Compact Magic Squares

Compact (from my Bibliography)Gakuho Abe used this term for a magic square where the four
cells of all 2x2 squares contained within it summed to 4/m of S. [1] Gakuho Abe, Fifty Problems of Magic Squares, Self published 1950. Later republished in Discrete Math, 127, 1994, pp 313. On April 15, 2007 Aale de Winkel emailed a proof stating that any {compact} square is {panmagic} for the sum of corners of even subsquares: On April 17, Aale sent a clarification; Harvey {compact}ness is a premise (startpoint) of the proof, the proof states in any {compact} square the sum of the four corners of an even subsquare sum to the same sum. I don't think the reverse is provable!. Putting it more direct you need to test a square for {compact}ness, you then know that it is panmagic for all possible figures you can form using even square corners, which means most (if not al) patterns on your Franklin page. (the statement is thus powerful. The same day, Walter Trump sent a message confirming Aale's proof. Also the same day, Donald Morris sent a message saying that corners of all rectangles (and squares) of a compact magic square summing correctly, as long as both dimensions are an even number, further confirming Aale's proof. Slightly digressing.
Then still more discussion. Woodruff pandiagonal square Woodruff (modified) Morris pandiagonal square 01 32 34 63 37 60 06 27 52 45 55 42 24 09 19 14 60 53 04 13 20 29 44 37 48 49 15 18 12 21 43 54 29 04 26 07 57 40 62 35 06 11 62 51 46 35 22 27 19 14 52 45 55 42 24 09 58 39 61 36 30 03 25 08 61 52 05 12 21 28 45 36 62 35 29 04 26 07 57 40 23 10 20 13 51 46 56 41 03 14 59 54 43 38 19 30 25 08 58 39 61 36 30 03 44 53 47 50 16 17 11 22 63 50 07 10 23 26 47 34 56 41 23 10 20 13 51 46 05 28 02 31 33 64 38 59 01 16 57 56 41 40 17 32 11 22 44 53 47 50 16 17 34 63 37 60 06 27 01 32 58 55 02 15 18 31 42 39 38 59 05 28 02 31 33 64 15 18 12 21 43 54 48 49 08 09 64 49 48 33 24 25
After checking Aales letters, I thought maybe the outside dimensions can be
anything, as long as the total number of cells used is a multiple of 4. Of
course, as Aale explains, the shape can be considered to be in an array of even
dimensions filled with required blank cells. Available for download is my Compact_8MS.xls. I used it to check out and compare the above 3 squares, plus an order 8 Franklin pandiagonal magic square by Peter Loly. The Loly square is similar to the above, with the extra feature that correct half rows start on all odd columns.
Shapes tested in above squares The letter patterns are some of those suggested by Aale de
Winkel in an email of April 20, 2007.
