Sunday, December 18, 2016

Discovering Magic Squares

  If one looks at the Indian and Dürer's magic one can see how pairs of sums are identical. I used Excel 2016 to compute these magic squares using the 6 unknowns and show the vertical, horizontal and diagonal sums for the partitioned magic squares.

It turns out that there are a dozen sums but half of them are 34's complements of the other half, i.e., a sum and its complement add up to 34. In the Indian magic square at the bottom left we see that 4+5=19 and the same is true for the upper row of the bottom right where 7+12=19. One can fill in a magic square by selecting pairs of numbers from a table of sums and also from the sum's complement.

In first magic square below the first sums 16=1+5 and 18=8+10 were used as a starting point. By changing the order of these pairs on can change the diagonal sum for the bottom left quarter. When filling the bottom right and upper left quarters one has to pick sums for which the sums are 16 and 18 and their diagonals also match. One can place an x next to a sum that was used to eliminate it for a later selection. One can also place x's in the table to note which of the 16 numbers have already been used to eliminate sums for the diagonals. Once the 6 unknowns (marked in green) are chosen one can compute the remainder of the magic square.

The same procedure was used for the magic square on the right. Here are a few more which I did by hand a couple of weeks ago.

One does not have to be able to solve a system of linear equations to find a magic square what follows the chosen rules.

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