Squaring the magic squares of order 4

In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete proof, pointing out the role of the action of the symmetric group. Moreover, we provide a new representation for magic squares of order 4. Such representation allows the construction of magic squares in a very simple way, using essentially only five particular 4X4 matrices

Some Thoughts on the Search for 5×55 \times 5 and 6×66 \times 6 Additive-Multiplicative Magic Squares

An additive-multiplicative magic square is a square grid of numbers whose rows, columns, and long diagonals all have the same sum (called the magic sum) and the same product (called the magic product). There are numerous open problems about magic squares by Christian Boyer on multimagie.com. One such problem is to construct or prove the impossibility of a $5 \times 5$ or $6 \times 6$ additive-multiplicative magic square of distinct positive integers. Here, we present a possible approach to this problem and some partial results. We observe that such a square can be described by a form determined by the prime factorizations of its entries and that identifying these forms might be helpful in finding such a square or ruling out specific magic products.Comment: This paper has 10 pages and 3 figures. It is currently accepted, finalized, and in press at the Minnesota Journal of Undergraduate Mathematic

Design and economic analysis of solar home system for urban areas of Mogadishu using homer software

In this century of enhanced progress by various spaces, some African states are still challenging lack of energy due to scarcity in some places. The most used energy (generation of electricity) is hydropower because heat and fuel are still on some scale. This problem comes about in less efficiency and financial decay of some nations such as Somalia which is among the country at very high speed in progress, the grid lines from distant places are stack and they are rare matched to the required of power in all areas of the nation, especially in remote or urban areas where each household needs electricity utilization instead of utilizing local, conventional and lighting at domestic. This issue can be illuminated utilizing other elective sources of renewable energy for provincial electrification such as Photovoltaic systems. Hence, this project basically focuses on the design of SHS that incorporate financial assessment and utilize of an individual SHS of 200W, so that the satisfaction of the people and the targets of the country can be effectively achieved. Under this project, the dedicated on the investigation of power utilization based on single house household family SHS has been taking a case study of one village in Mogadishu Somalia named Heliwaa placed in Benaadir region. The survey was conducted by assessing the average major load conditions for consecutive hours per day based on photovoltaic capacity. The purpose of this study was achieved the optimal size of the photovoltaic panel and the battery capacity that can be used to power the home. Ultimately, designed project and cost will be compared to other private sector electricity cost, it means which one is more reliable and economically for electricity generation. Therefore according to, the findings the cost of energy is 2.614 in $/KWh which is lower than the private sector. This was considered optimum solution. In this project the design and simulation tasks was achieved through the assistance of HOMER software. The electrification and economics information on combination of photovoltaic systems, in the form of SHS and other renewable energy like stand-alone systems, to provide a reliable and economic system

Quantum automorphism groups of small metric spaces

To any finite metric space $X$ we associate the universal Hopf \c^*-algebra $H$ coacting on $X$. We prove that spaces $X$ having at most 7 points fall into one of the following classes: (1) the coaction of $H$ is not transitive; (2) $H$ is the algebra of functions on the automorphism group of $X$; (3) $X$ is a simplex and $H$ corresponds to a Temperley-Lieb algebra; (4) $X$ is a product of simplexes and $H$ corresponds to a Fuss-Catalan algebra.Comment: 22 page