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15.0 Special Magic Squares, Bimagic Squares, Part 2
This method is based on the application of two Diagonal Euler Squares (B1/B2) as shown below: |
B1
b-c+d a+c b+d a a+c+d b-c a+d b a b+d a+c b-c+d b a+d b-c a+c+d a+c+d b-c a+d b b-c+d a+c b+d a b a+d b-c a+c+d a b+d a+c b-c+d b-c a+c+d b a+d a+c b-c+d a b+d a+d b a+c+d b-c b+d a b-c+d a+c a+c b-c+d a b+d b-c a+c+d b a+d b+d a b-c+d a+c a+d b a+c+d b-c
B2
cp-r(a-b) cp+cr cp+cs cp-r(a-b+c)+cs cp-r(a-b)+cs cp+cr+cs cp cp-r(a-b+c) cp-r(a-b)+cs cp+cr+cs cp cp-r(a-b+c) cp-r(a-b) cp+cr cp+cs cp-r(a-b+c)+cs cp+cr cp-r(a-b) cp-r(a-b+c)+cs cp+cs cp+cr+cs cp-r(a-b)+cs cp-r(a-b+c) cp cp+cr+cs cp-r(a-b)+cs cp-r(a-b+c) cp cp+cr cp-r(a-b) cp-r(a-b+c)+cs cp+cs cp cp-r(a-b+c) cp-r(a-b)+cs cp+cr+cs cp+cs cp-r(a-b+c)+cs cp-r(a-b) cp+cr cp+cs cp-r(a-b+c)+cs cp-r(a-b) cp+cr cp cp-r(a-b+c) cp-r(a-b)+cs cp+cr+cs cp-r(a-b+c) cp cp+cr+cs cp-r(a-b)+cs cp-r(a-b+c)+cs cp+cs cp+cr cp-r(a-b) cp-r(a-b+c)+cs cp+cs cp+cr cp-r(a-b) cp-r(a-b+c) cp cp+cr+cs cp-r(a-b)+cs
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With proper selected variables,
the resulting square M1 = B1 + B2
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
3 6 7 2 8 1 4 5 2 7 6 3 5 4 1 8 8 1 4 5 3 6 7 2 5 4 1 8 2 7 6 3 1 8 5 4 6 3 2 7 4 5 8 1 7 2 3 6 6 3 2 7 1 8 5 4 7 2 3 6 4 5 8 1 B2
32 40 24 16 48 56 8 0 48 56 8 0 32 40 24 16 40 32 16 24 56 48 0 8 56 48 0 8 40 32 16 24 8 0 48 56 24 16 32 40 24 16 32 40 8 0 48 56 0 8 56 48 16 24 40 32 16 24 40 32 0 8 56 48 M1 = B1 + B2
35 46 31 18 56 57 12 5 50 63 14 3 37 44 25 24 48 33 20 29 59 54 7 10 61 52 1 16 42 39 22 27 9 8 53 60 30 19 34 47 28 21 40 41 15 2 51 62 6 11 58 55 17 32 45 36 23 26 43 38 4 13 64 49
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While varying the variables {a, b, c, d, p, r, s},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of two Diagonal Euler Squares (B1/B2) as shown below: |
B1
a + c b-c a+d b+d a+c+d b-c+d a b b a b-c+d a+c+d b+d a+d b-c a+c b-c+d a+c+d b a b-c a+c b+d a+d a+d b+d a+c b-c a b a+c+d b-c+d b-c a+c b+d a+d b-c+d a+c+d b a a b a+c+d b-c+d a+d b+d a+c b-c a+c+d b-c+d a b a+c b-c a+d b+d b+d a+d b-c a+c b a b-c+d a+c+d
B2
cp + cs cp+cr+cs cp cp+cr cp-r(a-b) +cs cp-r(a-b+c) +cs cp-r(a-b) cp-r(a-b+c) cp-r(a-b) cp-r(a-b+c) cp-r(a-b) +cs cp-r(a-b+c) +cs cp cp+cr cp+cs cp+cr+cs cp+cr+cs cp+cs cp + cr cp cp-r(a-b+c) +cs cp-r(a-b)+cs cp-r(a-b+c) cp-r(a-b) cp-r(a-b+c) cp-r(a-b) cp-r(a-b+c) +cs cp-r(a-b) +cs cp+cr cp cp+cr+cs cp+cs cp cp+cr cp+cs cp+cr+cs cp-r(a-b) cp-r(a-b+c) cp-r(a-b)+cs cp-r(a-b+c) +cs cp-r(a-b)+cs cp-r(a-b+c) +cs cp-r(a-b) cp-r(a-b+c) cp+cs cp+cr+cs cp cp+cr cp+cr cp cp+cr+cs cp+cs cp-r(a-b+c) cp-r(a-b) cp-r(a-b+c) +cs cp-r(a-b)+cs cp-r(a-b+c)+cs cp-r(a-b) +cs cp-r(a-b+c) cp-r(a-b) cp+cr+cs cp+cs cp+cr cp
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With proper selected variables,
the resulting square M1 = B1 + B2
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
2 3 5 8 6 7 1 4 4 1 7 6 8 5 3 2 7 6 4 1 3 2 8 5 5 8 2 3 1 4 6 7 3 2 8 5 7 6 4 1 1 4 6 7 5 8 2 3 6 7 1 4 2 3 5 8 8 5 3 2 4 1 7 6 B2
32 40 0 8 56 48 24 16 24 16 56 48 0 8 32 40 40 32 8 0 48 56 16 24 16 24 48 56 8 0 40 32 0 8 32 40 24 16 56 48 56 48 24 16 32 40 0 8 8 0 40 32 16 24 48 56 48 56 16 24 40 32 8 0 M1 = B1 + B2
34 43 5 16 62 55 25 20 28 17 63 54 8 13 35 42 47 38 12 1 51 58 24 29 21 32 50 59 9 4 46 39 3 10 40 45 31 22 60 49 57 52 30 23 37 48 2 11 14 7 41 36 18 27 53 64 56 61 19 26 44 33 15 6
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While varying the variables {a, b, c, d, p, r, s},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of two Diagonal Euler Squares (B1/B2) as shown below: |
B1
a+c a+d b-c b+d b b-c+d a a+c+d a+c+d a b-c+d b b+d b-c a+d a+c b-c+d b a+c+d a a+d a+c b+d b-c b-c b+d a+c a+d a a+c+d b b-c+d a+d a+c b+d b-c b-c+d b a+c+d a a a+c+d b b-c+d b-c b+d a+c a+d b b-c+d a a+c+d a+c a+d b-c b+d b+d b-c a+d a+c a+c+d a b-c+d b
B2
cq-cr cq cp cp+cr cq+dr cq-cr+dr cp+cr+dr cp+dr cp+cr+dr cp+dr cq+dr cq-cr+dr cp cp+cr cq-cr cq cq cq-cr cp+cr cp cq-cr+dr cq+dr cp+dr cp+cr+dr cp+dr cp+cr+dr cq-cr+dr cq+dr cp+cr cp cq cq-cr cp cp+cr cq-cr cq cp+cr+dr cp+dr cq+dr cq-cr+dr cq+dr cq-cr+dr cp+cr+dr cp+dr cq-cr cq cp cp+cr cp+cr cp cq cq-cr cp+dr cp+cr+dr cq-cr+dr cq+dr cq-cr+dr cq+dr cp+dr cp+cr+dr cq cq-cr cp+cr cp
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With proper selected variables,
the resulting square M1 = B1 + B2
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
2 5 3 8 4 7 1 6 6 1 7 4 8 3 5 2 7 4 6 1 5 2 8 3 3 8 2 5 1 6 4 7 5 2 8 3 7 4 6 1 1 6 4 7 3 8 2 5 4 7 1 6 2 5 3 8 8 3 5 2 6 1 7 4 B2
16 24 0 8 56 48 40 32 40 32 56 48 0 8 16 24 24 16 8 0 48 56 32 40 32 40 48 56 8 0 24 16 0 8 16 24 40 32 56 48 56 48 40 32 16 24 0 8 8 0 24 16 32 40 48 56 48 56 32 40 24 16 8 0 M1 = B1 + B2
18 29 3 16 60 55 41 38 46 33 63 52 8 11 21 26 31 20 14 1 53 58 40 43 35 48 50 61 9 6 28 23 5 10 24 27 47 36 62 49 57 54 44 39 19 32 2 13 12 7 25 22 34 45 51 64 56 59 37 42 30 17 15 4
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While varying the variables {a, b, c, d, p, q, r},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of two Diagonal Euler Squares (B1/B2) as shown below: |
B1
b-c+d a+c b+d a b a+d b-c a+c+d a b+d a+c b-c+d a+c+d b-c a+d b b a+d b-c a+c+d b-c+d a+c b+d a a+c+d b-c a+d b a b+d a+c b-c+d a+d b a+c+d b-c a+c b-c+d a b+d b-c a+c+d b a+d b+d a b-c+d a+c a+c b-c+d a b+d a+d b a+c+d b-c b+d a b-c+d a+c b-c a+c+d b a+d
B2
cp+cr+dr cp+cr cq-cr cq-cr+dr cq+dr cq cp cp+dr cq+dr cq cp cp+dr cp+cr+dr cp+cr cq-cr cq-cr+dr cp+cr cp+cr+dr cq-cr+dr cq-cr cq cq+dr cp+dr cp cq cq+dr cp+dr cp cp+cr cp+cr+dr cq-cr+dr cq-cr cp cp+dr cq+dr cq cq-cr cq-cr+dr cp+cr+dr cp+cr cq-cr cq-cr+dr cp+cr+dr cp+cr cp cp+dr cq+dr cq cp+dr cp cq cq+dr cq-cr+dr cq-cr cp+cr cp+cr+dr cq-cr+dr cq-cr cp+cr cp+cr+dr cp+dr cp cq cq+dr
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With proper selected variables,
the resulting square M1 = B1 + B2
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
7 2 8 1 4 5 3 6 1 8 2 7 6 3 5 4 4 5 3 6 7 2 8 1 6 3 5 4 1 8 2 7 5 4 6 3 2 7 1 8 3 6 4 5 8 1 7 2 2 7 1 8 5 4 6 3 8 1 7 2 3 6 4 5 B2
40 8 16 48 56 24 0 32 56 24 0 32 40 8 16 48 8 40 48 16 24 56 32 0 24 56 32 0 8 40 48 16 0 32 56 24 16 48 40 8 16 48 40 8 0 32 56 24 32 0 24 56 48 16 8 40 48 16 8 40 32 0 24 56 M1 = B1 + B2
47 10 24 49 60 29 3 38 57 32 2 39 46 11 21 52 12 45 51 22 31 58 40 1 30 59 37 4 9 48 50 23 5 36 62 27 18 55 41 16 19 54 44 13 8 33 63 26 34 7 25 64 53 20 14 43 56 17 15 42 35 6 28 61
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While varying the variables {a, b, c, d, p, q, r},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of three Auxiliary Squares (B1/B21/B22) as shown below: |
B1
a+c b-c+d a b+d a+c+d b-c a+d b a b+d a+c b-c+d a+d b a+c+d b-c a+c+d b-c a+d b a+c b-c+d a b+d a+d b a+c+d b-c a b+d a+c b-c+d b-c a+c+d b a+d b-c+d a+c b+d a b a+d b-c a+c+d b+d a b-c+d a+c b-c+d a+c b+d a b-c a+c+d b a+d b+d a b-c+d a+c b a+d b-c a+c+d B21
cq cq cp cp cp+cr cp+cr cq-cr cq-cr cp+cr cp+cr cq-cr cq-cr cq cq cp cp cp cp cq cq cq-cr cq-cr cp+cr cp+cr cq-cr cq-cr cp+cr cp+cr cp cp cq cq cq-cr cq-cr cp+cr cp+cr cp cp cq cq cp cp cq cq cq-cr cq-cr cp+cr cp+cr cp+cr cp+cr cq-cr cq-cr cq cq cp cp cq cq cp cp cp+cr cp+cr cq-cr cq-cr B22
x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1
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With proper selected variables,
the resulting square M1 = B1 + (B21 + B22)
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
2 7 1 8 6 3 5 4 1 8 2 7 5 4 6 3 6 3 5 4 2 7 1 8 5 4 6 3 1 8 2 7 3 6 4 5 7 2 8 1 4 5 3 6 8 1 7 2 7 2 8 1 3 6 4 5 8 1 7 2 4 5 3 6 B2 = B21 + B22
56 24 0 32 8 40 48 16 8 40 48 16 56 24 0 32 32 0 24 56 16 48 40 8 16 48 40 8 32 0 24 56 48 16 8 40 0 32 56 24 0 32 56 24 48 16 8 40 40 8 16 48 24 56 32 0 24 56 32 0 40 8 16 48 M1 = B1 + B2
58 31 1 40 14 43 53 20 9 48 50 23 61 28 6 35 38 3 29 60 18 55 41 16 21 52 46 11 33 8 26 63 51 22 12 45 7 34 64 25 4 37 59 30 56 17 15 42 47 10 24 49 27 62 36 5 32 57 39 2 44 13 19 54
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While varying the variables {a, b, c, d, p, q, r},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of three Auxiliary Squares (B1/B21/B22) as shown below: |
B1
a+c a+c+d b b+d b-c b-c+d a a+d b b+d a+c a+c+d a a+d b-c b-c+d b-c+d b-c a+d a a+c+d a+c b+d b a+d a b-c+d b-c b+d b a+c+d a+c a+c+d a+c b+d b b-c+d b-c a+d a b+d b a+c+d a+c a+d a b-c+d b-c b-c b-c+d a a+d a+c a+c+d b b+d a a+d b-c b-c+d b b+d a+c a+c+d B21
cq cq-cr cp+cr cp cq-cr cq cp cp+cr cp cp+cr cq-cr cq cp+cr cp cq cq-cr cp+cr cp cq cq-cr cp cp+cr cq-cr cq cq-cr cq cp cp+cr cq cq-cr cp+cr cp cp+cr cp cq cq-cr cp cp+cr cq-cr cq cq-cr cq cp cp+cr cq cq-cr cp+cr cp cq cq-cr cp+cr cp cq-cr cq cp cp+cr cp cp+cr cq-cr cq cp+cr cp cq cq-cr B22
x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1
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With proper selected variables,
the resulting square M1 = B1 + (B21 + B22)
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
2 6 4 8 3 7 1 5 4 8 2 6 1 5 3 7 7 3 5 1 6 2 8 4 5 1 7 3 8 4 6 2 6 2 8 4 7 3 5 1 8 4 6 2 5 1 7 3 3 7 1 5 2 6 4 8 1 5 3 7 4 8 2 6 B2 = B21 + B22
56 16 8 32 48 24 0 40 0 40 48 24 8 32 56 16 40 0 24 48 32 8 16 56 16 56 32 8 24 48 40 0 8 32 56 16 0 40 48 24 48 24 0 40 56 16 8 32 24 48 40 0 16 56 32 8 32 8 16 56 40 0 24 48 M1 = B1 + B2
58 22 12 40 51 31 1 45 4 48 50 30 9 37 59 23 47 3 29 49 38 10 24 60 21 57 39 11 32 52 46 2 14 34 64 20 7 43 53 25 56 28 6 42 61 17 15 35 27 55 41 5 18 62 36 16 33 13 19 63 44 8 26 54
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While varying the variables {a, b, c, d, p, q, r},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of three Auxiliary Squares (B11/B12/B2) as shown below: |
B11
br br ar ar ar+cr ar+cr br-cr br-cr ar+cr ar+cr br-cr br-cr br br ar ar ar ar br br br-cr br-cr ar+cr ar+cr br-cr br-cr ar+cr ar+cr ar ar br br br-cr br-cr ar+cr ar+cr ar ar br br ar ar br br br-cr br-cr ar+cr ar+cr ar+cr ar+cr br-cr br-cr br br ar ar br br ar ar ar+cr ar+cr br-cr br-cr B12
x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 B2
p+r q-r+s p q+s p+r+s q-r p+s q p q+s p+r q-r+s p+s q p+r+s q-r p+r+s q-r p+s q p+r q-r+s p q+s p+s q p+r+s q-r p q+s p+r q-r+s q-r p+r+s q p+s q-r+s p+r q+s p q p+s q-r p+r+s q+s p q-r+s p+r q-r+s p+r q+s p q-r p+r+s q p+s q+s p q-r+s p+r q p+s q-r p+r+s
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With proper selected variables,
the resulting square M1 = (B11 + B12) + B2
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1 = B11 + B12
56 24 0 32 8 40 48 16 8 40 48 16 56 24 0 32 32 0 24 56 16 48 40 8 16 48 40 8 32 0 24 56 48 16 8 40 0 32 56 24 0 32 56 24 48 16 8 40 40 8 16 48 24 56 32 0 24 56 32 0 40 8 16 48 B2
2 7 1 8 6 3 5 4 1 8 2 7 5 4 6 3 6 3 5 4 2 7 1 8 5 4 6 3 1 8 2 7 3 6 4 5 7 2 8 1 4 5 3 6 8 1 7 2 7 2 8 1 3 6 4 5 8 1 7 2 4 5 3 6 M1 = B1 + B2
58 31 1 40 14 43 53 20 9 48 50 23 61 28 6 35 38 3 29 60 18 55 41 16 21 52 46 11 33 8 26 63 51 22 12 45 7 34 64 25 4 37 59 30 56 17 15 42 47 10 24 49 27 62 36 5 32 57 39 2 44 13 19 54
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While varying the variables {a, b, c, p, q, r, s},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of three Auxiliary Squares (B11/B12/B2) as shown below: |
B11
br br-cr ar+cr ar br-cr br ar ar+cr ar ar+cr br-cr br ar+cr ar br br-cr ar+cr ar br br-cr ar ar+cr br-cr br br-cr br ar ar+cr br br-cr ar+cr ar ar+cr ar br br-cr ar ar+cr br-cr br br-cr br ar ar+cr br br-cr ar+cr ar br br-cr ar+cr ar br-cr br ar ar+cr ar ar+cr br-cr br ar+cr ar br br-cr B12
x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 B2
p+r p+r+s q q+s q-r q-r+s p p+s q q+s p+r p+r+s p p+s q-r q-r+s q-r+s q-r p+s p p+r+s p+r q+s q p+s p q-r+s q-r q+s q p+r+s p+r p+r+s p+r q+s q q-r+s q-r p+s p q+s q p+r+s p+r p+s p q-r+s q-r q-r q-r+s p p+s p+r p+r+s q q+s p p+s q-r q-r+s q q+s p+r p+r+s
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With proper selected variables,
the resulting square M1 = (B11 + B12) + B2
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1 = B11 + B12
56 16 8 32 48 24 0 40 0 40 48 24 8 32 56 16 40 0 24 48 32 8 16 56 16 56 32 8 24 48 40 0 8 32 56 16 0 40 48 24 48 24 0 40 56 16 8 32 24 48 40 0 16 56 32 8 32 8 16 56 40 0 24 48 B2
2 6 4 8 3 7 1 5 4 8 2 6 1 5 3 7 7 3 5 1 6 2 8 4 5 1 7 3 8 4 6 2 6 2 8 4 7 3 5 1 8 4 6 2 5 1 7 3 3 7 1 5 2 6 4 8 1 5 3 7 4 8 2 6 M1 = B1 + B2
58 22 12 40 51 31 1 45 4 48 50 30 9 37 59 23 47 3 29 49 38 10 24 60 21 57 39 11 32 52 46 2 14 34 64 20 7 43 53 25 56 28 6 42 61 17 15 35 27 55 41 5 18 62 36 16 33 13 19 63 44 8 26 54
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While varying the variables {a, b, c, p, q, r, s},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of three Auxiliary Squares (B1/B21/B22) as shown below: |
B1
b-c b b-c+d b+d a+d a+c+d a a+c b-c+d b+d b-c b a a+c a+d a+c+d a+c+d a+d a+c a b b-c b+d b-c+d a+c a a+c+d a+d b+d b-c+d b b-c b b-c b+d b-c+d a+c+d a+d a+c a b+d b-c+d b b-c a+c a a+c+d a+d a+d a+c+d a a+c b-c b b-c+d b+d a a+c a+d a+c+d b-c+d b+d b-c b B22
x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 B21
(a-b+c)q (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)p (a-b+c)(q-r) (a-b+c)q (a-b+c)p (a-b+c)(p+r) (a-b+c)p (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)q (a-b+c)(p+r) (a-b+c)p (a-b+c)q (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)p (a-b+c)q (a-b+c)(q-r) (a-b+c)p (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)q (a-b+c)(q-r) (a-b+c)q (a-b+c)p (a-b+c)(p+r) (a-b+c)q (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)p (a-b+c)(p+r) (a-b+c)p (a-b+c)q (a-b+c)(q-r) (a-b+c)p (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)q (a-b+c)(q-r) (a-b+c)q (a-b+c)p (a-b+c)(p+r) (a-b+c)q (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)p (a-b+c)q (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)p (a-b+c)(q-r) (a-b+c)q (a-b+c)p (a-b+c)(p+r) (a-b+c)p (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)q (a-b+c)(p+r) (a-b+c)p (a-b+c)q (a-b+c)(q-r)
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With proper selected variables,
the resulting square M1 = B1 + (B21 + B22)
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
3 4 7 8 5 6 1 2 7 8 3 4 1 2 5 6 6 5 2 1 4 3 8 7 2 1 6 5 8 7 4 3 4 3 8 7 6 5 2 1 8 7 4 3 2 1 6 5 5 6 1 2 3 4 7 8 1 2 5 6 7 8 3 4 B2 = B21 + B22
0 40 48 24 8 32 56 16 56 16 8 32 48 24 0 40 16 56 32 8 24 48 40 0 40 0 24 48 32 8 16 56 48 24 0 40 56 16 8 32 8 32 56 16 0 40 48 24 32 8 16 56 40 0 24 48 24 48 40 0 16 56 32 8 M1 = B1 + B2
3 44 55 32 13 38 57 18 63 24 11 36 49 26 5 46 22 61 34 9 28 51 48 7 42 1 30 53 40 15 20 59 52 27 8 47 62 21 10 33 16 39 60 19 2 41 54 29 37 14 17 58 43 4 31 56 25 50 45 6 23 64 35 12
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While varying the variables {a, b, c, d, p, q, r},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
This method is based on the application of three Auxiliary Squares (B1/B21/B22) as shown below: |
B1
b-c a+c+d a b+d b a+d a+c b-c+d a b+d b-c a+c+d a+c b-c+d b a+d b a+d a+c b-c+d b-c a+c+d a b+d a+c b-c+d b a+d a b+d b-c a+c+d a+d b b-c+d a+c a+c+d b-c b+d a b-c+d a+c a+d b b+d a a+c+d b-c a+c+d b-c b+d a a+d b b-c+d a+c b+d a a+c+d b-c b-c+d a+c a+d b B22
x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 x1 0 0 x1 0 x1 x1 0 0 x1 x1 0 x1 0 0 x1 B21
(a-b+c)q (a-b+c)q (a-b+c)p (a-b+c)p (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)q (a-b+c)q (a-b+c)p (a-b+c)p (a-b+c)p (a-b+c)p (a-b+c)q (a-b+c)q (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)p (a-b+c)p (a-b+c)q (a-b+c)q (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)p (a-b+c)p (a-b+c)q (a-b+c)q (a-b+c)p (a-b+c)p (a-b+c)q (a-b+c)q (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)(q-r) (a-b+c)q (a-b+c)q (a-b+c)p (a-b+c)p (a-b+c)q (a-b+c)q (a-b+c)p (a-b+c)p (a-b+c)(p+r) (a-b+c)(p+r) (a-b+c)(q-r) (a-b+c)(q-r)
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With proper selected variables,
the resulting square M1 = B1 + (B21 + B22)
will be Pandiagonal and Bimagic, as illustrated in following numerical example:
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B1
3 6 1 8 4 5 2 7 1 8 3 6 2 7 4 5 4 5 2 7 3 6 1 8 2 7 4 5 1 8 3 6 5 4 7 2 6 3 8 1 7 2 5 4 8 1 6 3 6 3 8 1 5 4 7 2 8 1 6 3 7 2 5 4 B2 = B21 + B22
0 32 56 24 48 16 8 40 48 16 8 40 0 32 56 24 24 56 32 0 40 8 16 48 40 8 16 48 24 56 32 0 8 40 48 16 56 24 0 32 56 24 0 32 8 40 48 16 16 48 40 8 32 0 24 56 32 0 24 56 16 48 40 8 M1 = B1 + B2
3 38 57 32 52 21 10 47 49 24 11 46 2 39 60 29 28 61 34 7 43 14 17 56 42 15 20 53 25 64 35 6 13 44 55 18 62 27 8 33 63 26 5 36 16 41 54 19 22 51 48 9 37 4 31 58 40 1 30 59 23 50 45 12
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While varying the variables {a, b, c, d, p, q, r},
320 (80 unique) Pandiagonal Bimagic Squares can be found.
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