Bimagic and trimagic squares, from 17th to 64th-order

This table shows that bimagic squares are known for each order. The last bimagic gaps were filled by Li Wen in 2008.

Trimagic series are impossible for any 4k+2 order (i.e. 18, 22, 26,...). But there remains a lot of work for trimagic squares:

Hmmmm... and no tetramagic square is known for any order < 243!

If you have some new trimagic or tetramagic squares in this range 17-64, or any proof of impossibility, send me a message! I will be pleased to add your results in this page.

This page should be dedicated to orders <= 65. However... in January 2018, I received an impressive file compiling examples for all orders from 65 to 100, bimagic squares constructed from 2002 to 2015 by Chinese researchers: Pan Fengchu, Zhongming, Shixueliang, Niu Guoliang, Li Wen, Gao Zhiyuan. Here is this file. For some orders (i.e. 81), it would possible to do better and construct trimagic squares.

Edouard Barbette's bimagic square : order 25.

In his book "Les Carrés magiques du mième ordre", published in Liège in 1912, the Belgian Edouard Barbette was probably the very first to publish a bimagic square of order 25. In his book, we can find also bimagic squares of orders 8 and 9.

At the Mathematical Art Exhibition, held at the 2011 Joint Mathematics Meetings in New Orleans, USA, Margaret Kepner, independent artist http://mekvisysuals.yolasite.com, received First Place Award for her work, "Magic Square 25 Study". But as reported by Reijo Sund, Ka Lok Chu, and George P. H. Styan at the IWMS 2016 (25th International Workshop on Matrices and Statistics, Madeira, Portugal), her magic square of order 25 is a... bimagic square!

More info and sources of above images:

Jacques Guéron's bimagic squares: orders 17, 18, 19, 21, 22, 23.

Jacques Guéron (Nice 1948 - )

In March 2006, only one month after his 13th-order bimagic square, Jacques Guéron, French teacher of mathematics, was the first to construct 17th and 19th-order bimagic squares. His interesting construction method is explained in his Excel file on the 19th-order.

And in April & May 2006, Jacques Guéron was the first to construct 18th, 21st, 22nd, 23rd-order bimagic squares.

About his first 23rd-order bimagic squares, found May 2nd, incredible: Chen Qinwu / Chen Mutian constructed independently another 23rd-order bimagic square exactly the same day! Chen Qinwu/Chen Mutian also constructed 17th, 19th, 21st-order bimagic squares in 2006, but after Jacques Guéron.

Su Maoting's bimagic squares: orders 20, 28, 30, 33, 36 (and 24).
Gao Zhiyuan's bimagic squares: orders 39, 50, 54, 56 (and 48).
Gao Zhiyuan / Su Maoting's bimagic squares: orders 42, 51, 57.

In January 2006, Su Maoting was the first to construct a 20th-order bimagic square. In January-February 2006, he constructed the first known 28th and 36th-order bimagic squares. And in June 2006, he constructed the first known 30th and 33rd-order bimagic squares. He constructed also a 24th-order bimagic square, but after Chen Qinwu's square of the same order.

Su Maoting is also the author of the first known pandiagonal bimagic square.

Gao Zhiyuan

In May-June-July 2006, Gao Zhiyuan, Mathematics Department of Yan'an College, Shaanxi Province, China, author of the website http://www.zhghf.net/China, was the first to construct 39th, 50th, 54th and 56th-order bimagic squares. He constructed also a 48th-order bimagic square, but after Pan Fengchu's square of the same order.

And in July 2006, together, Gao Zhiyuan and Su Maoting were the first to construct 42th, 51st and 57th-order bimagic squares.

Chen Qinwu's bimagic squares: orders 24, 26, 31.
Chen Qinwu / Chen Mutian's bimagic squares: orders 23, 29 (and 17, 19, 21).
Chen Qinwu's trimagic square: order 48.

In June 2005, Chen Qinwu, Computer Science Department of the Shantou University, Guangdong Province, China, was the first to construct a 24th-order bimagic square. And it is a nearly trimagic square. In May-June 2006, one year after his 24th-order bimagic square, he constructed the first known 26th and 31st-order bimagic squares.

Incredible: May 2nd 2006, Chen Qinwu, associated with Chen Mutian, same University, found a 23rd-order bimagic square exactly the same day as Jacques Guéron! In April-May 2006, Chen Qinwu and Chen Mutian created also 17th, 19th, 21st and 29th-order squares, but only the 29th is a first known, because 17th, 19th, 21st were previously constructed by Jacques Guéron. Their 21st-order square came only 3 weeks after Guéron's square.

As described in this page, Pan Fengchu constructed a 48th-order bimagic square in 2004, Gao Zhiyuan constructed another 48th-order bimagic square in May 2006, and Louis Caya constructed a 48th-order nearly trimagic square in May 2006. But Chen Qiwun in February 2007 was the first to construct a 48th-order fully trimagic square. Today, the known trimagic squares are scarce: orders 12, 16, 32, 48, 64, 81,...

Chen Qinwu and Chen Mutian are also the authors of smaller first known squares: bimagic (13th, 14th, 15th-order) and trimagic (16th-order) squares.

Li Wen's bimagic squares: orders 34, 35, 37, 38, 41, 43, 46, 47, 53, 55, 58, 59, 61, 62 (and 63).
Li Wen's trimagic squares: orders 24, 40.
Pan Fengchu's bimagic squares: orders 40, 44, 45, 48, 52, 60, 63 (and 35, 50, 55, 56).

In the 1980's, Pan Fengchu was the first to construct a 40th-order bimagic square.
In 2003, Li Wen was the first to construct a 35th-order bimagic square.
In 2004, Pan Fengchu was the first to construct 45th and 48th-order bimagic squares. And he constructed another 35th-order bimagic square, different of Li Wen's square.

In May 2006, Li Wen was the first to construct a 55th-order bimagic square. He constructed also a 63rd-order bimagic square, but it seems that it was after Pan Fengchu's square.
Also in May 2006, Pan Fengchu was the first to construct 44th, 52nd, 60th, 63rd-order bimagic squares. He constructed also 50th and 56th-order bimagic squares, but it seems that it was after Gao Zhiyuan's squares. He constructed also a 55th-order bimagic square, but it seems that it was after Li Wen's square.

Between February and April 2008, Li Wen constructed several new bimagic squares: now, thanks to him, bimagic squares are known for any order ≤ 64. And in March 2008, he constructed the first known 24th and 40th-order trimagic squares.

Li Wen is also the author of the first known pentamagic square of order 729.
Pan Fengchu is also the author of the first known hexamagic square.

Louis Caya's bimagic square, nearly trimagic square: order 48.

In May 2006, Louis Caya, certified general accountant, Québec, Canada, created a square very close to a trimagic square: 48 trimagic rows, 48 trimagic columns, one trimagic diagonal... the other diagonal being "only" bimagic... A fully bimagic square, but also a very nearly trimagic square!

Some months later, in February 2007, Chen Qinwu constructed the first known 48th-order trimagic square, with its two trimagic diagonals. See above in this page.

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