Today's application is to magic! Not the friendship kind, though 1 ; instead, we're going to talk about magic squares, an incredibly old piece of mathematics that we can study using Latin squares. Definition. A magic square is a n × n grid filled with the integers {0, 1,. .. n 2 − 1}, such that • each number is used exactly once in our entire grid, and • the sum of all of the entries along any ...

We define a simple algorithm for creating large numbers of semi-magic n× n squares when n is odd. Special cases of the algorithm can also be used to easily create magic and extra magic (or panmagic) n× n squares when n is odd. The algorithm can also create extremely magic n × n squares when n ≥ 5 is prime. These extremely magic n × n squares have n! standard magic n-element subsets, and these n...

Objective. The aim of this study was to evaluate the radiopacity of two conventional cements (Zinc Cement and Ketac Cem Easymix), one resin-modified glass ionomer cement (RelyX Luting 2) and six resin cements (Multilink, Bistite II DC, RelyX ARC, Fill Magic Dual Cement, Enforce and Panavia F) by digitization of images. Methods. Five disc-shaped specimens (10 × 1.0 mm) were made for each materia...

Let $A$ be an abelian group. A graph $G=(V,E)$ is said to be $A$-barycentric-magic if there exists a labeling $l:E(G)longrightarrow Asetminuslbrace{0}rbrace$ such that the induced vertex set labeling $l^{+}:V(G)longrightarrow A$ defined by $l^{+}(v)=sum_{uvin E(G)}l(uv)$ is a constant map and also satisfies that $l^{+}(v)=deg(v)l(u_{v}v)$ for all $v in V$, and for some vertex $u_{v}$ adjacent t...

We define a semi-magic square to be a square matrix whose entries are nonnegative integers and whose rows and columns (that is, lines) sum up to the same number. A magic square is a semimagic square whose main diagonals also add up to the line sum. A symmetric magic square is a magic square which is a symmetric matrix. A pandiagonal magic square is a semi-magic square whose diagonals parallel t...

Magic Lounge is a virtual meeting application which will integrate a number of technologies most of which are already known, such as www and video conferencing. In addition, Magic Lounge will integrate the innovations necessary to achieve the envisioned Magic Lounge functionality as described in [Dybkjær and Bernsen 1998]. Figure 1 outlines the physical Magic Lounge environment, i.e. how users ...

For a positive integer k, a graph G (V, E) is £k-magic if there exists a function, namely, a labeling, I : E(G) -+ £k such that the induced vertex set labeling [+ : V(G) £k, where [+(v) is the sum of the labels of the edges incident with a vertex v is a constant map. The set of all positive integer k such that G is k-magic is denoted by IM(G). We call this set the integer-magic spectrum of G. I...

introduction: in radiotherapy, methods of treatment planning are becoming increasingly more complicated. this requires verification of the doses delivered to increasingly smaller and more precise regions. radiotherapy techniques are continuously employing smaller and smaller field sizes to deliver tighter radiation doses with higher therapeutic ratios, generating interest among researchers to p...