Compact Magic Squares

Compact_k

For an order n compact_k magic square, the sum of each k x k block, (including wrap-around), is equal to k²/n of the magic constant. A square that is compact_k is compact_jk for j=1,2,3, ...

Examples

Order 4

See compact₂.

Order 8

.txt

Here are some compact₂. The 2x2 block sums are 130, (half the magic constant).

Below are some not compact₂ but compact₄. The 4x4 block sums are 520, (twice the magic constant).

Of the 64 compact blocks, here are the first and last starting in the top row and the first and last starting in the bottom row:

Order 9

.txt

Here are some compact₃. The 3x3 block sums are 369, (the magic constant).

Order 12

.txt

Here is a compact₂. The 2x2 block sums are 290, (one third the magic constant).

Below are some not compact₂ but compact₄ and/or compact₆. The 4x4 block sums are 1160, (4/3 times the magic constant); the 6x6 block sums are 2610, (3 times the magic constant).

Order 15

.txt

Here are some compact₅. The 5x5 block sums are 2825, (5/3 times the magic constant).