Smallest multiplicative magic squares

[Ed Pegg Jr]

Here's a tantalizing sequence: 6^3, 7!, 9!

Smallest multiplicative constants for NxN squares.

10 24  4 42  9
28 54  5 16  3
  8  2 21 36 30
27 20 48  1 14
  6  7 18 15 32
magic product 9! = 362880 = 2^7.3^4.5.7
Found by Michael Kleber
Dudeney found a 5x5 with constant 60,466,176.

Fully magic 4x4
  1 12 30 14
10 42  3  4
21  5  8  6
24  2  7 15
magic product 7! = 5040
Found by Michael Kleber.  Possibly found much earlier.

Fully magic 3x3
2  9 12
36 6  1
3  4 18
magic product 216 = 6^3 (published by Boris Kordiemski).
In 1667 (a long time ago...), Arnauld gave a 3x3 multiplicative
square with powers of 2.
Also studied by Benjamin Franklin

The semimagic squares are also interesting.

Semimagic 4x4
  1 27  8 20
10 16  9  3
24  2 15  6
18  5  4 12
magic product 4320
Found by Ed Pegg Jr after David Wilson's analysis gave 4320 as
the only possible constant smaller than 5040.  The diagonals cannot
be made to work with 4320.

Semimagic 3x3
  1 20  6
12  2  5
10  3  4
magic product 120
Found by Christian Boyer

Minimality of K = 216 for 3x3 mulgic square (Richard Schroeppel):
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.

Michael Kleber note:
In fact the number of 4x4 magic squares with given magic
sum was calculated by Beck, Cohen, Cuomo, and Gribelyuk, in an
article in the Monthly (110, no. 8 (2003); the preprint is math.CO/0201013,
and came out just a week and a half before the Ahmed, De Loera, and
Hemmecke paper I mentioned before).  It's in the EIS, of course: A093199.
It's the value of one of two degree-7 polynomials, depending on whether
the magic sum is even or odd.  And the ADH paper gives a generating fn.

Edwin Clark note:
I'm not sure, but Googling "multiplicative magic square" gives 140 hits.
Several of which seem to be interesting.

for example: here is a 3x3 and a 4x4 at
http://en.wikipedia.org/wiki/Magic_square#Other_operations
with multiplicative constants 216 and 6720 respectively.

The question is raised at
http://www.math.duke.edu/~jose/math149/prob12.pdf
whether or not there is an nxn multiplicative magic square using the
numbers 1 to n^2.

http://math.washcoll.edu/resources/senior-oblig-problems.pdf
contain a problem: show that a 3x3 multiplicative magic square has 
common product = a perfect cube.

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There is also the addition-multiplication magic square:
http://mathworld.wolfram.com/Addition-MultiplicationMagicSquare.html

Is the 5x5 case minimal?  Can someone find the minimal 6x6?  Is this
an already well-traveled road?

Ed Pegg Jr