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 10.0   Magic Squares (10 x 10) 10.1   Medjig Solutions (10 x 10) 10.1.1 General As described in section 6.8, for any integer n, a magic square C of order 2n can be constructed from any n x n medjig-square A with each row, column, and main diagonal summing to 3n, and any n x n magic square B, by application of the equations: bi + n2 aj with i = 1, 2, ... n2 and j = 1, 2, ... 4n2. The Medjig method of constructing a Magic Square of order 10 is as follows: Construct a 5 x 5 Medjig-Square A (ignoring the original game's limit on the number of times that a given sequence is used) Construct a 5 x 5 (Pan) Magic Square B (Already 28800 possibilities for Pan Magic only, refer Attachment 5.2.6) Construct a 10 x 10 Magic Square C by applying the equations mentioned above.

B (5 x 5)

 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25

Medjig Square A (5 x 5)

 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

Magic Square C (10 x 10)

 b1+25*a1 b1+25*a2 b2+25*a3 b2+25*a4 b3+25*a5 b3+25*a6 b4+25*a7 b4+25*a8 b5+25*a9 b5+25*a10 b1+25*a11 b1+25*a12 b2+25*a13 b2+25*a14 b3+25*a15 b3+25*a16 b4+25*a17 b4+25*a18 b5+25*a19 b5+25*a20 b6+25*a21 b6+25*a22 b7+25*a23 b7+25*a24 b8+25*a25 b8+25*a26 b9+25*a27 b9+25*a28 b10+25*a29 b10+25*a30 b6+25*a31 b6+25*a32 b7+25*a33 b7+25*a34 b8+25*a35 b8+25*a36 b9+25*a37 b9+25*a38 b10+25*a39 b10+25*a40 b11+25*a41 b11+25*a42 b12+25*a43 b12+25*a44 b13+25*a45 b13+25*a46 b14+25*a47 b14+25*a48 b15+25*a49 b15+25*a50 b11+25*a51 b11+25*a52 b12+25*a53 b12+25*a54 b13+25*a55 b13+25*a56 b14+25*a57 b14+25*a58 b15+25*a59 b15+25*a60 b16+25*a61 b16+25*a62 b17+25*a63 b17+25*a64 b18+25*a65 b18+25*a66 b19+25*a67 b19+25*a68 b20+25*a69 b20+25*a70 b16+25*a71 b16+25*a72 b17+25*a73 b17+25*a74 b18+25*a75 b18+25*a76 b19+25*a77 b19+25*a78 b20+25*a79 b20+25*a80 b21+25*a81 b21+25*a82 b22+25*a83 b22+25*a84 b23+25*a85 b23+25*a86 b24+25*a87 b24+25*a88 b25+25*a89 b25+25*a90 b21+25*a91 b21+25*a92 b22+25*a93 b22+25*a94 b23+25*a95 b23+25*a96 b24+25*a97 b24+25*a98 b25+25*a99 b25+25*a100
 The rows, columns and main diagonals of Square C sum to 2 times the corresponding sum of Magic Square B plus 25 times the corresponding sum of Medjig square A which results in s1 = 2 * 65 + 25 * 15 = 505. As b(i) ≠ b(j) for i ≠ j with i, j = 1, 2, ... 25 it is obvious that also c(m) ≠ c(n) for n ≠ m with n, m = 1, 2, ... 100. A numerical example is shown below:

B (5 x 5)

 12 6 5 24 18 4 23 17 11 10 16 15 9 3 22 8 2 21 20 14 25 19 13 7 1

Medjig Square A (5 x 5)

 0 2 0 1 0 2 3 2 2 3 1 3 3 2 3 1 0 1 1 0 0 3 1 3 3 0 0 1 1 3 2 1 0 2 2 1 3 2 0 2 3 0 0 1 2 3 0 3 3 0 1 2 3 2 1 0 2 1 2 1 1 2 3 0 1 3 2 1 2 0 3 0 2 1 0 2 0 3 3 1 3 0 2 0 3 2 2 0 0 3 1 2 1 3 0 1 3 1 1 2

Magic Square C (10 x 10)

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

10.1.2 Further Analysis

Medjig Squares are described by the same linear equations as applicable for Magic Squares, however with magic sum 15:

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

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

a( 1)+a(12)+a(23)+a(34)+a(45)+a(56)+a(67)+a(78)+a(89)+a(100) = 15
a(10)+a(19)+a(28)+a(37)+a(46)+a(55)+a(64)+a(73)+a(82)+a( 91) = 15

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

Resulting in the matrix representation:

A * a = s

which can be reduced, by means of row and column manipulations, to the minimum number of linear equations:

 a(91) = 15 - a(92) - a(93) - a(94) - a(95) - a(96) - a(97) - a(98) - a(99) - a(100) a(89) =  6 - a(90) - a(99) - a(100) a(87) =  6 - a(88) - a(97) - a(98) a(85) =  6 - a(86) - a(95) - a(96) a(83) =  6 - a(84) - a(93) - a(94) a(81) =  6 - a(82) - a(91) - a(92) a(71) = 15 - a(72) - a(73) - a(74) - a(75) - a(76) - a(77) - a(78) - a(79) - a(80) a(69) =  6 - a(70) - a(79) - a(80) a(67) =  6 - a(68) - a(77) - a(78) a(65) =  6 - a(66) - a(75) - a(76) a(63) =  6 - a(64) - a(73) - a(74) a(61) =  6 - a(62) - a(71) - a(72) a(51) = 15 - a(52) - a(53) - a(54) - a(55) - a(56) - a(57) - a(58) - a(59) - a(60) a(49) =  6 - a(50) - a(59) - a(60) a(47) =  6 - a(48) - a(57) - a(58) a(45) =  6 - a(46) - a(55) - a(56) a(43) =  6 - a(44) - a(53) - a(54) a(41) =  6 - a(42) - a(51) - a(52) a(31) = 15 - a(32) - a(33) - a(34) - a(35) - a(36) - a(37) - a(38) - a(39) - a(40) a(29) =  6 - a(30) - a(39) - a(40) a(27) =  6 - a(28) - a(37) - a(38) a(25) =  6 - a(26) - a(35) - a(36) a(23) =  6 - a(24) - a(33) - a(34) a(21) =  6 - a(22) - a(31) - a(32) a(19) = 18 + a(20) - a(28) + a(30) - a(37) + a(40) - a(46) - a(49) - a(55) - a(59) - a(64) - a(69) - a(73) - a(79) +            - a(82) - a(89) - a(91) - a(99) a(12) = 0.5 * (33  - a(13) - a(14) - a(15) - a(16) - a(17) - a(18) - a(19) - a(20) - a(22) + a(24) - a(32) + a(33) +                    - a(42) - a(45) - a(52) - a(56) - a(62) - a(67) - a(72) - a(78) - a(82) - a(89) - a(92) - a(100)) a(11) =-18 + a(12) + a(22) - a(24) + a(32) - a(33) + a(42) + a(45) + a(52) + a(56) + a(62) + a(67) + a(72) + a(78) +            + a(82) + a(89) + a(92) + a(100) a(10) = 15 - a(20) - a(30) - a(40) - a(50) - a(60) - a(70) - a(80) - a(90) - a(100) a( 9) = -9 - a(20) + a(28) + a(37) + a(46) + a(55) + a(64) + a(73) + a(82) + a(91) a( 8) = 15 - a(18) - a(28) - a(38) - a(48) - a(58) - a(68) - a(78) - a(88) - a(98) a( 7) =  6 - a( 8) - a(17) - a(18) a( 6) = 15 - a(16) - a(26) - a(36) - a(46) - a(56) - a(66) - a(76) - a(86) - a(96) a( 5) =  6 - a( 6) - a(15) - a(16) a( 4) = 15 - a(14) - a(24) - a(34) - a(44) - a(54) - a(64) - a(74) - a(84) - a(94) a( 3) =  9 - a(13) + a(24) + a(34) - a(43) - a(53) - a(63) - a(73) - a(83) - a(93) a( 2) = 0.5 * (30  - a(11) - a(12) - a(19) + a(20) - a(22) - a(24) - a(28) + a(30) - a(32) - a(33) - a(37) + a(40) +                    - a(42) + a(45) - a(46) - a(49) - a(52) - a(55) + a(56) - a(59) - a(62) - a(64) + a(67) - a(69) +                    - a(72) - a(73)+ a(78) - a(79) - 2 * a(82) - a(91) - a(92) - a(99) + a(100)) a( 1) = -6 + a( 2) + a(22) + a(24) + a(32) + a(33) + a(42) - a(45) + a(52) - a(56) + a(62) - a(67) + a(72) - a(78) +                    + a(82) - a(89) + a(92) - a(100)
 The linear equations shown above are ready to be solved, for the magic constant 15. The solutions can be obtained by guessing a(i) for      i = 13 ... 18, 20, 22, 24, 26, 28, 30, 32 ... 40, 42, 44, 46, 48, 50, 52 ... 60                         62, 64, 66, 68, 70, 72 ... 80, 82, 84, 86, 88, 90, 92 ... 100 and filling out these guesses in the abovementioned equations. To obtain the integers 0, 1, 2 and 3 also following relations should be applied:      0 =< a(i) =< 3        for i = 1 ... 12, 19, 21, 23, 25, 27, 29, 31, 41, 43, 45, 47, 49, 51                                                  61, 63, 65, 67, 69, 71, 81, 83, 85, 87, 89, 91      Int(a(i)) = a(i)      for i = 2 and 12 which can be incorporated in a guessing routine, which can be used to generate a defined number of Medjig Squares within a reasonable time. With 18 of the 25 Medjig pieces constant, an optimized guessing routine (MgcSqr10a), produced 2112 Medjig Squares within 201 seconds, which are shown in Attachment 10.1.1. The resulting Magic Squares, based on the 5th order Magic Square shown in the numerical solution above are shown in Attachment 10.1.2. It should be noted that, although much faster, not all Magic Squares of the 10th order can be found by means of the Medjig Solution. 10.1.3 Concentric Magic Squares The Medjig method of constructing a Concentric Magic Square of order 10 is as follows: Construct a  5 x  5 Concentric Magic Medjig-Square A; Construct a  5 x  5 Concentric Magic Square B (ref. Attachment 5.5.2); Construct a 10 x 10 Concentric Magic Square C by applying the equations mentioned in Section 10.1.1 above. A numerical example is shown below:

B (5 x 5)

 25 5 8 24 3 4 16 11 12 22 6 9 13 17 20 7 14 15 10 19 23 21 18 2 1

Medjig Square A (5 x 5)

 1 2 0 1 0 2 2 3 1 3 0 3 3 2 3 1 1 0 2 0 0 3 2 1 0 3 3 0 1 2 2 1 0 3 2 1 2 1 0 3 3 0 1 0 0 3 2 3 3 0 1 2 3 2 2 1 1 0 2 1 1 2 3 2 2 0 0 2 2 1 3 0 0 1 3 1 1 3 3 0 3 0 2 0 3 2 0 2 0 3 1 2 1 3 0 1 3 1 1 2

Magic Square C (10 x 10)

 50 75 5 30 8 58 74 99 28 78 25 100 80 55 83 33 49 24 53 3 4 79 66 41 11 86 87 12 47 72 54 29 16 91 61 36 62 37 22 97 81 6 34 9 13 88 67 92 95 20 31 56 84 59 63 38 42 17 70 45 32 57 89 64 65 15 10 60 69 44 82 7 14 39 90 40 35 85 94 19 98 23 71 21 93 68 2 52 1 76 48 73 46 96 18 43 77 27 26 51
 The Concentric Magic Squares resulting from the Medjig Square A shown above and 128 of the 23040 possible 5th order Concentric Magic Squares, are shown in Attachment 10.1.3. 10.1.4 Eccentric Magic Squares The Medjig method of constructing an Eccentric Magic Square of order 10 is as follows: Construct a  5 x  5 Eccentric Magic Medjig-Square A; Construct a  5 x  5 Eccentric Magic Square B (ref. Attachment 5.5.3); Construct a 10 x 10 Eccentric Magic Square C by applying the equations mentioned in Section 10.1.1 above. A numerical example is shown below:

B (5 x 5)

 5 19 3 20 18 7 21 23 6 8 4 22 16 11 12 25 1 9 13 17 24 2 14 15 10

Medjig Square A (5 x 5)

 0 2 0 1 0 3 1 3 2 3 1 3 3 2 2 1 2 0 1 0 3 0 1 3 3 0 0 1 3 1 1 2 0 2 1 2 3 2 0 2 2 3 0 1 2 1 0 3 3 0 1 0 3 2 0 3 2 1 2 1 1 2 3 0 1 0 0 3 2 3 3 0 2 1 3 2 2 1 1 0 0 1 2 3 3 2 2 0 0 2 3 2 1 0 0 1 3 1 1 3

Magic Square C (10 x 10)

 5 55 19 44 3 78 45 95 68 93 30 80 94 69 53 28 70 20 43 18 82 7 46 96 98 23 6 31 83 33 32 57 21 71 48 73 81 56 8 58 54 79 22 47 66 41 11 86 87 12 29 4 97 72 16 91 61 36 62 37 50 75 76 1 34 9 13 88 67 92 100 25 51 26 84 59 63 38 42 17 24 49 52 77 89 64 65 15 10 60 99 74 27 2 14 39 90 40 35 85
 The Eccentric Magic Squares resulting from the Medjig Square A shown above and 128 of the 3072 possible 5th order Eccentric Magic Squares, are shown in Attachment 10.1.4. 10.1.5 Almost Associated Magic Squares The Medjig method of constructing an Almost Associated Magic Square of order 10 is as follows: Construct a  5 x  5 Almost Associated Medjig-Square A composed of: An Associated Medjig Border An Almost Associated Medjig Center Square   (ref. Section 6.6.4); Construct a  5 x  5 Associated Magic Square B (ref. Attachment 5.4.2); Construct a 10 x 10 Almost Associated Magic Square C by applying the equations mentioned in Section 10.1.1 above. A numerical example is shown below:

B (5 x 5)

 25 23 2 11 4 7 9 12 16 21 6 8 13 18 20 5 10 14 17 19 22 15 24 3 1

Medjig Square A (5 x 5)

 3 2 3 2 2 0 2 0 1 0 1 0 0 1 1 3 1 3 3 2 3 2 3 2 0 3 1 0 1 0 0 1 1 0 2 1 3 2 2 3 1 2 1 3 0 1 1 3 0 3 0 3 0 2 2 3 0 2 1 2 0 1 1 0 2 1 3 2 2 3 3 2 3 2 3 0 1 0 1 0 1 0 0 2 0 2 2 3 3 2 3 2 3 1 3 1 1 0 1 0

Magic Square C (10 x 10)

 100 75 98 73 52 2 61 11 29 4 50 25 23 48 27 77 36 86 79 54 82 57 84 59 12 87 41 16 46 21 7 32 34 9 62 37 91 66 71 96 31 56 33 83 13 38 43 93 20 95 6 81 8 58 63 88 18 68 45 70 5 30 35 10 64 39 92 67 69 94 80 55 85 60 89 14 42 17 44 19 47 22 15 65 24 74 53 78 76 51 97 72 90 40 99 49 28 3 26 1
 Attachment 10.2.5 shows a few examples of Almost Associated Magic Squares resulting from the Associated Magic Square B shown above and miscellaneous Almost Associated Medjig Squares. The Almost Associated Medjig Squares are based on the border of Square A above and the 120 Almost Associated Medjig Center Squares shown in Attachment 6.8.5. It should be noted that although the Almost Associated Medjig Square A is bordered, the resulting Square C will be only Almost Associated as the 5 x 5 Square B is Associated. 10.1.6 Magic Squares with Bimagic Main Diagonals It can be proven that the Medjig method is not suitable for the construction of Bimagic Squares of order 10. However Magic Squares of order 10 with Bimagic Main Diagonals can be constructed as follows: Select order 5 Magic Lines suitable for the construction of order 10 Bimagic Lines (ref. Attachment 10.1.5); Generate order 5 Magic Squares B using these lines as diagonals (ref. Attachment 10.1.6); Generate the order 10 Magic Squares C with an appropriate guessing routine, which checks the Bimagic Diagonals while generating A and calculating C (ref. MgcSqr10a2). A numerical example is shown below:

B (5 x 5)

 25 10 3 5 22 4 23 7 15 16 12 11 14 19 9 18 8 20 2 17 6 13 21 24 1

Medjig Square A (5 x 5)

 3 2 0 2 0 2 0 2 1 3 0 1 1 3 1 3 1 3 2 0 0 2 2 3 0 1 2 3 0 2 1 3 1 0 3 2 1 0 1 3 0 1 1 0 3 2 1 2 2 3 3 2 3 2 0 1 3 0 1 0 1 0 3 1 1 0 2 3 3 1 3 2 0 2 3 2 0 1 2 0 3 2 1 0 1 0 2 1 2 3 1 0 3 2 3 2 3 0 1 0

Magic Square C (10 x 10)

 100 75 10 60 3 53 5 55 47 97 25 50 35 85 28 78 30 80 72 22 4 54 73 98 7 32 65 90 16 66 29 79 48 23 82 57 40 15 41 91 12 37 36 11 89 64 44 69 59 84 87 62 86 61 14 39 94 19 34 9 43 18 83 33 45 20 52 77 92 42 93 68 8 58 95 70 2 27 67 17 81 56 38 13 46 21 74 49 51 76 31 6 88 63 96 71 99 24 26 1
 Attachment 10.1.7 shows a few 10 x 10 Magic Squares with Bimagic Diagonals, which could be constructed based on the 5 x 5 Magic Squares described above. 10.1.7 Magic Squares with Bimagic Center Lines Eight of the order 5 Magic Lines shown in Attachment 10.1.5 might return two adjacent order 10 Bimagic Lines. Order 5 Magic Squares with these Magic Lines as center lines will return order 10 Magic Squares with Bimagic Center Lines. The Medjig method of constructing order 10 Magic Squares with Bimagic Center Lines is as follows: Generate, based on preselected center lines as described above, order 5 Magic Squares B (ref. MgcSqr5k); Generate the order 10 Magic Squares C with an appropriate guessing routine, which checks the Bimagic Center Lines while generating A and calculating C (ref. MgcSqr10a3). A numerical example is shown below:

B (5 x 5)

 7 19 1 13 25 15 23 10 6 11 2 8 14 21 20 24 3 18 16 4 17 12 22 9 5

Medjig Square A (5 x 5)

 1 2 0 2 2 3 0 1 1 3 0 3 1 3 0 1 2 3 0 2 0 2 1 2 2 3 0 2 0 3 1 3 0 3 0 1 1 3 2 1 2 1 2 0 1 0 3 1 3 2 3 0 3 1 3 2 2 0 1 0 1 0 1 0 2 3 0 3 3 2 3 2 3 2 1 0 2 1 1 0 1 0 1 0 3 2 2 1 3 2 3 2 3 2 1 0 3 0 1 0

Magic Square C (10 x 10)

 32 57 19 69 51 76 13 38 50 100 7 82 44 94 1 26 63 88 25 75 15 65 48 73 60 85 6 56 11 86 40 90 23 98 10 35 31 81 61 36 52 27 58 8 39 14 96 46 95 70 77 2 83 33 89 64 71 21 45 20 49 24 28 3 68 93 16 91 79 54 99 74 78 53 43 18 66 41 29 4 42 17 37 12 97 72 59 34 80 55 92 67 87 62 47 22 84 9 30 5
 Attachment 10.1.9 shows a few 10 x 10 Magic Squares with Bimagic Center Lines, which could be constructed based on the 5 x 5 Magic Squares described above. 10.1.8 Spreadsheet Solution The linear equations shown in section 10.1.2 above can be applied in an Excel spreadsheet (Ref. CnstrSngl10a). The red figures have to be “guessed” to construct a 5 x 5 Medjig Square (wrong solutions are obvious).