I know- there is no direct way to find this, using additive squares. That's the open problems of my 2 tables at www.multimagie.com/English/Multiplicative.htm and www.multimagie.com/English/MultiplicCubes.htm where the max nbs and products are the "smallest known". Only few of them are proved to be the smallest possible. I am sure that numerous results of these tables can be improved. Another problem: pandiagonal additive squares (normal magic square, using consecutive integers) of orders 4k+2 are impossible, as first proved by Frost in 1878. But we know now that 6x6 and 10x10 pandiagonal multiplicative square are possible. Because normal pandiag. squares of such orders cannot exist, for sure we can't use them for pandiag. multiplicative squares! Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de dasimov@earthlink.net Envoyé : dimanche 23 avril 2006 16:48 À : math-fun Objet : Re: [math-fun] 6x6 pandiag. multiplic. squares ARE NOT impossible! Michael Kleber's latest brilliant analysis of Christian's new multiplicative magic square -- in terms of bit planes -- is quite reminiscent of (what I think it) Conway & Coxeter's analysis of additive magic squares as it appears in (some of) the most recent edition(s) of W.W. Rouse Ball's "Mathematical Recreations and Essays" (updated by Coxeter). Let N := the set of nonnegative integers. Then {additive magic squares of elements of N^k for some k} is in 1-1 correspondence with {multiplicative magic squares} -- just letting the components of v in N^k correspond to prime exponents. I was wondering if there could be a multiplicative magic square that was somehow not a consequence of additive ones . . . but now I see that's impossible. --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Christian Boyer -
dasimov＠earthlink.net -
Michael Kleber