Office Applications and Entertainment, Magic Squares of Subtraction

Construction of Semi Magic Squares (11 x 11)
s11 = 671, Res11 = 61

Because of the extremely high number of possible permutations within the rows of order 11 Generators, a non-iterative procedure has been used for the construction of Semi Magic Squares (ref. SemiSqrs11).

Unfortunately this results (generally) in Semi Magic Squares for which the last column is non-subtractive.

This can however be corrected with a procedure, which recalculates the last four columns (ref. CnstrSqrs11a), suitable for Semi Magic Squares with 11 subtractive rows and columns (Res11 = 61), as illustrated below:

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

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

Recalculated Columns 81 83 105 111 113 10 39 17 19 9 84 94 114 89 103 16 18 20 7 72 37 101 112 102 115 73 24 41 27 47 11 22 97 104 95 88 96 6 28 8 70 35 23 118

The resulting Semi Magic Square shown above, has been used in the construction example of previous page.

Notes

1. Based on a limited collection of 95 Generators, 18 (suitable) Semi Magic Squares could be obtained.
2. Based on the Semi Magic Square shown above (left), 5 sets of 'Recalculated Columns' could be generated.