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:

 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:

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

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).

 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!

 66 1921 98 1913 56 1834 1457 1226 1342 1330 1284 1431 1756 132 1839 36 1939 14 339 385 217 918 2109 888 2128 711 2118 2066 773 2110 812 2183 944 295 397 281 100 54 1473 78 1297 1899 1218 1850 1804 1786 1902 1292 1961 1375 2 1415 80 88 962 824 353 733 323 2129 262 2158 2175 2117 2080 250 2055 297 751 429 876 900 1345 1481 1281 133 1826 38 1838 34 1943 1917 46 1914 96 1764 55 1229 1407 1327 703 2063 813 2113 940 2133 417 294 341 279 218 365 2159 922 2125 887 2121 781 1808 1293 1882 1305 1959 1418 1 85 30 104 103 79 1470 1901 1367 1870 1217 1782 2081 2115 2105 230 748 301 879 428 892 970 354 821 319 736 282 2079 2177 2157 1893 1915 49 1766 93 1249 125 1323 1406 1482 1349 63 1261 41 1824 31 1837 1967 219 349 2155 362 2145 925 2123 837 704 780 863 2061 937 2093 420 2137 271 293 29 9 107 1904 1450 1867 1365 1832 1216 1294 1758 1307 1885 1438 1956 81 71 105 404 969 316 819 285 716 2107 2083 2082 2156 2101 2185 768 227 881 304 893 378 1405 65 1299 61 1264 27 1821 1968 1907 1845 1892 1769 53 1246 73 1373 123 1483 859 779 957 2131 422 2090 272 2140 269 243 2152 348 2148 360 2053 905 705 841 1286 1310 1757 1435 1889 131 1936 106 69 11 28 1924 57 1863 1453 1833 1362 1224 2098 2106 771 2184 811 225 894 284 400 382 336 968 287 889 2108 713 2132 2086 1905 1789 1891 1242 3 1374 76 1413 120 68 1475 58 1298 77 1268 1969 1801 1847 2172 247 2150 347 2054 430 755 902 856 844 960 729 352 2130 273 2088 265 2120

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.

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.

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)