Wednesday, November 9, 2016

Magic Squares

  A magic square is an n x n grid of numbers whose rows, columns and diagonals all have the same sum. One can create a magic square by filling a 4 x 4 matrix with 7 numbers in the positions below. I included the numbers of the election date, 11/8/2016, in the bottom left corner. The remaining 3 numbers were chosen at random. If one knows the value of the sums, Σ, one can immediately solve for some of the unknowns.

We can use the unused equations to separate the the known variables from the unknown variables as follows and solve the system of equations for the unknowns k, l, o & p. One of the equations is redundant so we choose 4 for which the determinant of matrix, M, is nonzero. Here the determinant D = |M| = 2.

Filling the calculated values into our magic we get the following matrix which checks out correctly.

Each value of Σ generates a new magic square. One can try to choose the minimum value for Σ for which all the numbers are non-negative.

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