Composite magic squares

A magic square of order m · n is called composite, when it can be decomposed into m2 magic subsquares, each of order n.

It is to be noted that the minimal composite magic square must be of order n=9. This is the minimal number with two divisors, for which magic squares exist.

Zusammengesetztes Quadrat 1

This means that the fundamental magic square can be decomposed into m2=32=9 magic subsquares, each of them with order n=. Of course it is impossible that all subsquares are normalized, i.e. are composed of consecutive integers 1, 2, … , n2.

716667202524293433
646872272319363228
697065222126313035
834403944747978
159454137817773
672384342767580
475449566358111615
525048615957181410
514653605562131217

The next composite magic square is found for order m · n=12. This order can be divided in the following manner:

Zusammengesetztes Quadrat 2

This means that the composite square can be decomposed in m2=32=9 magic subsquares of order n=4.

1731302013214214312961566049
2822232513713513414051585463
2426272113313913813650595562
2919183214413013114164535752
1001051011126873698036413748
110103107987871756646394334
111102106997970746747384235
971081041096576727733444045
84898596414151116121117128
9487918297612126119123114
95869083511108127118122115
81928893162313113124120125

But, also another decomposition is possible.

Zusammengesetztes Quadrat 3

This composite magic square is decomposed into m2=42=16 magic subsquares, each of order n=3.

834119124123130135128293631
159126122118129131133343230
672121120125134127132332835
105106101585762495447748176
100104108635955485052797775
107102103566160534651787380
716469929994898485443742
666870979593828690394143
677265969198878883404538
114115110131217262122137142141
109113117181410192327144140136
116111112111615242520139138143