Related papers: A construction of magic 24-cells
In this paper, we give a characterization of unicyclic graphs with diameter at most 4 which are A-vertex magic. Moreover, let G be a bicyclic graph of diameter 3, then G is group vertex magic if and only if G = M11(0, 0).
The aim of this note is to introduce fastest new general methods for the construction of double and single even order magic squares. As in [5], the method for double even order magic squares is fairly straight-forward but some adjustments…
It is unknown at present whether a magic square of squared integers exists. Such an object is defined to be a 3 by 3 grid of 9 distinct integer squares, such that the entries of each row, column, and two main diagonals sum to the same…
A magic rectangle of order $m\times n$ with precisely $r$ filled cells in each row and precisely $s$ filled cells in each column, denoted $MR(m,n;r,s)$, is an arrangement of the numbers from 0 to $mr-1$ in an $m\times n$ array such that…
A magic square of order n is an nxn square (matrix) whose entries are distinct nonnegative integers such that the sum of the numbers of any row and column is the same number, the magic constant. In this paper we introduce the concept of…
We study 4-by-4 squares formed by cards from the EvenQuads deck. EvenQuads is a card game with 64 cards where cards have 3 attributes with 4 values in each attribute. A quad is four cards with all attributes the same, all different, or half…
A simple graph is called triangular if every edge of it belongs to a triangle. We conjecture that any graphical degree sequence all terms of which are greater than or equal to 4 has a triangular realisation, and establish this conjecture…
Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami…
A supermagic labeling (often also called supermagic labeling) of a graph $G(V,E)$ with $|E|=k$ is a bijection from $E$ to the set of first $k$ positive integers such that the sum of labels of all incident edges of every vertex $x\in V$ is…
A graph $G$ is said to be distance magic if there exists a bijection $f:V\rightarrow \{1,2, \ldots , v\}$ and a constant {\sf k} such that for any vertex $x$, $\sum_{y\in N(x)} f(y) ={\sf k}$, where $N_(x)$ is the set of all neighbours of…
A magic series is a set of natural numbers that, by virtue of its size, sum, and maximum value, could fill a row of a normal magic square. In this paper, we derive an exact two-dimensional integral representation for the number of magic…
This manuscript introduces Diophantine labeling, a new way of labeling of the vertices for finite simple undirected graphs with some divisibility condition on the edges. Maximal graphs admitting Diophantine labeling are investigated and…
In this short note we have produced different kinds of bimagic squares using only the digits 0, 1, 2, 5 and 8. The universal bimagic squares presented are of order 8x8, 9x9, 16x16 and 25x25. In order to bring universal bimagic square of…
Let $G=(V,E)$ be a graph of order $n$. A distance magic labeling of $G$ is a bijection $\ell \colon V\rightarrow {1,...,n}$ for which there exists a positive integer $k$ such that $\sum_{x\in N(v)}\ell (x)=k$ for all $v\in V $, where $N(v)$…
We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding…
Odd, positive-definite, integral, unimodular lattices N of rank 24 were classified by Borcherds. There are 273 isometry classes of such lattices. Associated to them are vertex superalgebras $V_N$ of central charge c=24. We show that at…
A distance magic labeling of an $n$-dimensional hypercube is a labeling of its vertices by natural numbers from $\{0, \ldots, 2^n-1\}$, such that for all vertices $v$ the sum of the labels of the neighbors of $v$ is the same. Such a…
Using computational algebraic geometry techniques and Hilbert bases of polyhedral cones we derive explicit formulas and generating functions for the number of magic squares and magic cubes.
This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5…
Associated to any complex simple Lie algebra is a non-reductive complex Lie algebra which we call the intermediate Lie algebra. We propose that these algebras can be included in both the magic square and the magic triangle to give an…