I have decided to publish my results for the 12x12, 14x14 and 16x16 boards here rather than go to all the trouble of writing the subject up for publication in a journal. I no longer have the patience for that frankly. These were solved in reverse order, the largest 16x16 first because I thought its division into 16 areas 4x4 would make the solution easier by use of the squares and diamonds method in each quarter, and this proved to be the case.
12x12
16x16
14x14
In the 14x14 case the sequences of squares (red) and the sequence of complements (green) have no points of intersection, but in the 16x16 case there are two (16 and 144 differing by 128), while in the 12x12 case there are four (two pairs, 9 and 81, 49, and 121, with difference of 72).
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