Office Applications and Entertainment, Magic Squares  
Index  About the Author 
18.0 Special Magic Squares, Lozenge Squares
Lozenge Squares are Magic Squares (odd order) with the even numbers in the corners, as illustrated below:
Lozenge Squares of a certain order, can be generated  relatively fast  with comparable routines as discussed in the corresponding sections.
18.2 Lozenge Squares (5 x 5)
Routine MgcSqr5b2 counted with a(13) = 13, 306416 (38302 unique) Lozenge Squares of order 5 within about half an hour.
Following sections will describe some interesting sub sets.
18.2.2 Pan Magic Squares, Don't Exist
The equations defining a Pan Magic Square of the fifth order as deducted in Section 3.1 proof that order 5 Pan Magic Lozenge Squares don't exist e.g.:
18.2.3 Associated Magic Squares
Associated Lozenge Squares of the fifth order can be generated with routine MgcSqr5c2, which produced 6912 Associated Lozenge Squares within 132 seconds.
18.2.4 Concentric Magic Squares
The 3 x 3 Center Square of a Concentric Lozenge Square of the fifth order contains only odd numbers, as illustrated below:
Possible Center Squares can be generated with routine Priem3a which generated 32 (4 unique) order 3 Magic Squares with odd numbers.
The table to the right side provides a breakdown of the number of Concentric Lozenge Squares for each of the Center Squares shown.
Diamond Inlay (General):
The table to the right side provides a breakdown of the number of Lozenge Squares for each of the Diamond Inlay shown.
Possible Square Inlays can be generated with routine Priem3b, which generated 72 (9 unique) order 3 Square Inlays with even corner numbers.
The table to the right side provides a breakdown of the number of Inlaid Lozenge Squares for each of the Square Inlays shown.
18.3.1 Concentric Magic Squares
The 5 x 5 Center Square of a Concentric Lozenge Square of the seventh order contains only 4 even corner numbers, as illustrated below:
Possible Center Squares (3 x 3) could be generated with routine Priem3a,
which generated 208 (26 unique) order 3 Magic Squares with odd numbers.
A set of unique squares is shown in Attachment 18.3.1.
Both values are only a fraction of the total possible number of order 7 Concentric Lozenge Squares.
18.3.2 Associated Magic Squares (Diamond Inlays)
Lozenge Squares of the seventh order, with Diamond Inlays of order 3 and 4, can be generated with routine MgcSqr7c, which counted 53980 of subject Lozenge Squares with a(28) = 1.
The order 3 Diamonds can be considered as a transformation of the Center Squares discussed in Section 18.3.1 above,
as shown in Attachment 18.3.2.
18.4.1 Concentric Magic Squares
The Concentric Center Squares of a Concentric Lozenge Square of the ninth order have following properties:
as illustrated in following example:
Concentric Lozenge Squares of order 9 can be constructed as follows:
Routine Priem3a generated 672 (84 unique) order 3 Center Squares with odd numbers (s1 = 123).
Both values are only a fraction of the total possible number of order 9 Concentric Lozenge Squares.
18.4.2 Concentric Magic Squares (Diamond Inlays)
The defining equations for a 5 x 5 Diamond Inlay suitable for order 9 Concentric Lozenge Squares are a(58) =  s1/9  a(59)  a(60) + a(67) + a(69) + 2 * a(77) a(57) = 6*s1/9  a(61)  a(67)  a(69)  2 * a(77) a(50) = 3*s1/9 + 2 * a(59) + a(67) + a(69) a(49) = 6*s1/9  a(51)  2 * a(59)  a(67)  a(69) a(45) = 5*s1/9  a(53)  a(61)  a(69)  a(77) a(43) = 5*s1/9  a(51)  a(53)  a(59)  a(69) a(42) = 7*s1/9  2 * a(51)  2 * a(59)  a(67)  a(69) a(35) = a(53)  a(67) + a(69) a(34) = 4*s1/9 + a(51)  a(52) + a(53) + a(59)  2 * a(61)  a(67)  2 * a(77)
With a(44), a(51), a(52), a(53), a(59), a(60), a(61), a(67), a(68), a(69), a(77) the independent variables,
Concentric Lozenge Squares with order 5 Diamond Inlays can be constructed as follows:
Attachment 18.4.5 shows miscellaneous suitable order 5 Diamond Inlays with odd numbers
(ref. Diamond5).
18.4.3 Associated Magic Squares (Diamond Inlays)
Associated Lozenge Squares of order 9, with Associated Diamond Inlays of order 4 and 5 as shown in following example (L.S. Frierson):
can be constructed as follows:
Both the order 4 and 5 Diamond Inlays contain only odd numbers, as can be seen in the example above.
a9( 7) = 369  a9(16)  a9(25)  a9(34)  a9(43)  a9(52)  a9(61)  a9(70)  a9(79) a9( 8) = 369  a9(17)  a9(26)  a9(35)  a9(44)  a9(53)  a9(62)  a9(71)  a9(80) a9(18) = 369  a9( 9)  a9(27)  a9(36)  a9(45)  a9(54)  a9(63)  a9(72)  a9(81) a9(19) = 369  a9(20)  a9(21)  a9(22)  a9(23)  a9(24)  a9(25)  a9(26)  a9(27) a9(54) = 369  a9(46)  a9(47)  a9(48)  a9(49)  a9(50)  a9(51)  a9(52)  a9(53) a9(72) = (410  a9( 9)  a9(81) + a9(17)  a9(71)  a9(45) + a9(46)  a9(54) + a9(55)  a9(63)  a9(66)  a9(67)  a9(68)  a9(69)  a9(70)) / 2 a9(73) = 369  a9(74)  a9(75)  a9(76)  a9(77)  a9(78)  a9(79)  a9(80)  a9(81) a9(76) = 369  a9( 4)  a9(13)  a9(22)  a9(31)  a9(40)  a9(49)  a9(58)  a9(67)
with a9(16), a9(17), a9(20), a9(26), a9(27), a9(36), a9(70), a9(71) and a9(78) thru a9(81) the independent variables.
18.4.4 Associated Magic Squares (L.S. Frierson)
A more detailed observation of the Associated Lozenge Square with order 4 and 5 Diamond Inlays, as published by L.S. Frierson:
learns that subject square has following additional properties:
After incorporation of subject properties, routine Priem4d2 generated numerous
order 4 Diamond Inlays with the properties described above, of which a few are shown in Attachment 18.4.22.
18.4.5 Associated Magic Squares (La Hire)
Associated Lozenge Squares can be constructed based on Latin Squares (B1/B2) as illustrated below (La Hire): 
B1
5 6 7 8 0 1 2 3 4 6 7 8 0 1 2 3 4 5 7 8 0 1 2 3 4 5 6 8 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 0 2 3 4 5 6 7 8 0 1 3 4 5 6 7 8 0 1 2 4 5 6 7 8 0 1 2 3 B2 (B1 Mirrored)
4 3 2 1 0 8 7 6 5 5 4 3 2 1 0 8 7 6 6 5 4 3 2 1 0 8 7 7 6 5 4 3 2 1 0 8 8 7 6 5 4 3 2 1 0 0 8 7 6 5 4 3 2 1 1 0 8 7 6 5 4 3 2 2 1 0 8 7 6 5 4 3 3 2 1 0 8 7 6 5 4 M = B1 + 9 * B2 + 1
42 34 26 18 1 74 66 58 50 52 44 36 19 11 3 76 68 60 62 54 37 29 21 13 5 78 70 72 55 47 39 31 23 15 7 80 73 65 57 49 41 33 25 17 9 2 75 67 59 51 43 35 27 10 12 4 77 69 61 53 45 28 20 22 14 6 79 71 63 46 38 30 32 24 16 8 81 64 56 48 40
The resulting Lozenge Square M corresponds with 16 Lozenge Squares, which
can be obtained by exchange of row and column n with (10  n), as shown in Attachment Lozenge 9.3.
18.6 Lozenge Squares (11 x 11)
18.6.1 Concentric Magic Squares
The Concentric Center Squares of a Concentric Lozenge Square of the eleventh order have following properties:
as illustrated in following example:
Concentric Lozenge Squares of order 11 can be constructed as follows:
Routine Priem3a generated 1600 (200 unique) order 3 Center Squares with odd numbers (s1 = 183).
Both values are only a fraction of the total possible number of order 11 Concentric Lozenge Squares.
18.6.2 Concentric Magic Squares (Diamond Inlays)
The defining equations for a 6 x 6 Diamond Inlay suitable for order 11 Concentric Lozenge Squares are a(92) = 6*s1/11  a(96)  a(104)  a(106)  2 * a(116) a(91) = s1/11  a(93)  a(94)  a(95)  a(97) + a(104) + a(106) + 2 * a(116) a(82) = 6*s1/11  a(84)  2 * a(94)  a(104)  a(106) a(81) =  s1/11  a(83)  a(85) + 2 * a(94) + a(104) + a(106) a(73) =  s1/11 + 0.5 * a(74) + 0.5 * a(84) + 0.5 * a(86) + 0.5 * a(96) a(72) = 3*s1/11 + a(94) + a(104) + a(106) + a(116) a(71) = 7*s1/11  0.5 * a(74)  0.5 * a(84)  0.5 * a(86)  a(94)  0.5 * a(96)  a(104)  a(106)  a(116) a(66) = 6*s1/11  a(76)  a(86)  a(96)  a(106)  a(116) a(63) = p2 + a(64)  a(74) + a(76)  a(83)  a(84)  2 * a(85) + a(94) + a(106) a(62) = 9*s1/11  a(74)  a(84)  a(86)  a(94)  a(96)  a(104)  a(106)  a(116) a(54) = 6*s1/11  a(64)  a(74)  a(84)  a(94)  a(104) a(53) = 17*s1/11  2 * a(64)  a(74)  a(75)  a(76)  a(84)  2*a(86) + a(91) +  a(94)  2*a(96)  a(97)  a(104)  2*a(106)  2*a(116) a(52) =  a(64)  a(76) + a(84) + a(94) + a(104) a(42) = 12*s1/11 + a(64) + a(74) + a(76) + a(84) + a(86) + a(94) + 2 * a(96) + a(104) + 2 * a(106) + 2 * a(116)
With the independent variables:
Concentric Lozenge Squares with order 6 Diamond Inlays can be constructed as follows:
Attachment 18.4.7 shows miscellaneous suitable order 6 Diamond Inlays with odd numbers and
the four (even) corner points of the order 7 concentric squares
(ref. Diamond6).
18.6.3 Associated Magic Squares (Diamond Inlays)
Associated Lozenge Squares of order 11, with Associated Diamond Inlays of order 5 and 6 as shown in following example:
can be constructed as follows:
Both the order 5 and 6 Diamond Inlays contain only odd numbers, as can be seen in the example above.
a(115) = s1  a( 5)  a( 16)  a( 27)  a( 38)  a( 49)  a( 60)  a( 71)  a( 82)  a( 93)  a(104) a( 55) = s1  a( 45)  a( 46)  a( 47)  a( 48)  a( 49)  a( 50)  a( 51)  a( 52)  a( 53)  a( 54) a(118) = s4  a(119)  a(120)  a(121) a(103) = s1  a( 4)  a( 15)  a( 26)  a( 37)  a( 48)  a( 59)  a( 70)  a( 81)  a( 92)  a(114) a( 88) = s4  a( 99)  a(110)  a(121) a( 87) = s1  a( 78)  a( 79)  a( 80)  a( 81)  a( 82)  a( 83)  a( 84)  a( 85)  a( 86)  a( 88) a(113) = s1  a(111)  a(112)  a(114)  a(115)  a(116)  a(117)  a(118)  a(119)  a(120)  a(121) a( 89) = s1  a( 1)  a( 12)  a( 23)  a( 34)  a( 45)  a( 56)  a( 67)  a( 78)  a(100)  a(111) a(101) = s1  a(100)  a(102)  a(103)  a(104)  a(105)  a(106)  a(107)  a(108)  a(109)  a(110) a( 90) = s1  a( 2)  a( 13)  a( 24)  a( 35)  a( 46)  a( 57)  a( 68)  a( 79)  a(101)  a(112) a( 97) =(2*s1/11  a(9) a( 20) + a( 23) + a( 24) + a( 26) + a( 27) + a( 28) + a( 29) + a( 30) + a( 32) + + a(33)  a(42)  a(53)  a(64)  a(75)  a(86)  a(108)  a(119)) / 2 a( 91) = s1  a( 97)  a( 89)  a( 90)  a( 92)  a( 93)  a( 94)  a( 95)  a( 96)  a( 98)  a( 99)
with the independent variables:
18.6.4 Associated Magic Squares (La Hire)
Associated Lozenge Squares can be constructed based on Latin Squares (B1/B2) as illustrated below (La Hire): 
B1
6 7 8 9 10 0 1 2 3 4 5 7 8 9 10 0 1 2 3 4 5 6 8 9 10 0 1 2 3 4 5 6 7 9 10 0 1 2 3 4 5 6 7 8 10 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0 2 3 4 5 6 7 8 9 10 0 1 3 4 5 6 7 8 9 10 0 1 2 4 5 6 7 8 9 10 0 1 2 3 5 6 7 8 9 10 0 1 2 3 4 B2 (B1 Mirrored)
5 4 3 2 1 0 10 9 8 7 6 6 5 4 3 2 1 0 10 9 8 7 7 6 5 4 3 2 1 0 10 9 8 8 7 6 5 4 3 2 1 0 10 9 9 8 7 6 5 4 3 2 1 0 10 10 9 8 7 6 5 4 3 2 1 0 0 10 9 8 7 6 5 4 3 2 1 1 0 10 9 8 7 6 5 4 3 2 2 1 0 10 9 8 7 6 5 4 3 3 2 1 0 10 9 8 7 6 5 4 4 3 2 1 0 10 9 8 7 6 5 M = B1 + 11* B2 + 1
62 52 42 32 22 1 112 102 92 82 72 74 64 54 44 23 13 3 114 104 94 84 86 76 66 45 35 25 15 5 116 106 96 98 88 67 57 47 37 27 17 7 118 108 110 89 79 69 59 49 39 29 19 9 120 111 101 91 81 71 61 51 41 31 21 11 2 113 103 93 83 73 63 53 43 33 12 14 4 115 105 95 85 75 65 55 34 24 26 16 6 117 107 97 87 77 56 46 36 38 28 18 8 119 109 99 78 68 58 48 50 40 30 20 10 121 100 90 80 70 60
The resulting Lozenge Square M corresponds with 32 Lozenge Squares, which
can be obtained by exchange of row and column n with (12  n), as shown in Attachment Lozenge 11.1.
The obtained results regarding the miscellaneous types of Lozenge Squares as deducted and discussed in previous sections are summarized in following table: 
Order
Characteristics
Subroutine
Examples
Total Number
Notes
5
Simple Magic

306416
a(13)=13
Associated
6912

Concentric
3264
Note 1
Magic/Diamond Inlay
1856

Associated/Diamond Inlay
48

Concentric/Diamond Inlay
992

Magic/Square Inlay
27136

Magic/Square + Diamond Inlay
160

7
Concentric


Associated, Diamond Inlays
53980
a(28)=1
9
Concentric


Concentric, Diamond Inlays


Associated, Diamond Inlays (1)


Associated, Diamond Inlays (2)


Associated, Latin Square Based



11
Concentric


Concentric, Diamond Inlays


Associated, Diamond Inlays


Associated, Latin Square Based



Note 1: Based on 4 unique Center Squares

Index  About the Author 