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:

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

1903: A family of 8x8 pandiagonal magic squares which are also bimagic squares, by Gaston Tarry, France
Necessary condition for bimagic diagonals: r(a-b) = c(p-q).
On the right, his example with (a, b, c, d) = (1, 4, 1, 4) and (p, q, r, s) = (0, 24, 8, 32).

In Tarry's example above:

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).

In Schots's square of order 8 above:

2000: A non-normal pandiagonal bimagic square with all its broken diagonals bimagic,
S1 = 19 674, S2 = 31 866 762, by Su Maoting, China

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:

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).

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:

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. 479-485. Here is their abstract:

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

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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