Bimagic squares, 10th and 11th-order

Fredrik Jansson (Turku 1982 - )

Numerous 8th and 9th-order bimagic squares have been constructed since Pfeffermann, a century ago, but none 10th or 11th-order have been successfully constructed.

The first known 10th-order bimagic square was constructed in January 2004 by Fredrik Jansson, Finland. Fredrik is a young student, currently a second year physics student at the Åbo Akademi University, in Turku. He also studies mathematics and computer science.

 2 19 70 1 66 74 73 60 68 72 58 77 15 3 65 4 67 69 71 76 62 63 82 75 61 59 79 6 5 13 49 18 14 78 98 40 25 96 43 44 94 41 27 42 35 91 21 95 37 22 93 39 23 38 31 90 33 30 29 99 34 100 36 83 45 24 26 28 97 32 8 85 64 57 7 56 80 48 16 84 54 11 86 47 87 12 92 20 50 46 51 52 88 81 10 55 9 53 89 17

He used the fact that there are so many series of order 10, as mentioned in the old version of this page: if we list the series of 10 different integers, from 1 to 100, having for magic sums S1 =  505, and S2 =  33,835, we find 24,643,236 series. And if we limit ourselves to series having the number 100 among their 10 integers, there are still 2,240,776 series! Thinking of these enormous numbers, the probability was very high that 10th-order bimagic squares existed... but nobody had succeeded. Congratulations to Fredrik!

Here are other bimagic squares of order 10, constructed later, in 2006 and 2007:

 81 44 41 63 88 3 49 53 1 82 26 38 92 90 25 45 42 62 2 83 96 97 31 46 68 8 22 24 57 56 16 100 9 75 11 71 43 54 65 61 28 48 7 51 34 91 95 59 77 15 13 27 87 14 60 89 55 64 79 17 72 36 52 18 86 47 23 6 66 99 58 10 74 30 84 50 5 94 67 33 80 76 39 98 37 32 78 4 21 40 35 29 73 20 12 69 93 85 70 19
 89 51 52 88 53 55 10 9 17 81 59 82 62 61 13 6 79 75 63 5 1 2 66 68 72 74 70 73 60 19 42 41 27 22 91 21 35 37 95 94 57 80 64 8 16 85 56 48 7 84 54 20 86 92 11 50 12 87 46 47 78 98 14 40 43 18 44 96 25 49 3 65 4 77 71 58 76 15 67 69 39 30 33 23 90 38 99 31 93 29 83 36 97 26 45 100 24 34 32 28

G. Pfeffermann, again him, and again in Les Tablettes du Chercheur, had published 3 non-normal bimagic squares of order 10, in 1894 and 1896. Here is the last published, which uses non-consecutive integers between 1 and 137.

 24 133 108 51 129 19 71 42 13 100 130 28 35 134 72 52 14 111 94 20 1 7 84 113 106 40 37 64 117 121 107 65 2 18 23 99 128 122 85 41 83 112 116 29 15 135 95 6 36 63 75 102 132 43 3 123 109 22 26 55 97 53 16 10 39 115 120 136 73 31 17 21 74 101 98 32 25 54 131 137 118 44 27 124 86 66 4 103 110 8 38 125 96 67 119 9 87 30 5 114

11th-order bimagic squares

Also in January 2004, only 18 days later than his 10th-order bimagic square, Fredrik Jansson constructed the first 11th-order bimagic square, using similar methods (combining bimagic series).

 84 80 88 2 82 10 81 74 1 86 83 53 114 118 35 47 26 27 55 58 113 25 119 45 40 51 116 38 42 29 33 117 41 21 109 20 66 60 37 115 111 54 59 19 69 87 85 4 79 89 94 8 3 75 78 39 15 14 105 96 64 103 61 98 63 13 121 34 44 57 46 120 30 48 108 31 32 73 91 90 71 7 92 95 5 76 6 65 52 36 17 107 16 104 18 77 70 62 112 12 11 99 72 100 67 43 97 68 9 93 28 49 56 101 22 24 23 106 102 50 110

Another 11th-order bimagic square was constructed later by Chen Mutian, China, in May 2005. This square is symmetrical around its centre, meaning that two cells symmetrical (around the centre) have always the same sum, here 122. An interesting consequence is that the 4 lines of 11 numbers going through the centre are trimagic: the central row, the central column, and the 2 diagonals.

 9 19 65 30 72 76 106 121 93 47 33 101 97 88 20 56 4 27 74 70 108 26 86 87 51 109 112 41 12 54 3 37 79 84 11 58 94 8 91 40 78 120 24 63 6 104 23 115 22 105 69 60 55 73 39 15 32 117 80 45 61 77 42 5 90 107 83 49 67 62 53 17 100 7 99 18 116 59 98 2 44 82 31 114 28 64 111 38 43 85 119 68 110 81 10 13 71 35 36 96 14 52 48 95 118 66 102 34 25 21 89 75 29 1 16 46 50 92 57 103 113