[math-fun] Frost visits the Gulf on a blustery evening
A front-page Sat., Sept. 17 story on the NY Times website, found at http://snipurl.com/hqxi, claims that FEMA has delivered 177 million tons of ice to Louisiana, Mississippi and Alabama in the aftermath of hurricane Katrina.
Some say the Gulf will end in water, Some say in ice.
From what I've witnessed of Katrina, I hold with those who favor FEMA. But if it had to perish twice, I think I know enough of weight, To say that for destruction ice Is also great And would suffice.
Hilarie, With apologies to RF.
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 Is there a list of smallest multiplicative constants for various NxN squares somewhere? Ed Pegg Jr
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.
On 9/21/05, Michael Kleber <michael.kleber@gmail.com> wrote:
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.
Well, 2^a q^b anyway, where q is the prime that makes them all distinct.
--Michael Kleber
-- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
BTW, any additive magic square can be turned into infinitely many multiplicative ones just by exponentiating by a common base. On 9/21/05, ed pegg <ed@mathpuzzle.com> 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
Is there a list of smallest multiplicative constants for various NxN squares somewhere?
Ed Pegg Jr
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
From ed pegg Is there a list of smallest multiplicative constants for various NxN squares somewhere?
Back to the original question asked by Ed, here is the answer! You will find below the 10 smallest multiplicative constants for 3x3 and for 4x4 MAGIC squares. Ed, I can provide you a longer list if you want. For SEMI-MAGIC squares (not taking care of the diagonals), the smallest magic product is 120 for 3x3 with for example 1 20 6 12 2 5 10 3 4 with 3 magic rows, 3 magic columns (and 1 magic diagonal!!... but it was not asked for semi-magic squares), and is 4320 for semi-magic 4x4 as we have seen last week. Best regards. Christian. ---- 3x3 multiplicative squares Prod. = 2^ * 3^ * 5^ * 7^ * 11^ 216 3 3 0 0 0 1000 3 0 3 0 0 1728 6 3 0 0 0 2744 3 0 0 3 0 3375 0 3 3 0 0 4096 12 0 0 0 0 5832 3 6 0 0 0 8000 6 0 3 0 0 9261 0 3 0 3 0 10648 3 0 0 0 3 ---- 4x4 multiplicative squares Prod. = 2^ * 3^ * 5^ * 7^ * 11^ 5040 4 2 1 1 0 6480 4 4 1 0 0 6720 6 1 1 1 0 7200 5 2 2 0 0 7560 3 3 1 1 0 7920 4 2 1 0 1 8400 4 1 2 1 0 8640 6 3 1 0 0 9072 4 4 0 1 0 9240 3 1 1 1 1 ----
Heard on the radio: Half of Louisiana is under water; the other half is under indictment. At 10:38 AM 9/21/2005, Hilarie Orman wrote:
A front-page Sat., Sept. 17 story on the NY Times website, found at http://snipurl.com/hqxi, claims that FEMA has delivered 177 million tons of ice to Louisiana, Mississippi and Alabama in the aftermath of hurricane Katrina.
Some say the Gulf will end in water, Some say in ice.
From what I've witnessed of Katrina, I hold with those who favor FEMA. But if it had to perish twice, I think I know enough of weight, To say that for destruction ice Is also great And would suffice.
Hilarie,
With apologies to RF.
participants (7)
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Christian Boyer -
ed pegg -
Henry Baker -
Hilarie Orman -
Michael D Beeler -
Michael Kleber -
Mike Stay