Related papers: DIGITAL ERA: Universal Bimagic Squares
An elementary construction using binary codes gives new record kissing numbers in dimensions from 32 to 128.
This paper presents algorithms for computing the length of a sum of squares and a Pythagoras element in a global field $K$ of characteristic different from $2$. In the first part of the paper, we present algorithms for computing the length…
We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes, and extend previously published results to include information on the number…
In this paper we have established relations of the genetic tables with magic and bimagic squares. Connections with Hamming distances, binomial coefficients are established. The idea of Gray code is applied. Shannon entropy of magic squares…
The question of which triangular numbers have a decimal representation containing a single repeated digit seamed to be settled since at least the 1970s: Ballew and Weger provided a complete list and a proof that these are the only numbers…
We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be…
We have implemented the six series of three commuting measurement of the Mermin-Peres magic square on an online, five qubit, quantum computer. The magic square tests if the measurements of the system can be described by physical realism (in…
All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four…
R. L. Graham and H. O. Pollak observed that the sequence $$u_1=1,\qquad u_{n+1}=\lfloor \sqrt{2} (u_n+1/2)\rfloor, \quad n\geq 1,$$ has the curious property that the sequence of numbers $(u_{2n+1}-2u_{2n-1})_{n\geq 1}$ denotes the binary…
We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. We show that $\mathbb Q(\sqrt 5)$ is the only such real quadratic field, and that among fields of degrees 3, 4, 5, and 7…
Explicit algorithms are developed for constructing odd order n pandiagonal latin cubes in 3 and 4 dimensions, and these are used to construct pandiagonal magic cubes and 4 dimensional hypercubes, respectively. It is established that these…
The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used…
Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1 to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication,…
For $m=3,4,\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\binom n2+n\ (n=0,1,2,\ldots)$. For positive integers $a,b,c$ and $i,j,k\ge3$ with $\max\{i,j,k\}\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for…
We introduce Magic Gems, a geometric representation of magic squares as three-dimensional polyhedra. By mapping an n times n magic square onto a centered coordinate grid with cell values as vertical displacements, we construct a point cloud…
This article is meant to provide an additional point of view, applying known knowledge, to supply keys that have a series of non-repeating digits, in a manner that is not usually thought of. Traditionally, prime numbers are used in…
We present and discuss several old and new methods for mapping a circular disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside the circular disc to a point (x,y) inside a square region.…
The calculation of many and large Perrin pseudoprimes is a challenge. This is mainly due to their rarity. Perrin pseudoprimes are one of the rarest known pseudoprimes. In order to calculate many such large numbers, one needs not only a fast…
We consider how many random edges need to be added to a graph of order $n$ with minimum degree $\alpha n$ in order that it contains the square of a Hamilton cycle w.h.p..
Tile the unit square with $n$ small squares. We determine the minimum of the sum of the side lengths of the $n$ small squares, where the minimum is taken over all tilings of the unit square with $n$ squares.