Cartesian magicness of 3-dimensional boards
Authors :
Gee-Choon Lau 1 *, Ho-Kuen Ng 2 and Wai-Chee Shiu 3
Author Address :
1 Faculty of Computer & Mathematical Sciences, Universiti Teknologi MARA (Johor Branch), 85000, Segamat, Malaysia.
2 Department of Mathematics, San José State University, San José CA 95192 USA.
3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China; & College of Global Talents, Beijing Institute of Technology, Zhuhai, P.R. China.
*Corresponding author.
Abstract :
A $(p,q,r)$-board that has $pq+pr+qr$ 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 $f : S \to [1,pq+pr+qr]$. Firstly, for $1 \le i \le p$, 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 $1 \le j \le q$, 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 $1\le k\le r$, 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 $\{x_i : 1\le i\le p\}=\{c_1\}$ for some constant $c_1$, $\{y_j : 1\le j\le q\}=\{c_2\}$ for some constant $c_2$, and $\{z_k : 1\le k\le r\}=\{c_3\}$ 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.
Keywords :
3-dimensional boards, Cartesian tri-magic, Cartesian bi-magic, Cartesian magic.
DOI :
Article Info :
Received : June 24, 2019; Accepted : July 09, 2020.
