| Abstract: | Since the complete solution for the existence of magic 2‐dimensional rectangles in 1881, much attention has been paid on the existence of magic l‐dimensional rectangles for  . The existence problem for magic l‐dimensional rectangles with even sizes has been solved completely for all integers  . However, very little is known for the existence of magic l‐dimensional rectangles ( ) with odd sizes except for some families and a few sporadic examples. In this paper, we focus our attention on the existence of magic 3‐dimensional rectangles and prove that the necessary conditions for the existence of magic 3‐dimensional rectangles are also sufficient. Our construction method is mainly based on a new concept, symmetric zero‐sum subset partition, which plays a crucial role in the recursive constructions of magic 3‐rectangles similar to that of PBD in the PBD‐closure construction in combinatorial design theory. |
