Editing Magic square (section)

===Japan===
Japan and China have similar mathematical traditions and have repeatedly influenced each other in the history of magic squares.<ref>[https://core.ac.uk/download/pdf/161528551.pdf The Influence of Chinese Mathematical Arts on Seki Kowa] by Shigeru Jochi, MA, School of Oriental and African Studies, University of London, 1993</ref> The Japanese interest in magic squares began after the dissemination of Chinese works—Yang Hui's ''Suanfa'' and Cheng Dawei's ''Suanfa tongzong''—in the 17th century, and as a result, almost all the ''[[Japanese mathematics|wasans]]'' devoted their time to its study.

In the 1660 edition of ''Ketsugi-sho'', Isomura Kittoku gave both odd and even ordered bordered magic squares as well as magic circles; while the 1684 edition of the same book contained a large section on magic squares, demonstrating that he had a general method for constructing bordered magic squares.<ref name="DESmith69">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n74 69]–75| url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> In ''Jinko-ki'' (1665) by Muramatsu Kudayu Mosei, both magic squares and magic circles are displayed. The largest square Mosei constructs is of 19th order. Various magic squares and magic circles were also published by Nozawa Teicho in ''Dokai-sho'' (1666), Sato Seiko in ''Kongenki'' (1666), and Hosino Sanenobu in ''Ko-ko-gen Sho'' (1673).<ref name="DESmith79">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n84 79]–80 | url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> One of [[Seki Takakazu]]'s ''Seven Books'' (''Hojin Yensan'') (1683) is devoted completely to magic squares and circles. This is the first Japanese book to give a general treatment of magic squares in which the algorithms for constructing odd, singly even and doubly even bordered magic squares are clearly described.<ref name="DESmith116">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n121 116]–122| url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> In 1694 and 1695, Yueki Ando gave different methods to create the magic squares and displayed squares of order 3 to 30. A fourth-order magic cube was constructed by Yoshizane Tanaka (1651–1719) in ''Rakusho-kikan'' (1683). The study of magic squares was continued by Seki's pupils, notably by Katahiro Takebe, whose squares were displayed in the fourth volume of ''Ichigen Kappo'' by Shukei Irie, Yoshisuke Matsunaga in ''Hojin-Shin-jutsu'', Yoshihiro Kurushima in ''Kyushi Iko'' who rediscovered a method to produce the odd squares given by Agrippa,<ref name="DESmith178">{{cite book |last1=Smith |first1=David Eugene | last2 = Mikami | first2= Yoshio |title=A history of Japanese mathematics |publisher=Open Court Publishing Company| date=1914| page=[https://archive.org/details/in.ernet.dli.2015.161063/page/n183 178]| url=https://archive.org/details/in.ernet.dli.2015.161063|quote=Isomura Kittoku.}}</ref> and [[Ajima Naonobu|Naonobu Ajima]].<ref name="Michiwaki">{{cite book |last=Michiwaki |first=Yoshimasa | series=Encyopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures | date=2008 | edition=2 | pages=1252–1259| publisher=Springer |doi=10.1007/978-1-4020-4425-0_9154 |title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |isbn=978-1-4020-4559-2 |chapter=Magic Squares in Japanese Mathematics }}</ref><ref name="Mikami">{{cite book |last=Mikami |first=Yoshio | title=Magic squares in Japanese mathematics |publisher=Imperial Academy of Science| date=1917 | location=Tokyo | language=ja | url=https://books.google.com/books?id=sU1tAAAAMAAJ&q=Isomura++Kittoku}}</ref> Thus by the beginning of the 18th century, the Japanese mathematicians were in possession of methods to construct magic squares of arbitrary order. After this, attempts at enumerating the magic squares was initiated by Nushizumi Yamaji.<ref name="Mikami"/>