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*.
*Note that this is a requirement for Ollerenshaw’s most-perfect magic
squares.

[1] Gakuho Abe,

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. 4 outside corners of a magic square summing to 4S/m does not
imply that the square is compact, or even pandiagonal! |
Trump
pandiagonal, not compact |

Then still more discussion.

Walter Trump sent an order-8 square constructed by a Mr. Woodruff in 1916.

I uncovered a square with almost similar features sent me by Donald Morris in
2005.

Aale de Winkel reported that the Woodruff square could be converted to a
bent-diagonal type square simply by shifting the square 2 rows and columns down
to the right.

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

Feature Comparison |
Woodruff |
Morris |
Woodruff (modified) |

Associated |
Yes |
No |
No |

Pandiagonal magic |
Yes |
Yes |
Yes |

Compact all 2x2 squares sum to S |
Yes |
Yes |
Yes |

Half rows and half columns = S |
Yes. Starting on 1 |
Yes. Starting on 1 |
Yes. Starting on 3rd & 7 |

Complete bent diagonals |
No – horizontal only,
starting on 3rd & 7 |
Yes – starting on 1 |
Yes – starting on 1 |

4x4, 6x6, 8x8 circles |
Yes |
Yes |
Yes |

Corners of 2x2, 2x4, 2x6, 2x8, 4x4,4x6, 4x8, 6x6, 6x8, 8x8 |
Yes |
Yes |
Yes |

Aale’s letters A, a, C, c, E, e, F, f |
Yes |
Yes |
Yes |

Cross & diamond |
No |
No |
No |

Aale’s proof showed that corners of all rectilinear shapes with even dimensions
sum to 4/m of S.

Circles of even order sum correctly to xS/m (where x = number of cells in
figure and m = order)

All of Aale’s letters that I tested also summed correctly.

In all cases, this includes wrap-around. The corner of the figure may start on
*any *cell in the square!

After checking Aale’s 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.

The array size of the ‘letter’s’ checked were 6x4, 5x4,and 5x3.

After checking the cross (8x5) and diamond (8x7), I realize that this is
incorrect.

I leave the reason for this, for someone else to explain .

Available for download is my Compact_8-MS.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.

Addendum (July 12, 2008: Compact_8-MS.xls has been
expanded to compare 7 different order-8 magic squares.One of these is from Dwane Campbell who has a comprehensive new
site containing
hypercube generators. |

**Shapes tested in above squares**

The letter patterns are some of those suggested by Aale de Winkel in an email of April 20, 2007.

The above shapes are some of the preliminary ones first suggested by Aale.

He has a comprehensive entry on this subject (and more refined shapes) in The Magic Encyclepedia. (Look at Compact under C.)

**Please send me Feedback about my Web
site!**

Harvey Heinz harveyheinz@shaw.ca

This page last updated
September 01, 2009

Copyright © 2002 by Harvey D. Heinz