COUNTING SEVEN BY SEVEN ‘MAGIC’ SQUARES
Here is a really perfect square. I have decided to call these exceptionally perfect squares (EP)!
Not only do all rows and columns sum to 175, but so do all diagonals and all blocks of seven in a playing card pattern! E.g 16+7+40+25+10+43+34 = 175. See the template illustrated next below to see where these squares are found. I am still searching for more of these EP squares. Let me know if you know of any! email@example.com
Below is illustrated part of the reference pattern for generating all the squares in each family. An attempt has been made to remove all duplicates at this stage, but a checking system will hopefully eliminate other pattern repeats, and any unrecognised duplication routes. Glenn Duff generated a unique hexadecimal number for each square of low order, but it was not perfect. It failed at a seven x seven square because it was based on a false initial assumption – my fault!
In the template below A with B shown as coloured squares:-
Comparison of squares shows that there are many duplications involving rotations and reflections. Elimination of these shows that there are 777,600 Perfect squares of which 259,200 are exceptionally perfect as illustrated above. The green shading shows established repeats, there thus appear to be only 3,758,400 Imperfect squares including all the arrangements possible from each of the basic squares. Or 648,000 different imperfect basic arrangements.
For a different slant on eliminating duplicates see my article on “Generating and Counting Magic Squares”, Google finds it if you put my name in the search. Viz:- Generating and Counting Magic Squares Taudevin For those who would like to see some results from Glen Duff’s Square generating programme, below are illustrated various types of square with their metrics. The Metrics show that changing the number sequence does not always produce the same result in the metrics; this enables some patterns to be eliminated early and means that large squares can be analysed with less effort (I have yet to persuade Glenn to re-write the programme to do this).
The sequence will follow the Knight (2,1) pattern in the NNE direction. The continuation sequence will follow the Extended Knight (7,1) pattern in the NNE direction. This pattern does not produce magic squares as the R column (Rows) in the metrics does not sum to the required value. You will see that the metric is constant at 14.0, but the rows sum to differing values from a fixed selection! (I feel this may be interesting!!)