Cartesian magicness of 3-dimensional boards

Abstract

A $(p, q, r)$-board that has $p q+p r+q r$ squares consists of a $(p, q)-$, a $(p, r)-$, and a $(q, r)$-rectangle. Let $S$ be the set of the squares. Consider a bijection $f: S \rightarrow[1, p q+p r+q r]$. Firstly, for $1 \leq i \leq p$, let $x_i$ be the sum of all the $q+r$ integers in the $i$-th row of the $(p, q+r)$-rectangle. Secondly, for $1 \leq j \leq q$, let $y_j$ be the sum of all the $p+r$ integers in the $j$-th row of the $(q, p+r)$-rectangle. Finally, for $1 \leq k \leq r$, let $z_k$ be the the sum of all the $p+q$ integers in the $k$-th row of the $(r, p+q)$-rectangle. Such an assignment is called a $(p, q, r)$-design if $\left\{x_i: 1 \leq i \leq p\right\}=\left\{c_1\right\}$ for some constant $c_1,\left\{y_j: 1 \leq j \leq q\right\}=\left\{c_2\right\}$ for some constant $c_2$, and $\left\{z_k: 1 \leq k \leq r\right\}=\left\{c_3\right\}$ for some constant $c_3$. A $(p, q, r)$-board that admits a $(p, q, r)$-design is called (1) Cartesian tri-magic if $c_1, c_2$ and $c_3$ are all distinct; (2) Cartesian bi-magic if $c_1, c_2$ and $c_3$ assume exactly 2 distinct values; (3) Cartesian magic if $c_1=c_2=c_3$ (which is equivalent to supermagic labeling of $K(p, q, r)$ ). Thus, Cartesian magicness is a generalization of magic rectangles into 3-dimensional space. In this paper, we study the Cartesian magicness of various $(p, q, r)$-board by matrix approach involving magic squares or rectangles. In Section 2, we obtained various sufficient conditions for $(p, q, r)$-boards to admit a Cartesian tri-magic design. In Sections 3 and 4 , we obtained many necessary and (or) sufficient conditions for various ( $p, q, r)$-boards to admit (or not admit) a Cartesian bi-magic and magic design. In particular, it is known that $K(p, p, p)$ is supermagic and thus every $(p, p, p)$-board is Cartesian magic. We gave a short and simpler proof that every $(p, p, p)$-board is Cartesian magic.