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 CompteRendu de la 32ème Session (Angers) de l'AFAS:
a+p+r 
bc+qr+s 
b+d+p 
a+c+d+q+s 
b+p+r+s 
a+c+qr 
a+d+p+s 
bc+d+q 
>>> 
9 
51 
8 
62 
44 
18 
37 
31 
b+p 
a+c+q+s 
a+d+p+r 
bc+d+qr+s 
a+p+s 
bc+q 
b+d+p+r+s 
a+c+d+qr 
4 
58 
13 
55 
33 
27 
48 
22 

a+c+d+p+r+s 
b+d+qr 
bc+p+s 
a+q 
bc+d+p+r 
a+d+qr+s 
a+c+p 
b+q+s 
46 
24 
35 
25 
15 
53 
2 
60 

bc+d+p+s 
a+d+q 
a+c+p+r+s 
b+qr 
a+c+d+p 
b+d+q+s 
bc+p+r 
a+qr+s 
39 
29 
42 
20 
6 
64 
11 
49 

a+d+qr 
bc+d+p+r+s 
b+q 
a+c+p+s 
b+d+qr+s 
a+c+d+p+r 
a+q+s 
bc+p 
21 
47 
28 
34 
56 
14 
57 
3 

b+d+q 
a+c+d+p+s 
a+qr 
bc+p+r+s 
a+d+q+s 
bc+d+p 
b+qr+s 
a+c+p+r 
32 
38 
17 
43 
61 
7 
52 
10 

a+c+qr+s 
b+p+r 
bc+d+q+s 
a+d+p 
bc+qr 
a+p+r+s 
a+c+d+q 
b+d+p+s 
50 
12 
63 
5 
19 
41 
30 
40 

bc+q+s 
a+p 
a+c+d+qr+s 
b+d+p+r 
a+c+q 
b+p+s 
bc+d+qr 
a+d+p+r+s 
59 
1 
54 
16 
26 
36 
23 
45 
In Tarry's example above:
In JanuaryFebruary 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 OctoberNovember 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:
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 nonnormal magic square: it uses nonconsecutive numbers.
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 FebruaryApril 2009, Li Wen, China, constructed other normal pandiagonal bimagic squares of bigger orders:
and in February 2009, Li Wen was the first to succeed in constructing a nonnormal 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).
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. 427438. Here is their abstract:
In this article we give a construction of pandiagonal bimagic squares by means of fourdimensional 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 n^{4} 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:
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. 479485. 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.
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