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## Friday, July 22, 2011

### Magic squares - Revisited

Regular readers of this blog know that I am fascinated by magic squares. Finding the 3 by 3 magic square with digits 1,2, ..., 9 can be formulated as an Integer Programming problem. An Integer Programming problem is a Linear Programming problem with the additional constraints that all variables must have integer values.
 c={1,1,1,1,1,1,1,1,1}; m={ {1,1,1,0,0,0,0,0,0}, {0,0,0,1,1,1,0,0,0}, {0,0,0,0,0,0,1,1,1}, {1,0,0,1,0,0,1,0,0}, {0,1,0,0,1,0,0,1,0}, {0,0,1,0,0,1,0,0,1}, {1,0,0,0,1,0,0,0,1}, {0,0,1,0,1,0,1,0,0}, {-1,-1,-1,0,0,0,0,0,0}, {0,0,0,-1,-1,-1,0,0,0}, {0,0,0,0,0,0,-1,-1,-1}, {-1,0,0,-1,0,0,-1,0,0}, {0,-1,0,0,-1,0,0,-1,0}, {0,0,-1,0,0,-1,0,0,-1}, {-1,0,0,0,-1,0,0,0,-1}, {0,0,-1,0,-1,0,-1,0,0}, {1,0,0,0,0,0,0,0,0}, {0,1,0,0,0,0,0,0,0}, {0,0,1,0,0,0,0,0,0}, {0,0,0,1,0,0,0,0,0}, {0,0,0,0,1,0,0,0,0}, {0,0,0,0,0,1,0,0,0}, {0,0,0,0,0,0,1,0,0}, {0,0,0,0,0,0,0,1,0}, {0,0,0,0,0,0,0,0,1}, {-1,0,0,0,0,0,0,0,0}, {0,-1,0,0,0,0,0,0,0}, {0,0,-1,0,0,0,0,0,0}, {0,0,0,-1,0,0,0,0,0}, {0,0,0,0,-1,0,0,0,0}, {0,0,0,0,0,-1,0,0,0}, {0,0,0,0,0,0,-1,0,0}, {0,0,0,0,0,0,0,-1,0}, {0,0,0,0,0,0,0,0,-1} }; b={15,15,15,15,15,15,15,15,-15,-15,-15,-15,-15,-15,-15,-15,9,9,9,9,9,9,9,9,9,-1,-1,-6,-1,-1,-1,-1,-1,-1}; LinearProgramming[-c,-m,-b] {2, 7, 6, 9, 5, 1, 4, 3, 8} 
Mathematica does find a solution in less than a second. An interesting ( Mathematica ) programming exercise would be to generate the code for solving this integer programming problem for magic squares of size n. With increasing n the number of constraints increase fast. This could become an interesting benchmark. - For a 3 by 3 magic square 34 constraints are used above ( although 32 would have sufficed, probably even less ) but these constraints can be systematically generated.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)