Office Applications and Entertainment, Magic Squares

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18.0   Special Magic Squares, Lozenge Squares

18.1   Introduction

Lozenge Squares are Magic Squares (odd order) with the even numbers in the corners, as illustrated below:

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

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)

18.2.1 Simple Magic Squares

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.:

      a(1) + a(16) = a(23) + a(24)

can't be thru with a(1), a(16) and a(24) even and a(23) odd.

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.

Attachment Lozenge 5.2 shows the first occurring Associated Lozenge Squares for a(25) = 2, 4 ... 24.

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:

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

Possible Center Squares can be generated with routine Priem3a which generated 32 (4 unique) order 3 Magic Squares with odd numbers.

Based on the four unique Center Squares shown below, 3264 Concentric Lozenge Squares could be generated with routine MgcSqr5c3 within 72 seconds.

1
23 5 11
1 13 25
15 21 3
3
21 7 11
3 13 23
15 19 5

Sqr Nr n9
1 992
3 736
5 864
6 672
5
19 11 9
3 13 23
17 15 7
6
19 9 11
5 13 21
15 17 7

The table to the right side provides a breakdown of the number of Concentric Lozenge Squares for each of the Center Squares shown.

Attachment Lozenge 5.3 shows the first occurring Concentric Lozenge Squares for a(25) = 2, 4 ... 24, for each of the Center Squares shown above.

18.2.5 Inlaid Magic Squares

Diamond Inlay (General):

Lozenge Squares of the fifth order with Diamond Inlay can be generated with routine MgcSqr5g1, which produced 1856 of subject Lozenge Squares within 34 seconds.

The Diamonds of Inlaid Lozenge Squares of the fifth order contain only odd numbers and can be considered as a transformation of the Center Squares discussed in Section 18.2.4 above.

Based on the four unique Diamonds shown below, 232 (= 1856 / 8) Lozenge Squares with Diamond Inlay could be generated with the same routine, which are shown in Attachment Lozenge 5.4.

1
- - 23 - -
- 5 - 1 -
11 - 13 - 15
- 25 - 21 -
- - 3 - -
3
- - 21 - -
- 7 - 3 -
11 - 13 - 15
- 23 - 19 -
- - 5 - -

Inlay Nr n9
1 10
3 34
5 157
6 31
5
- - 19 - -
- 11 - 3 -
9 - 13 - 17
- 23 - 15 -
- - 7 - -
6
- - 19 - -
- 9 - 5 -
11 - 13 - 15
- 21 - 17 -
- - 7 - -

The table to the right side provides a breakdown of the number of Lozenge Squares for each of the Diamond Inlay shown.

Diamond Inlay (Associated):

Associated Lozenge Squares of the fifth order with Diamond Inlay can be generated with routine MgcSqr5g2.

Attachment Lozenge 5.5 shows the 48 Associated Lozenge Squares with Diamond Inlay which could be generated with subject routine in 1,5 seconds.

Diamond Inlay (Concentric):

Concentric Lozenge Squares of the fifth order with Diamond Inlay can be generated with routine MgcSqr5g3, which produced 992 of subject Lozenge Squares within 22,6 seconds.

Attachment Lozenge 5.6 shows the first occurring Lozenge Squares with Diamond Inlay for a(25) = 2, 4 ... 24.

Square Inlay:

Lozenge Squares of the fifth order with Squares Inlay can be generated with routine MgcSqr5g4, which produced 27136 of subject Lozenge Squares within 15 minutes.

The Square Inlays of Inlaid Lozenge Squares of the fifth order contain even corner numbers, as illustrated below:

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

Possible Square Inlays can be generated with routine Priem3b, which generated 72 (9 unique) order 3 Square Inlays with even corner numbers.

Based on the nine unique Square Inlays shown below, 3392 (= 27136 / 8) Inlaid Lozenge Squares could be generated with the same routine.

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

Sqr Nr n9
1 576
3 384
7 480
4 448
8 224
9 352
14 416
15 320
22 192
4
22 5 12
3 13 23
14 21 4
8
20 9 10
3 13 23
16 17 6
9
20 7 12
5 13 21
14 19 6
14
18 11 10
5 13 21
16 15 8
15
18 9 12
7 13 19
14 17 8
22
16 11 12
9 13 17
14 15 10

The table to the right side provides a breakdown of the number of Inlaid Lozenge Squares for each of the Square Inlays shown.

Attachment Lozenge 5.7 shows the first occurring Inlaid Lozenge Squares for a(25) = 2, 4 ... 24, for each of the Square Inlays shown above.

Square - and Diamond Inlay:

Also Lozenge Squares of the fifth order with Square - and Diamond Inlays can be generated with routine MgcSqr5g4 if the option 'Check Diamond Inlay' is enabled.

Attachment Lozenge 5.8 shows the 160 Lozenge Squares with Square - and Diamond Inlays which could be generated with subject routine in 8,2 seconds.

18.3   Lozenge Squares (7 x 7)

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:

44 10 12 21 30 42 16
14 48 15 17 41 4 36
18 7 47 5 23 43 32
19 11 1 25 49 39 31
22 13 27 45 3 37 28
24 46 35 33 9 2 26
34 40 38 29 20 8 6

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.

Routine MgcSqr7b constructs, based on the Center Squares (3 x 3), suitable Center Squares (5 x 5) and completes the Concentric Lozenge Squares (7 x 7).

Attachment Lozenge 7.1 shows the first occurring Concentric Lozenge Square (7 x 7) for each of the order 3 Center Squares with odd numbers shown in Attachment 18.3.1.

Notes:

  1. By means of permutation of the border pairs of the main - and order 5 sub square, each of the order 7 Concentric Lozenge Squares shown correspond with 62 * 242 = 20736 squares.
  2. While varying the border corner points of the main - and sub square, this number has at least to be multiplied with 24 * 20 = 480 resulting in 107 squares, for each unique order 3 Center Square.

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.

Both the order 3 and 4 Diamonds of Inlaid Lozenge Squares of the seventh order contain only odd numbers, as illustrated below:

38 36 44 7 4 16 30
8 24 9 27 47 28 32
2 35 29 37 17 45 10
49 19 11 25 39 31 1
40 5 33 13 21 15 48
18 22 3 23 41 26 42
20 34 46 43 6 14 12

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.

The order 4 Diamonds can be generated with routine Priem4d, which generated 13056 (34 unique) order 4 Diamonds with odd numbers (s4 = 164). A set of unique order 4 Diamonds is shown in Attachment 18.3.3.

Attachment Lozenge 7.2 shows the first occurring 108 Associated Lozenge Squares with Diamond Inlays, which are based on order 4 Diamond Inlay Nr. 1.

18.4   Lozenge Squares (9 x 9)

18.4.1 Concentric Magic Squares

The Concentric Center Squares of a Concentric Lozenge Square of the ninth order have following properties:

  • The 3 x 3 and 5 x 5 Center Square have only odd numbers
  • The 7 x 7 Concentric Square has 4 x 3 even corner numbers

as illustrated in following example:

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

Concentric Lozenge Squares of order 9 can be constructed as follows:

  • Generate order 3 Center Squares (odd numbers);
  • Generate order 5 Concentric Squares (odd numbers), based on a selection of the order 3 Center Squares;
  • Generate order 7 Concentric Squares (4 x 3 even corner numbers), based on a selection of the order 5 Concentric Squares;
  • Generate order 9 Concentric Lozenge Squares, based on a selection of the order 7 Concentric Squares.

Routine Priem3a generated 672 (84 unique) order 3 Center Squares with odd numbers (s1 = 123).
Attachment 18.4.1 shows a set of unique order 3 Center Squares.

Attachment 18.4.3 shows the first occurring order 5 Concentric Squares with odd numbers for each of the order 3 Center Squares with odd numbers shown in Attachment 18.4.1 (ref. Priem5c).

Attachment 18.4.4 shows the first occurring order 7 Concentric Squares with 4 x 3 even corner numbers for each of the order 5 Center Squares with odd numbers shown in Attachment 18.4.3 (ref. Priem7c).

Attachment Lozenge 9.1 shows the first occurring order 9 Concentric Lozenge Squares for each of the order 7 Concentric Squares with 4 x 3 even corner numbers shown in Attachment 18.4.4 (ref. MgcSqr9a).

Notes:

  1. By means of permutation of the border pairs of the main - and sub squares, each of the order 9 Concentric Lozenge Squares shown correspond with 62 * 122 * 7202 = 2,7 109 squares.
  2. While varying the border corner points of the main - and sub squares, this number has at least to be multiplied with 32 * 40 * 28 = 35840 resulting in 9,6 1013 squares, for each unique order 3 Center Square.

Both values are only a fraction of the total possible number of order 9 Concentric Lozenge Squares.

18.4.2 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):

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

can be constructed as follows:

  • Read the previously generated Order 4 Associated Magic Diamonds with Magic Sum s4 = 164;
  • Generate Order 5 Associated Magic Diamonds with Magic Sum s5 = 205;
  • Complete the Order 9 Associated Lozenge Squares with the remaining Border Pairs.

Both the order 4 and 5 Diamond Inlays contain only odd numbers, as can be seen in the example above.

Routine Priem4d generated 278400 (725 unique) order 4 Diamond Inlays with odd numbers, of which a few are shown in Attachment 18.4.2.

The border (corners) can be described by following equations:

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)

a9( 1) = 82 - a9(81)
a9( 2) = 82 - a9(80)
a9( 3) = 82 - a9(79)
a9( 4) = 82 - a9(78)
a9( 6) = 82 - a9(76)

a9( 9) = 82 - a9(73)
a9(10) = 82 - a9(72)
a9(11) = 82 - a9(71)
a9(12) = 82 - a9(70)
a9(28) = 82 - a9(54)

a9(46) = 82 - a9(36)
a9(55) = 82 - a9(27)
a9(56) = 82 - a9(26)
a9(62) = 82 - a9(20)
a9(63) = 82 - a9(19)

a9(64) = 82 - a9(18)
a9(65) = 82 - a9(17)
a9(66) = 82 - a9(16)
a9(74) = 82 - a9( 8)
a9(75) = 82 - a9( 7)

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.

Subject equations can be incorporated in guessing routine to generate the defined Associated Lozenge Squares (ref. MgcSqr9b).

Attachment Lozenge 9.2 shows the first occurring Associated Lozenge Squares for the order 4 Diamond Inlays shown in Attachment 18.4.2.

Note:
With a9(79), a9(78), a9(36) and the order 4 and 5 Diamond Inlays fixed the same routine generated 239996 squares within about 3 hours, which is only a fraction of the total possible number of order 9 Associated Lozenge Squares.

18.4.3 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.

Note:
The resulting collection contains 24 = 16 squares, however for each square is the 180o rotated aspect included.

18.5   Lozenge Squares (11 x 11)

18.5.1 Concentric Magic Squares

The Concentric Center Squares of a Concentric Lozenge Square of the eleventh order have following properties:

  • The 3 x 3 and 5 x 5 Center Square have only odd numbers
  • The 7 x 7 Concentric Square has 4 even corner numbers
  • The 9 x 9 Concentric Square has 4 x 5 even corner numbers

as illustrated in following example:

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

Concentric Lozenge Squares of order 11 can be constructed as follows:

  • Generate order 3 Center Squares (odd numbers);
  • Generate order 5 Concentric Squares (odd numbers), based on a selection of the order 3 Center Squares;
  • Generate order 7 Concentric Squares (4 even corner numbers), based on a selection of the order 5 Concentric Squares;
  • Generate order 9 Concentric Squares (4 x 5 even corner numbers), based on a selection of the order 7 Concentric Squares;
  • Generate order 11 Concentric Lozenge Squares, based on a selection of the order 9 Concentric Squares.

Routine Priem3a generated 1600 (200 unique) order 3 Center Squares with odd numbers (s1 = 183).
Attachment 18.5.1 shows a set of unique order 3 Center Squares.

Attachment 18.5.3 shows the first occurring order 5 Concentric Squares with odd numbers for each of the order 3 Center Squares with odd numbers shown in Attachment 18.5.1 (ref. Priem5c).

Attachment 18.5.4 shows the first occurring order 7 Concentric Squares with 4 even corner numbers for each of the order 5 Concentric Squares with odd numbers shown in Attachment 18.5.3 (ref. Priem7d).

Attachment 18.5.5 shows the first occurring order 9 Concentric Squares with 4 x 5 even corner numbers for each of the order 7 Concentric Squares with 4 even corner numbers shown in Attachment 18.5.4 (ref. Priem9c).

Attachment Lozenge 11.1 shows the first occurring order 11 Concentric Lozenge Squares for each of the order 9 Concentric Squares with 4 x 5 even corner numbers shown in Attachment 18.5.5 (ref. MgcSqr11a).

Notes:

  1. By means of permutation of the border pairs of the main - and sub squares, each of the order 11 Concentric Lozenge Squares shown correspond with 62 * 1202 * 1442 * (8!)2 = 1,75 1019 squares.
  2. While varying the border corner points of the main - and sub squares, this number has at least to be multiplied with 52 * 60 * 56 * 36 = 6,2 * 106 resulting in 1,1 * 1026 squares, for each unique order 3 Center Square.

Both values are only a fraction of the total possible number of order 11 Concentric Lozenge Squares.

18.5.2 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.

Note:
The resulting collection contains 25 = 32 squares, however for each square is the 180o rotated aspect included.

18.6   Summary

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

MgcSqr5b2

-

306416

a(13)=13

Associated

MgcSqr5c2

Lozenge 5.2

6912

-

Concentric

MgcSqr5c3

Lozenge 5.3

3264

Note 1

Magic/Diamond Inlay

MgcSqr5g1

Lozenge 5.4

1856

-

Associated/Diamond Inlay

MgcSqr5g2

Lozenge 5.5

48

-

Concentric/Diamond Inlay

MgcSqr5g3

Lozenge 5.6

992

-

Magic/Square Inlay

MgcSqr5g4

Lozenge 5.7

27136

-

Magic/Square + Diamond Inlay

MgcSqr5g4

Lozenge 5.8

160

-

7

Concentric

MgcSqr7b

Lozenge 7.1

-

-

Associated, Diamond Inlays

MgcSqr7c

Lozenge 7.2

53980

a(28)=1

9

Concentric

MgcSqr9a

Lozenge 9.1

-

-

Associated, Diamond Inlays

MgcSqr9b

Lozenge 9.2

-

-

Associated, Latin Square Based

-

Lozenge 9.3

-

-

11

Concentric

MgcSqr11a

Lozenge 11.1

-

-

Associated, Latin Square Based

-

Lozenge 11.2

-

-

Note 1: Based on 4 unique Center Squares


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