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<h2 class="text-medium" id="program">History of Magic Squares </h2>

<B>Leonhard Euler</B> (1707-1783) was a Swiss mathematician who made pioneering and influential discoveries such as analytic number theory, complex analysis and infinitesimat calculus. He introduced much of modern mathematical therminology and notatin, including the notation of mathematical function. He spent most of his adult life in St. Petersburg, Russia, and in Berlin, Preussia. Euler's interest in number theory and logic lead him to the construction of the Euler square, later named the latin square. It says that each row, column and diagonal should have only one letter a...d and only on number of each 1...4 in a square of order 4. This combination of letter and integer in rows, columns and diagonals are named Euler square. His original letter was Latin and Greek letter and when the conditions above where carry out, the square was named magic square. The result says that all rows, columns and diagonals have the same sum. Today it´s possibly to show Euler square who are bimagic at order 8, 9, 16, 25, 32. George Pfeffermann (1838-1914) was born on Frankfurt, Germany. But from age 21, he lived in France. He was a bank employee in Paris. Married in 1888 to a French women, and he obtained French citizenship in 1891. He authored numerous articles on magic square published in French, mainly between 1890 and 1896. He invented the first bimagic square of order 8. He made a puzzle of the world first known bimagic square and published in the French paper: "Les Tablettes du Chercheur", in 1890. In year 1891 he published also the worlds first known bimagic square of order 9. In 1907, he was named "Officer d'Académie" (Order of Academie palms) by the French Minister Aristide Briand. From 1909 until his death, he published one regular column titled "Divertissments" (Amusements) in the French paper: "Le Moniteur d'Issoire". Gaston Terry (1843-1913) a French mathematicain who spent his working life in Algeria. He find different method to prove that a Euler-latin square of order 6 does not exist. Together with his friend Brutus Portier he could find out several panmagic squares of order 8. They are bimagic and have trimagic diagonals. Tarry was the first to find an example of a trimagic square. It had order 128 and he named the method for construction: "Cabalistic condensator". He said that the discharge can only stay in a good conductive environment, a magic field. André Gérardin (1879-1953) a French mathematician who dedicated a theory of number he later called "Diophante". The paper was edited in Nancy, France, from 1948 to 1952. He made many solutions of bimagic squares of order 8. He did also many other published works in the mathematicial area. William H. Benson (1902-1984) an American who was in the U.S. Navy Forces until 1955. Later he was made an associate professor of mathematics in 1957. He is most famus for the construction of the first known trimagic square of order 32, together with Jacoby Oswald (1902-1984). Jacoby who was in early age a U.S. Champion in professional bridge playing. Later he was diverted for service as a U. S. Naval intelligence officer in World War II and Korea War. The whole square n32 was published for the first time 1976 in the book: "New recreations with Magic Squares". William H. Benson and Jacoby Oswald wrote about the method they had used. Her is an extract of their talks: <cite>"So far is known, this is the first trimagic square ever to be constructed of an order lower than 64. It has been completely checked by the use of IBM equipment and proved to be correct. The method is perfectly general and flexible. Any number of trimagic squares of the 32nd (64th, 128th, etc...) order can be constructed by its use of 32nd"</cite>. Today in modern time the trimagic square of order 32 from William H. Benson and Jacoby Oswald, 1976, is running in hightech equipment from Nikon Lithography and Windows operative systems from Microsoft Corporation.</p>

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<abbr title="Download Leonhard Euler - On Magic Squares, pdf-file."><a href="../Squarespace_3/Leonhard_Euler.pdf" target="_blank" download class="button">Leonhard Euler</a></abbr>

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<h2 class="text-medium" id="program">Modern History</h2>

<B>Modern Magic Squares</B>. Numerous 8th and 9th-order of bimagic squares have been constructed since Pfeffermann, a century ago, but none 10th or 11th-order have yet been successfully constructed. The first known 10th-order bimagic square was constructed in January 2004 by Fredrik Jansson, Finland. Fredrik where a young student, currently a second-year physics student at the Åbo Akademi University, in Turku. He also studies mathematics and computer science. Also in January 2004, only 18 days later than his 10th-order bimagic square, Fredrik Jansson constructed the first 11th-order bimagic square, using similar methods (combining bimagic series). In 2011, Chen Kenju, Li Wen, and Pan Fengchu published "A family of pandiagonal bimagic squares based on orthogonal arrays" in the Journal of Combinatorial Designs, Vol. 19, Issue 6, November 2011, pp. 427-438. Here is their abstract: <cite>In this article we give a construction of pandiagonal bimagic squares by means of four-dimensional bimagic rectangles, which can be obtained from orthogonal arrays with special properties. In particular, we show that there exists a normal pandiagonal bimagic square of order n4 for all positive integer n ≥ 7 such that gcd(n,30) = 1, which gives an answer to problem 22 of Abe in [Discrete Math 127 (1994), p.3–13].</cite> Another 11th-order bimagic square was constructed later by Chen Mutian, China, in May 2005. This magic square is symmetrical around its centre, meaning that two cells symmetrical around the centre have always the same sum, here 122. An interesting consequence is that the 4 lines of 11 numbers going through the centre are trimagic: the central row, the central column, and the 2 diagonals. Chen Qinwu and Chen Mutian, China, where first to constructed in year 2005 a trimagic square of order 16. Three year later in year 2008, Li Wen, China, constructed a second trimagic square of the same order 16. Chen Qinwu and Chen Mutian where after these work on the trimagic squares of order 16 appointed to be professors at the Computer Science Department of the Shantou University, Guangdong Province, China. First know prime bimagic square of order 11 was created and constructed in year 2006 from Christian Boyer, France, a former <cite>Microsoft</cite> employeer with responsibility over the French district in Europe. Nicolas Rouanet, France, is an engineer working at LATMOS, one of the optical department in France. This “Laboratoire ATmosphéres, Miliex, Observation Spatiales” (Atmospheres, Environments, Space Observations Laboratory) is a joint research unit of CNRS +University of Versailles +Sorbonne University. From January to November 2018, he worked on bimagic squares of primes, in orders from 8 to 25. Su Maoting, China, was about 45 years old when he created and constructed all his bimagic squares of higher orders you can find on the page of squaremagie space. He lives and works in Fujian Province, China. Walter Trump is named in a article about trimagic squares.

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