Pandiagonal bimagic and trimagic squares

A pandiagonal magic square is a magic square with a supplemental property: all its broken diagonals are magic. A lot of pandiagonal magic squares of various orders are known. The smallest possible panmagic squares are of order 4. Among the 880 different magic squares of order 4, only 48 are pandiagonal. Here is one of them:

A pandiagonal magic square

3	6	15	10
16	9	4	5
2	7	14	11
13	12	1	8

In the above square, the two diagonals 3+9+14+8 and 10+4+7+13 sum to 34. But also all the broken diagonals, for example 10+16+7+1, 15+5+2+12, 6+4+11+13,...

It is more difficult to create a pandiagonal magic square which is also a bimagic square. The first one was published in 1903 by Gaston Tarry, in Compte-Rendu de la 32ème Session (Angers) de l'AFAS:

In Tarry's example above:

• the rows, the columns, the two standard diagonals, and the broken diagonals are magic.
• the rows, the columns, the two standard diagonals are bimagic... BUT only two broken diagonals are bimagic.

In January-February 2012, Francis Gaspalou enumerated the squares obtained by Tarry's method: see Gaspalou's PDF file, and also his study of Coccoz's method here.
In October-November 2013, Holger Danielsson worked on ten other similar families of 8x8 squares, also invented by Tarry: see Danielssson's PDF file#1 and PDF file#2 (list of squares).

1939: A pandiagonal magic square with all its broken diagonals bimagic,
by H. Schots, Belgium

1	2	60	59	7	8	62	61
15	40	32	49	9	34	26	55
18	42	45	21	24	48	43	19
54	27	35	12	52	29	37	14
64	63	5	6	58	57	3	4
50	25	33	16	56	31	39	10
47	23	20	44	41	17	22	46
11	38	30	53	13	36	28	51

In Schots's square of order 8 above:

• the rows, the columns, the two standard diagonals, and the broken diagonals are magic.
• the two standard diagonals and the broken diagonals are bimagic... BUT the rows and the columns are NOT bimagic!
   

The above square of order 18 constructed by Su Maoting in 2000 is a very interesting pandiagonal bimagic square with all its broken diagonals bimagic... BUT it is a non-normal magic square: it uses non-consecutive numbers.

• Download the above square as an Excel file (42Kb)

 Su Maoting

Six years later, in February 2006, Su Maoting was the first to succeed in constructing a normal pandiagonal bimagic square, using consecutive integers, with all its broken diagonals bimagic. A lot of people (including me...) thought that this problem was perhaps impossible. A difficult problem unsolved for more than one century. Congratulations! His square is of order 32. Su Maoting, 45 years old, lives in Fujian province, China. He works in an automobile transport company.

• Download the first known pandiagonal bimagic square (Excel file, 137Kb)

Is it possible to construct a pandiagonal bimagic square smaller than the order 32 used by Su Maoting? If you have some results on this problem, send me a message! I will be pleased to add your results to this page.

In February-April 2009, Li Wen, China, constructed other normal pandiagonal bimagic squares of bigger orders:

• Download Li Wen's pandiagonal bimagic square of order 77 (zipped Excel file, 211Kb)
• Download Li Wen's pandiagonal bimagic squares of orders 91 and 125 (zipped Excel file, 70Kb)

and in February 2009, Li Wen was the first to succeed in constructing a non-normal pandiagonal TRImagic square, with all its broken diagonals trimagic. The 156816 integers used are distinct but not consecutive: the biggest integer is 278259381. And an incredible supplemental property: this is also a PENTAmagic square, meaning that its rows, columns and two main diagonals are magic up to the 5th power!!! Li Wen was already famous for constructing in 2003 a pentamagic square of order 729, which is still today the smallest known normal pentamagic square (see the multimagic records).

• Download Li Wen's non-normal pandiagonal trimagic square, and pentamagic square, of order 396 (zipped Excel file, 693Kb)

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:

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), 3–13].

The same year, Pan Fengchu constructed pandiagonal bimagic squares of order n ≥ 32 with gcd(n,72) = 1 or 9:

• Download Pan Fengchu's pandiagonal bimagic squares of orders 32, 40, 56, 64, 72, 80, 88, 104, 112, 128, 136, 144, 152, 160, 176, 184, 200, 208, 216, 224, 232, 248, 256 (zipped Excel file, 1.7Mb)

In 2012, Li Wen, Wu Dianhua, and Pan Fengchu published "A construction for doubly pandiagonal magic squares" in Discrete Mathematics, Vol. 312, Issue 2, 28 January 2012, pp. 479-485. Here is their abstract:

In this note, a doubly magic rectangle is introduced to construct a doubly pandiagonal magic square. A product construction for doubly magic rectangles is also presented. Infinite classes of doubly pandiagonal magic squares are then obtained, and an answer to problem 22 of [G. Abe, Unsolved problems on magic squares, Discrete Math. 127 (1994) 3] is given.

"Doubly magic" means here "bimagic". The important part of their paper is this theorem:

Theorem 1.3. For each integer n ≥ 1, and (p, q) ∈ E = {(11, 7), (13, 7), (19, 7), (13, 11), (17, 11)}, there exists a doubly pandiagonal magic square of order (pq)ⁿ.

It explains for example why the squares above of orders 77 = 11*7 and 91 = 13*7, constructed by Li Wen in 2009, are possible.

In 2015, Li Wen constructed a pandiagonal bimagic square of order 385. It is difficult to construct such squares of order pqr, with p, q, r distinct primes! Here 385 = 5*7*11.

• Download Li Wen's pandiagonal bimagic square of order 385 (Excel file, 1Mb)

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