Once finding that these triads exist, then in order to see if a magic square of squares exist, we have to check the differences between the highest member of i., ii. or iii. In this example, we find that the difference between i. and ii. So seven out of the eight conditions are satisfied. In light of the 16 Semi-Magic squares found above, it becomes interesting to conjecture that perhaps a Magic Square of all squares might exist. After all, there are an infinite number of Semi-Magic squares.
Further computer searching for more semi-magic squares is in progress.
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