COUNTING FIFTEEN BY FIFTEEN ‘MAGIC’ SQUARES
By the time I reach here, the title Magic seems a bit superfluous; they are now just regular squares with fascinating properties. I contacted David Brée to say I had found a most perfect 15 x15, but he noted that I had misinterpreted his definition! So here is my first ever ‘exceptionally’ perfect square ( E with P; see reference pattern below ) for your enjoyment. There are lots and lots of them as you can imagine.
So far only one of the above family has been generated, but an estimate is, that there are 6,974 for each sequence of numbers (single yellow cells above), and at least 6,974 sequences of follow on numbers (pale green cells above) . I estimate that the number of sequences that might work is over 18,714,000, but I have along way to go and need a faster computer and a system to detect duplicates. The screen shots further on show the problem; the duplicates found are not real as they are based on earlier work with five by fives and seven by sevens and do not work perfectly for
them, but certainly work well enough at that size of square.
The yellow boxes containing 1,695 are the sums of blocks of 3 x 5 cells. Five horizontal and three down, wherever you try!! Best use a spread sheet to produce the sums!
So far there are just two families of these exceptionally perfect squares. This family, as shown above, ( which includes E w U) and the other where the square blocks are five vertical and three horizontal; shown here below (D with K). I do not think you can rotate them to match the patterns and line up the blocks!!! The patterns are very similar, and reminiscent of a Hendrix or a Margossian nine by nine square.
So far I have generated 36, of the above family of squares, in only 17,890 cycles. This means there could be over 18,714,000 in each sequence and at least 3,874 sequences.
Using Glenn Duff’s fantastic square generating programme ( which he ran for 250,000,975 iterations on one sequence set, generating only 20 perfects and 794 Imperfects); I have generated a total of 530,994 normally perfect squares (including exceptionally perfects as above) and estimate that there might be 5,185,701,553 per follow on sequence and the minimum number of follow ons as 3,874, but it is far too early to say with any real idea. Anyone wishing to help would be more than welcome to join in the quest.
This chart was an attempt to establish a figure for the average rate of generation of perfect squares, but it shows the widely diverging rates that occur in practice. If one complete set of results can be run, then an average might emerge, but the number of iterations required is too high without help. If a system can be established, then volunteers to run a part sequences would be most welcome. The lps 5 is a designation for a sequence that produced a perfect square (the fifth found) at number 44,364,840 of the iteration cycle (an algorithm dependent number).
The figure of 87,178,291,200 is the number of iterations to cover all the sequences that can be found with the number one at the centre of the square, other sequences will be repeats; it is probable that, in most cases, only half this number will be unique! (?)
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 above a seven x seven square because it was based on a false initial assumption – my fault!
The green shaded squares show proven (I hope) duplications. ‘L’ seems to be the only one below ‘H’ to have nonduplicated partners.
Within the above generating pattern there seem to be four types of square. Those that generate Very Few, Few, Many, and Lots, of Perfect squares.