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 sums 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 created in 1903 by Gaston Tarry:

1903: A pandiagonal magic square which is also a bimagic square, by Gaston Tarry, France

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

In the above Tarry's square of order 8:

• 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.
   

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 the above Schots's square of order 8:

• 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 Su Maoting's square of order 18 constructed 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, and having all its broken diagonals bimagic. A lot of people (including me...) thought that this problem was perhaps impossible. A difficult problem asked for more than one century. Congratulations! His square is of order 32. Su Maoting, 45 years old, is living 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 in this page.

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

• Download the Li Wen's pandiagonal bimagic square of order 77 (zipped Excel file, 211Kb)
• Download the 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, all its broken diagonals are trimagic. Its 156816 used integers are distinct but not consecutive: the biggest used 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 in constructing in 2003 a pentamagic square of order 729, which is still today the smallest known normal pentamagic square (see the multimagic records).

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

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