Office Applications and Entertainment, Magic Squares of Subtraction

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1.0 Introduction

The "quadrata subtractionis" were published by Adam Kochanski (Poland) in 1686.

The old paper "Considerationes quaedam circa Quadrata et Cubos Magicos" was rediscovered by Henryk Fuks (Canada) and translated from Latin to English (2020).

In his paper "Magic Squares of Subtraction of Adam Adamandy Kochanski" Henryk Fuks explains these special magic squares, some theory and a few construction methods.

Recently Walter Trump proved the existence of (Additive) Magic Squares which are also Magic Squares of Subtraction (2024).

Following sections will describe for miscellaneous orders possible construction methods for this type of Magic Squares.

2.0 Definition and Terminology

A Magic Square of Subtraction of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, columns and diagonals have the same residuum.

Let the elements of a row, column or diagonal, sorted in decreasing order, be a(n) ... a(1), then the residuum for a square of order n can be defined as Res(n) = a(n) - a(n-1) + a(n-2) - .... a(1).

4.0 Squares of Subtraction (4 x 4)

The number of magic series for order 4 (Additive) Magic Squares (s4 = 34) is 86. The frequency (n4) of the occurring residuum values for these magic series is shown below.

Although for the majority of the occurring residuum values n4 > 10, no (Additive) Magic Squares can be found which are also Squares of Subtraction.

The enumeration of order 4 Magic Squares of Subtraction (non additive) has been summarised in Exhibit 4.

5.0 Squares of Subtraction (5 x 5)

The number of magic series for order 5 (Additive) Magic Squares (s5 = 65) is 1394. The frequency (n5) of the occurring residuum values for these magic series is shown below.

Although for the majority of the occurring residuum values n5 > 12, no (Additive) Magic Squares can be found which are also Squares of Subtraction.

The enumeration of order 5 Associated Magic Squares of Subtraction (non additive) has been summarised in Exhibit 5.

6.0 Squares of Subtraction (6 x 6)

The number of magic series for order 6 (Additive) Magic Squares (s6 = 111) is 32134. Walter Trump selected from these magic series 2025 suitable series with Residuum 15.

By means of a bit-vector program he found 1933 essential different (Additive) Magic Squares which are also Squares of Subtraction.

Attachment 6.4 shows a few (unique) Partly Crosswise Symmetric Magic Squares with s6 = 111 and Res6 = 15, which could be found within this collection.

6.1 Generator Method (6 x 6)

Alternatively subject squares can be constructed with following procedure:

  • Based on (a sub collection of) the magic series mentioned above, 'Generators' with 6 magic rows (s6 = 111, Res6 = 15) can be obtained (ref. CnstrGen6);
  • Based on the Generators obtained above, numerous (transposed) Semi Magic Squares with 6 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators (ref. CnstrSqrs6);
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained.

The procedure described above is illustrated below for the first occurring Semi Magic Square based on the third occurring Generator.

Generator
1 4 17 24 30 35
2 5 18 22 28 36
3 7 16 23 29 33
6 9 15 20 27 34
8 12 14 19 26 32
10 11 13 21 25 31
Semi Magic Square
1 28 29 9 19 25
4 2 33 15 26 31
17 36 7 27 14 10
24 18 16 34 8 11
30 5 23 20 12 21
35 22 3 6 32 13
Simple Magic Square
1 19 9 25 28 29
4 26 15 31 2 33
24 8 34 11 18 16
30 12 20 21 5 23
35 32 6 13 22 3
17 14 27 10 36 7

and is a variation on the methods used by e.g. Achille Rilly (1901) for the construction of Bimagic Squares of order 8 based on limited amounts of Bimagic Series (ref. Section 15.3.1).

6.2 Half Generator Method (6 x 6)

Based on the sub collection of complement free magic series (1280 ea), 115 complement free Generators with 3 magic rows (s6 = 111, Res6 = 15) can be obtained, which can be completed with their complements (blue).

The procedure is as described in Section 6.1 above and is illustrated below for the first occurring Semi Magic Square based on the third occurring (completed) Generator.

Generator
1 4 19 24 28 35
3 10 15 21 30 32
6 11 17 23 25 29
36 33 18 13 9 2
34 27 22 16 7 5
31 26 20 14 12 8
Semi Magic Square
1 32 29 2 16 31
4 30 11 18 34 14
19 3 23 33 7 26
24 10 25 13 27 12
28 15 17 9 22 20
35 21 6 36 5 8
Simple Magic Square
16 32 1 2 29 31
34 30 4 18 11 14
27 10 24 13 25 12
22 15 28 9 17 20
5 21 35 36 6 8
7 3 19 33 23 26

Based on the collection of 115 completed Generators, 244 essential different (Additive) Magic Squares could be obtained with procedure CnstrSqrs6.

Attachment 6.1 shows four (essential different) Almost Associated Magic Squares with s6 = 111 and Res6 = 15, which could be found within this (sub) collection.

Attachment 6.2 shows a few (unique) Almost Crosswise Symmetric Magic Squares with s6 = 111 and Res6 = 15, which could be found within this (sub) collection.

6.3 Other Residuum Values

Attachment 6.3 shows a few Simple Magic Squares of Subtraction with s6 = 111 and different residuum values, which could be obtained with the Half Generator Method described in Section 6.2 above.

7.0 Squares of Subtraction (7 x 7)

7.1 Generator Method (7 x 7)

For order 7 (Additive) Magic Squares (s7 = 175), 55534 complement free magic series with Residuum 25 can be found.

This enables a comparable construction method, for order 7 Magic Squares which are also Squares of Subtraction, as described in Section 6.2 above:

  • Based on the magic series mentioned above, Generators can be constructed with CnstrGen7:
    complement free sets of three complement free rows,
    which can be amended with three rows with their complements (blue) and
    one symmetrical row with the remaining seven integers (green)
  • Based on the Generators obtained above, numerous (transposed) Semi Magic Squares with 7 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators (ref. CnstrSqrs7);
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained.

The procedure described above is illustrated below for the first Semi Magic Square which returned an (Almost Associated) Simple Magic Square, based on the first occurring Generator.

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

Notes

  1. Numerous (Simple) Magic Squares can be generated with the procedure described above. Based on the first Generator 3526 (suitable) Semi Magic Squares were returned within a few minutes (terminated).
  2. Within this (sub) collection 312 Semi Magic Squares could be transformed to Associated Magic Squares, of which a few are shown in Attachment 7.1

7.2 Inlaid Magic Squares of Subtraction (7 x 7)

The collection of Associated Squares of Subtraction contains some interesting sub-collections of which a few examples are shown below:

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

Attachment 7.2 shows miscellaneous Associated Magic Squares of Subtraction with Square Inlays (s3 = 75, s4 = 100), as generated with routine SqrInlay7.

Attachment 7.3a shows miscellaneous Associated Magic Squares of Subtraction with Diamond Inlays (s3 = 75, s4 = 100), as generated with routine DiaInlay7a.

Attachment 7.3b shows miscellaneous Associated Lozenge Squares of Subtraction with one Diamond Inlay (s4 = 100), as generated with routine DiaInlay7b (note 2).

Attachment 7.4 shows miscellaneous Associated Magic Squares of Subtraction with Overlapping Sub Squares (s4 = 100), as generated with routine OverlSqrs7.

Notes

  1. The routines applied above for Inlaid Magic Squares of Subtraction have been adopted from comparable routines for (Additive) Inlaid Magic Squares as deducted in Section 7.6.3. Section 7.6.2 and Section 7.7.3.
  2. It can be proven - by means of exhaustion - that Associated Lozenge Squares of Subtraction with two Diamond Inlays (s3 = 75, s4 = 100) don't exist.

8.0 Squares of Subtraction (8 x 8)

8.1 Half Generator Method (8 x 8)

For order 8 (Additive) Magic Squares (s8 = 260), 3888 complement free Magic Euler Series with Residuum 32 can be found.

This enables a comparable construction method, for order 8 Magic Squares which are also Squares of Subtraction, as described in Section 6.2 above:

  • Based on the Magic Euler Series mentioned above, Generators can be constructed with CnstrGen8:
    complement free sets of four complement free rows,
    which can be amended with four rows with their complements (blue)
  • Based on the Generators obtained above, (transposed) Semi Magic Squares with 8 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators (ref. CnstrSqrs8);
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained.

The procedure described above is illustrated below for the first Semi Magic Square which returned a Simple Magic Square, based on the first occurring Generator.

Generator
1 10 19 28 38 45 56 63
3 12 17 26 40 47 54 61
5 14 23 32 34 41 52 59
7 16 21 30 36 43 50 57
64 55 46 37 27 20 9 2
62 53 48 39 25 18 11 4
60 51 42 33 31 24 13 6
58 49 44 35 29 22 15 8
Semi Magic Square
1 3 5 21 64 48 60 58
10 12 14 30 55 62 42 35
19 17 23 57 46 25 24 49
28 40 34 43 9 53 31 22
38 26 32 36 37 11 51 29
45 47 52 16 20 39 33 8
56 54 41 50 27 4 13 15
63 61 59 7 2 18 6 44
Simple Magic Square
48 1 21 64 3 58 5 60
62 10 30 55 12 35 14 42
11 38 36 37 26 29 32 51
4 56 50 27 54 15 41 13
18 63 7 2 61 44 59 6
53 28 43 9 40 22 34 31
25 19 57 46 17 49 23 24
39 45 16 20 47 8 52 33

Note
Numerous Generators can be generated with the procedure described above, of which the majority results in a few Semi Magic Squares.

8.2 Medjig Method (8 x 8)

Order 8 Magic Squares which are also Squares of Subtraction, can be constructed with the Medjig Construction Method, as described in detail in Section 8.3.1 and illustrated below (s8 = 260, Res8 = 32):

B (4 x 4)
12 13 3 6
7 2 16 9
14 11 5 4
1 8 10 15
A Medjig Square (4 x 4)
3 1 1 3 0 1 1 2
0 2 2 0 3 2 0 3
1 2 0 3 1 3 0 2
0 3 1 2 0 2 3 1
0 1 2 3 0 1 3 2
3 2 1 0 3 2 1 0
2 0 2 0 2 1 3 2
3 1 3 1 3 0 1 0
C Magic Square (8 x 8)
60 28 29 61 3 19 22 38
12 44 45 13 51 35 6 54
23 39 2 50 32 64 9 41
7 55 18 34 16 48 57 25
14 30 43 59 5 21 52 36
62 46 27 11 53 37 20 4
33 1 40 8 42 26 63 47
49 17 56 24 58 10 31 15

Attachment 8.7 shows miscellaneous (Medjig Method Based) Simple Magic Squares of Subtraction, as generated with routine MgcSqr8e.

Attachment 8.8 shows miscellaneous (Medjig Method Based) Associated Magic Squares of Subtraction, as generated with routine MgcSqr8a1.

Attachment 8.9 shows miscellaneous (Medjig Method Based) Pan Magic and Complete Squares of Subtraction, as generated with routine MgcSqr8b1.

8.3 Pan Magic Squares of Subtraction (8 x 8)

The collection of Pan Magic Squares of Subtraction (s8 = 260, Res8 = 16) contains some interesting sub-collections of which a few examples are shown below:

PM, Composed
25 32 34 39 21 20 46 43
36 37 27 30 42 47 17 24
31 26 40 33 19 22 44 45
38 35 29 28 48 41 23 18
10 15 49 56 6 3 61 60
51 54 12 13 57 64 2 7
16 9 55 50 4 5 59 62
53 52 14 11 63 58 8 1
PM, Inlaid
25 21 32 20 34 46 39 43
10 6 15 3 49 61 56 60
36 42 37 47 27 17 30 24
51 57 54 64 12 2 13 7
31 19 26 22 40 44 33 45
16 4 9 5 55 59 50 62
38 48 35 41 29 23 28 18
53 63 52 58 14 8 11 1
PM, Associated
64 19 38 9 15 36 21 58
2 45 28 55 49 30 43 8
51 32 41 6 4 47 26 53
13 34 23 60 62 17 40 11
54 25 48 3 5 42 31 52
12 39 18 61 59 24 33 14
57 22 35 16 10 37 20 63
7 44 29 50 56 27 46 1

Attachment 8.4 shows miscellaneous Pan Magic Squares of Subtraction composed of Pan Magic Sub Squares (s4 = 130), as generated with routine MgcSqr8b.

Attachment 8.5 shows miscellaneous Pan Magic Squares of Subtraction with four Pan Magic Square Inlays (s4 = 130), which are a transformation of the Composed Pan Magic Squares mentioned above.

Attachment 8.6 shows miscellaneous Ultra Magic Squares of Subtraction, as generated with routine MgcSqr8a.

Note

The routines applied above for Pan Magic Squares of Subtraction have been adopted from comparable routines for (Additive) Pan Magic Squares as deducted in Section 8.2.1 and Section 8.6.8.

8.4 Bimagic Squares of Subtraction (8 x 8)

The 136244 Essential Different Bimagic Squares of order eight, as published by Walter Trump and Francis Gaspalou (April 2014), can be stored in an Excel Workbook.

With procedure TestSqrs8 608 Bimagic Squares (s1 = 260, s2 = 11180) appeared to be Squares of Subtraction with Res8 = 32, being the only occurring residuum for Bimagig Squares of order 8.

Attachment 8.1 shows the 32 Associated Bimagic Squares enclosed in subject collection.

Attachment 8.2 shows the 16 Pan Magic and Complete Bimagic Squares enclosed in subject collection.

8.5 Other Residuum Values

Attachment 8.3 shows a few Magic Squares of Subtraction with s8 = 260 and different residuum values, which could be obtained with the methods described above.

9.0 Squares of Subtraction (9 x 9)

9.1 Generator Method (9 x 9)

Euler Series

The number of magic series for order 9 (Additive) Magic Squares (s9 = 369) with Residuum 41 would be inconvenient high. A more controllable collection can be obtained by limiting the collection to complement free Euler Series.

This results in a collection of 19688 magic series (s9 = 369, Res9 = 41) and enables, for order 9 Magic Squares which are also Squares of Subtraction, a comparable construction as described in Section 7.0 above:

  • Based on the magic series mentioned above, Generators can be constructed with CnstrGen9:
    complement free sets of four complement free rows,
    which can be amended with four rows with their complements (blue) and
    one symmetrical row with the remaining nine integers (green)
  • Based on the Generators obtained above, numerous (transposed) Semi Magic Squares with 9 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators (ref. CnstrSqrs9);
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained.

Subject procedure is illustrated below for the first occurring Semi Magic Square based on the first occurring (completed) Generator.

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

Note
Numerous Generators can be generated with the procedure described above, which result each in a number of Semi Magic Squares (e.g. based on the first Generator 21 (suitable) Semi Magic Squares were returned).

9.2 Generator Method (9 x 9)

Bimagic Euler Series

The collection of 19688 magic series described in Section 9.1 above contains a sub collection of 864 complement free bimagic series, which allows for the construction method described in Section 9.1 above.

Subject procedure is illustrated below for the first occurring Semi Magic Square based on the first occurring (completed) Generator.

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

Notes

  1. The Generator shown above has bimagic rows, the resulting Semi- and Simple Magic Squares have bimagic columns.
  2. Based on the (sub) collection described above, 44 Generators can be generated, which result each in a number of Semi Magic Squares (e.g. based on the first Generator 36 (suitable) Semi Magic Squares were returned).

9.3 Inlaid Magic Squares of Subtraction (9 x 9)

The collection of Associated Squares of Subtraction contains some interesting sub-collections of which a few examples are shown below:

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

Attachment 9.2 shows miscellaneous Associated Magic Squares of Subtraction with Square Inlays (s4 = 164, s5 = 205), obtained by means of transformation of order 9 Composed Semi Magic Squares of Subtraction (ref. SqrInlay9).

Attachment 9.3 shows a few Associated Lozenge Squares of Subtraction with Diamond Inlays (s4 = 164, s5 = 205), as generated with routine DiaInlay9.

Attachment 9.4 shows miscellaneous Associated Magic Squares of Subtraction with Overlapping Sub Squares (s5 = 205), as generated with routine OverlSqrs9.

Note
The routines applied above for Inlaid Magic Squares of Subtraction have been adopted from comparable routines for (Additive) Inlaid Magic Squares as deducted in Section 9.7.4. Section 18.4.3 and Section 9.7.3.

10.0 Squares of Subtraction (10 x 10)

10.1 Generator Method (10 x 10)

The number of complement free Euler Series for order 10 (Additive) Magic Squares (s10 = 505) with Residuum 49 would be inconvenient high.

A more controllable collection can be obtained by means of the Medjig Method as described for bimagic series in Section 10.1.6/7.

This results in a (sub) collection of 872 magic series (s10 = 505, Res10 = 49) and enables, for order 10 Magic Squares which are also Squares of Subtraction, following construction method:

  • Based on the (sub) collection of magic series mentioned above, Generators with 10 magic rows (s10 = 505, Res10 = 49) can be obtained (ref. CnstrGen10a);
  • Based on the Generators obtained above, Semi Magic Squares with 10 magic rows and columns can be constructed by permutating the numbers within the rows of the Generators.
  • By permutation of the rows and columns within the Semi Magic Squares, (Additive) Magic Squares which are also Squares of Subtraction can be obtained (ref. CnstrSqrs10b)

Subject procedure is illustrated below for one of the Generators and resulting Semi- and Simple Magic Square.

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

Notes

  1. Based on the (sub) collection of magic series described above, numerous Generators can be generated, within a reasonable time.
  2. The construction of Semi Magic Squares (s10 = 505, Res10 = 49) is illustrated and described in Exhibit 10.

10.2 Half Generator Method (10 x 10)

Based on a sub collection of 1766 complement free magic series, 379 complement free Generators with 5 magic rows (s10 = 505, Res10 = 49) could be obtained, which can be completed with their complements (blue).

The procedure is as described in Section 10.1 above and is illustrated below for the first occurring Semi Magic Square based on the first occurring (completed) Generator.

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

Based on the collection of 379 completed Generators, numerous essential different Magic Squares of Subtraction can be obtained with procedure CnstrSqrs10b.

10.3 Other Residuum Values

Attachment 10.3 shows a few Magic Squares of Subtraction with s10 = 505 and different residuum values, which could be obtained with the methods described above.

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