Cartesian Magicness of 3-Dimensional Boards

TL;DR

This paper explores the concept of Cartesian magicness in 3D boards, generalizing magic rectangles into three dimensions, and provides conditions for their existence using matrix approaches involving magic squares and rectangles.

Contribution

It introduces the notions of Cartesian tri-magic, bi-magic, and magic boards, and establishes conditions for their existence, including a simplified proof that all (p,p,p)-boards are Cartesian magic.

Findings

Established sufficient conditions for Cartesian tri-magic boards.

Derived necessary and sufficient conditions for Cartesian bi-magic and magic boards.

Proved that all (p,p,p)-boards are Cartesian magic with a simplified proof.

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.