Office Applications and Entertainment, Magic Squares of Subtraction

Construction of Semi Magic Squares (10 x 10)
s10 = 505, Res10 = 49

Because of the extremely high number of possible permutations within the rows of order 10 Generators, a non-iterative procedure has been used for the construction of Semi Magic Squares (ref. SemiSqrs10).

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. CnstrSqrs10a), suitable for Semi Magic Squares with 10 subtractive rows and columns (Res10 = 49), as illustrated below:

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

Recalculated Columns 99 74 61 86 3 40 21 39 44 38 24 18 60 36 97 28 64 83 63 32 68 49 85 10 72 47 45 52 51 26 93 43 11 35 34 22 2 96 75 94

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

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 286 Generators, 92 (suitable) Semi Magic Squares could be obtained.
2. Based on the Semi Magic Square shown above (left), 43 sets of 'Recalculated Columns' could be generated.