One thing that doesn't appeal to me in the usual definition of a magic square (an nxn matrix of distinct integers such that the sum of each row, column, *and main diagonal* must be equal) is that no permutation of the matrix elements can carry a row or column onto a main diagonal. (Except in the 2x2 case, which admits no solutions even without the main diagonals rule.)
SO: What I'd find much more interesting is the following:
Let a "magic template" T = (K,S) "of shape (r,s)" denote a set K of r unknowns R_i, 1 <= i <= r, and a set S of subsets S_j of K, each of size s, such that:
* for any two of the subsets, there is a permutation of K carrying one onto the other.
* a "magic pattern" (nomenclature pending) modeled on T is defined as an assignment of distinct integers to the K(i)'s such that each subset S_j has the same sum.
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The more symmetric the configuration of subsets, the more appealing the the magic template.
QUESTION: What are good ways to find highly-symmetric magic templates,
especially ones that have at least one magic pattern?
(I guess the cases where the number of unknowns is no smaller than the number of subsets -- i.e., equations -- have the best chance of being realizable.)
Some candidates might include
a) Use an n-dimensional cube of side L (so L^n unknowns) and its C(n,d)*L^(n-d) e-dimensional slices (e-cubes of size L^e) parallel to the coordinate axes.
b) Use n unknowns and all subsets of size s -- the subsimplices of fixed dimension of an (n-1)-simplex.
c) If these don't work, then perhaps highly-symmetrical subsets of the sets of subsets in a) or b).
Are there standard ways of finding such things?
--Danne thing that doesn't appeal to me in the usual definition of a magic square (an nxn matrix of distinct integers such that the sum of each row, column, *and main diagonal* must be equal) is the main diagonals rule.
One way to think of a magic square template is as a set of n^2 integer unknowns K_(i,j), 1 <= i,j <=n required to all be unequal, and having 2n+2 specified subsets (of size n) required to all have the same sum.
Clearly, for any two row or column subsets S_1, S_2, there's at least one permutation of the K(i,j)'s carrying S_1 onto S_2. But (except for the 2x2 case -- which is magic-squarefree even without the main diagonals rule) there is no permutation of the K(i,j)'s taking a row or column onto a main diagonal.
This lack of symmetry is what I find less than interesting about traditional (ordinary) magic squares.
SO: What I'd find much more interesting is the following:
Let a "magic template" T = (K,S) "of shape (r,s)" denote a set K of r unknowns R_i, 1 <= i <= r, and a set S of subsets S_j of K, each of size s, such that:
* for any two of the subsets, there is a permutation of K carrying one onto the other.
* a "magic pattern" (nomenclature pending) modeled on T is defined as an assignment of distinct integers to the K(i)'s such that each subset S_j has the same sum.
--------------------------------------------------------------
The more symmetric the configuration of subsets, the more appealing the the magic template.
QUESTION: What are good ways to find highly-symmetric magic templates, especially ones that have at least one magic pattern?
(I guess the cases where the number of unknowns is no smaller than the number of subsets -- i.e., equations -- have the best chance of being realizable.)
Some candidates might include
a) Use an n-dimensional cube of side L (so L^n unknowns) and its C(n,d)*L^(n-d) e-dimensional slices (e-cubes of size L^e) parallel to the coordinate axes.
b) Use n unknowns and all subsets of size s -- the subsimplices of fixed dimension of an (n-1)-simplex.
c) If these don't work, then perhaps highly-symmetrical subsets of the sets of subsets in a) or b).
Are there standard ways of finding such things?
--Dan
P.S. I'm well aware that people have come up with such generalizations of magic squares before -- e.g., in Clifford Pickover's book "The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions".
