Magic Cubes Research Articles

On the Construction of Pandiagonal Magic Cubes

This paper investigates the construction method of pandiagonal magic cube. First, we define a pandiagonal Latin cube. According to this definition, the cube can be constructed by simple methods. After designing a set of orthogonal pandiagonal Latin cubes, the corresponding order pandiagonal magic cube can be constructed. In addition, we give the algebraic conditions of the universal diagonal Latin cube orthogonality and the strict theoretical proof. Based on the proposed method, it can be shown that at least 6(n!)3 pandiagonal magic cubes of order n is formed through a pandiagonal Latin cube. Moreover, our method is easy to implement by computer program.

Affine Magic Cubes of Order 4: Concepts and Applications

We convert a classical magic cube of order 4, which is an arrangement of 1,2 , … , 64 , to a cube of order 4 whose entries belong to F 2 6 . By using finite-dimensional vector spaces over the field F 2 , we introduce the notion of affine magic cubes and study their properties. The obtained results can be applied to describe some features of various types of magic cubes of order 4.

Pseudo magic cubes: A multidimensional data hiding scheme exploiting modification directions for large payloads

In this paper, a multidimensional data hiding scheme exploiting modification directions (EMD) that can embed high payload information without causing much distortion, is proposed. Traditional, EMD embeds a secret digit represented in the (2n+1)−ary system has a maximum 1.161 bpp payload. In the proposed work, a secret-digit represented in the n3−ary notational system is hidden, on a group of 3 pixels that are chosen based on a random sequence yields a payload of log2(n). Hence, an increase in the dimension, n− of the neighborhood set results in the increased payload. We have conducted extensive experiments on the standard image set to verify the embedding efficiency and visual quality of the stego image. The PSNR of the proposed algorithm for embedding 2.3219 bpp in a cover image of 512 × 512 is 45.13 dB, which implies that the visual quality of the stego image has not degraded even for large payloads. We have also verified that the proposed technique is undetectable during steganalysis.

Magic cube puzzle approach for image encryption

In principle, the image encryption algorithm produces an encrypted image. The encrypted image is composed of arbitrary patterns that do not provide any clues about the plain image and its cipher key. Ideally, the encrypted image is entirely independent of its plain image. Many functions can be used to achieve this goal. Based on the functions used, image encryption techniques are categorized into: (1) Block-based; (2) Chaotic-based; (3) Transformation-based; (4) Conventional-based; and (5) Miscellaneous based. This study proposes a magic cube puzzle approach to encrypt an 8-bit grayscale image. This approach transforms a plain image into a particular size magic cube puzzle, which is consists of a set of blocks. The magic cube puzzle algorithm will diffuse the pixels of the plain image as in a Rubik’s Cube game, by rotating each block in a particular direction called the transposition orientation. The block’s transposition orientation is used as the key seed, while the generation of the cipher key uses a random permutation of the key seed with a certain key length. Several performance metrics have been used to assess the goals, and the results have been compared to several standard encryption methods. This study showed that the proposed method was better than the other methods, except for entropy metrics. For further studies, modification of the method will be carried out in such a way as to be able to increase its entropy value to very close to 8 and its application to true color images. In essence, the magic cube puzzle approach has a large space for pixel diffusion that is possibly supposed to get bigger as a series of data has transformed into several magic cubes. Then, each magic cube has transposed with a different technique. This proposed approach is expected to add to a wealth of knowledge in the field of data encryption.

Vulnerability assessment of atmospheric storage tanks to floods based on logistic regression

Atmospheric storage tanks damaged by floods may lead to severe NaTech accident scenarios, and the vulnerability assessment of process equipment suffered by natural events is a crucial point in NaTech risk analysis. In the present study, limit state equations for failure modes of displacement and buckling are introduced based on load-resistance relationships. The parameterized fragility models that can be used in a wide variety of atmospheric unanchored storage tanks and floods have been developed based on logistic regression (LR), and moreover, the models are assessed and validated by receiver operating characteristic (ROC) curves and available accident data in literature. The effects of filling level and density of storaged liquid, the diameter and height of tank, and the inundation height and velocity of flood on the vulnerability of different failure modes are analyzed using fragility curves. Furthermore, fragility magic cubes are first proposed to obtain critical disaster conditions and critical filling levels in different cases. Finally, corresponding quantitative mitigation measures in different stages such as site selection, design, and operation are proposed based on the results of vulnerability assessment.

A Deployable Quasi-Yagi Monopole Antenna Using Three Origami Magic Spiral Cubes

We propose a deployable quasi-Yagi monopole antenna using origami magic spiral cubes, constructed from low-cost paper substrate. The proposed quasi-Yagi antenna consists of an L-shaped driven monopole, an L-shaped reflector, and two L-shaped directors. These driven monopole, reflector, and directors are designed in L-shaped geometries to achieve maximum gain in a limited cube size. The driven element and reflector are realized on the origami magic cube #1, while the two directors are designed on the magic cubes #2 and #3, respectively. The cubes are stacked together using a thin adhesive film. The presented geometry of the origami magic spiral cube antenna can be easily folded and unfolded. The dimensions in the unfolded state are 44 mm × 44 mm × 132 mm while the dimensions in the folded state are 44 mm × 44 mm × 8 mm. Therefore, in the folded state, the antenna occupies only 6% of the unfolded volume. We have numerically analyzed the antenna performance and verified the outcomes experimentally. The L-shaped reflector has increased the monopole antenna gain from 1.9 to 5.7 dBi. With the addition of two directors, the gain is further increased to 7.3 dBi at 1.9 GHz.

A generalization of Trenkler’s magic cubes formula

Abstract A Magic Cube of order p is a p×p×p cubical array with non-repeated entries from the set {1, 2, . . . , p3} such that all rows, columns, pillars and space diagonals have the same sum. In this paper, we show that a formula introduced in The Mathematical Gazette 84(2000), by M. Trenkler, for generating odd order magic cubes is a special case of a more general class of formulas. We derive sufficient conditions for the formulas in the new class to generate magic cubes, and we refer to the resulting class as regular magic cubes. We illustrate these ideas by deriving three new formulas that generate magic cubes of odd order that differ from each other and from the magic cubes generated with Trenkler’s rule.

3D seismic edge detection using magic squares and cubes

Edge detection is a category of geometric seismic attributes that has the capability to delineate vital information from seismic reflection data that can be used to aid qualitative and quantitative interpretation. We have evaluated a new method for geologic interpretation based on templates derived from magic squares and cubes. These are discrete differential operators that approximately calculate the spatial derivative of seismic amplitude through 2D and 3D convolution to locate edges and/or geologic features in seismic data. The new operator benefits from multidirectional scanning leading to efficient detection of different edge locations and their respective orientations. We have tested the new operators against the commonly used Sobel filter using two 3D seismic data volumes. Results of the [Formula: see text] magic cube operators provided better definition of seismic features than the [Formula: see text] magic cube operators. The overall results compared favorably with the Sobel operator, which suggests that the method can serve as a complementary tool to other existing seismic attributes.

A magic cube based information hiding scheme of large payload

Hiding information on grayscale images has been drawing much attention in recent years. In pervious techniques, many schemes like Zhang and Wang's method or Sudoku method use magic matrix as a plane reference to modify the LSBs of cover images. To further improve the hiding capacity and visual quality, we propose the so-called magic cube based (MCB) information hiding scheme. MCB is an early attempt to use three-dimensional reference to achieve information hiding in grayscale image. Secret data are embedded in the cover pixel LSBs by utilizing spatial coordinates. MCB has a larger probability to modify fewer pixels in the whole image. The comparison experiments show that MCB achieves higher hiding capacity and better visual quality with impressive undetectability. Moreover, our method can easily avoid the overflow problem in the embedding process and the hiding capacity is flexible by operating on magic cubes with different scales.

A Solar Cell Based on a Hybrid Film of P3HT and Magic Cubic-Like PbS

Lead sulfide (PbS) magic cubes were prepared by a simple hydrothermal method without any organic solvent. The product was characterized by X-ray diffraction (XRD), Scanning electron microscopy (SEM) and absorption spectrum. A solar cell, with a structure of Al/P3HT:PbS/PEDOT:PSS/ITO/Glass, was made. By a spin coating method, a hybrid film of poly(3-hexylthiophene) (P3HT) and PbS was prepared on the PEDOT:PSS layer. The solar cells are photosensitive in a large spectral range (visible and near infrared regions). The cells, with the area of 0.15 cm2 without any special treatment, have shown the values of open-circuit voltage (Voc) of 242 mV, short circuit current (Jsc) of 0.01 mA/cm2, energy conversion efficiency (η) of 0.01 % and fill factor (FF) of 0.31 under illumination intensity of 100 mW/cm2.

SOME PROPERTIES OF MAGIC CUBES

Abstract. In this paper we introduce an algorithm to construct an in- nite family of magic cubes and investigate properties of magic cubes,which extend the results in [6]. 1. IntroductionThe study of magic squares has a long history([4]). Some remarkable prop-erties of magic squares have been shown and some generalizations have drawnon interesting mathematics([1] and [3]). In [2] an operation on the set of allmagic squares was introduced which makes the set of magic squares into a freemonoid. In [6] a product on the set of all matrices was introduced and an al-gorithm was given to construct an in nite family of magic squares. Under theproduct operation, it was shown that the set of all magic squares formed a freemonoid as in [2] and moreover the product preserved the symmetrical propertyintroduced in [5] to give in nitely many square-palindromic magic squares. Inthis article, we extend the results in [6] to construct an in nite family of magiccubes and investigate properties of magic cubes.2. Cube algebraAn n-cube is an n-dimensional array of real numbers. When n= 2;we haveusual matrices. For an n-cube A;every component can be written by an n-aryexpression a

Franklin Squares: A Chapter in the Scientific Studies of Magical Squares

Several aspects of magic(al) squares studies fall within the computational universe. Experimental computation has revealed patterns, some of which have lead to analytic insights, theorems, or combinatorial results. Other numerical experiments have provided statistical results for some very difficult problems. While classical nth order magic squares with the entries 1..n² must have the magic sum for each row, column, and the main diagonals, there are some interesting relatives for which these restrictions are increased or relaxed. These include: serial squares of all orders with sequential filling of rows which are always pandiagonal (i.e., having all parallel diagonals to the main ones on tiling with the same magic sum, also called broken diagonals); pandiagonal logic squares of orders 2n derived from Karnaugh maps; Franklin squares of orders 8n which are not required to have any diagonal properties, but have equal half row and column sums and 2-by-2 quartets; as well as sets of parallel magical bent diagonals. Our early explorations of magic squares, considered as square matrices, used Mathematica® to study their eigenproperties. We have also studied the moment of inertia and multipole moments of magic squares and cubes (treating the numerical entries as masses or charges), finding some elegant theorems. We have also shown how to easily compound smaller squares into very high order squares. There are patents proposing the use of magical squares for cryptography. Other possible applications include dither matrices for image processing and providing tests for developing constraint satisfaction problem (CSP) solvers for difficult problems.

The electric multipole expansion for a magic cube

Recent studies of the rotational properties of magic cubes are extended to a corresponding electrical problem. While expecting the dipole moments of magic cubes to vanish, it was a surprise to find their quadrupole moments also vanished. These properties are shown to follow from the RCP (row, column, pillar) symmetry of the orthogonal line sums, namely the semi-magic property of the cubes, which opens up the conclusions to a much wider class of charge distributions.

The inertia tensor of a magic cube

Magic cubes are shown to have maximally symmetric inertia tensors if they are interpreted as rigid body mass distributions. This symmetry is due to their semi-magic property where each row, column, and pillar has the same mass sum. The moment of inertia depends only on this property and the number of point masses in each row, column, and pillar. Because magic cubes do not possess detailed cubic symmetry, other scenarios that result in maximally symmetric inertia tensors are discussed.

Ways of constructing optimal magic cube of order n when (n, 2·3·5·7)=1

An optimal magic cube of order n is a magic cube whose row sums, column sums and oblique sums of 9n layers are n(n 3 + 1)/2. The author proved that optimal magic cubes of order n may be constructed as long as n and 2, 3, 5, 7 are relatively prime, and a formula for making optimal magic cubes by using optimal Latin squares and optimal magic squares was given.

The properties of the the magic cube of fourth order

In this article we present the solution of the magic cube of fourth order made up of natural numbers using the general algorithm introduced by us, and the properties of this cube were analyzed. We noticed that the properties of magic cubes are valid for any nurnbers (rational, irrational, ete). These properties of the magic cube that made up rnagic square slices have shown that there is a strong relationship among the numbers that were assigned to the squares in the cube. Because of this relationship in magic cubes and magic squares the rnass and the geometrical centers are overlaps each other. These are the important findings of the general algorithm that we found.

SMARANDACHE RELATIONSHIPS AND SUBSEQUENCES

Some Smarandache relationships between the terms of a given sequence are studied in the fIrst paragraph. In the second paragraph, are studied Smarandache subsequences (whose terms have the same property as the initial sequence) . In the third paragraph are studied the Smarandache magic squares and cubes of order n and some conjectures in number theory.

A construction of magic cubes

In this paper a magic cube of order n is a 3-dimensional matrix Qn = {qn(i,j,k); 1 ≤ i,j,k ≤ n} containing natural numbers 1,2, ... n3 such that the sums of the numbers along each row (n-tuple of elements having the same coordinates on two places) and also along each of its four great diagonals are the same. This contrasts with a previous article [1] which considered magic cubes with no requirement for the diagonal sums.

Magic p-dimensional cubes of order n ≢ 2 (mod 4)

Magic p-dimensional cubes of order n ≢ 2 (mod 4)

Magic N-Cubes Form a Free Monoid

In this paper we prove a conjecture stated in an earlier paper [A-L]. The conjecture states that with respect to a rather natural operation, the set of $N$-dimensional magic cubes forms a free monoid for every integer $N>1$. A consequence of this conjecture is a certain identity of formal Dirichlet series. These series and the associated power series are shown to diverge. Generalizations of the underlying ideas are presented. We also prove variants of the main results for magic cubes with remarkable power sum properties.