7.7 Overlapping Sub Squares
7.7.1 Introduction
On Harvey Heinz's site I found following 7^{th} order Magic Square, which contains a 5^{th} order Pan Magic Corner Square (bottom/right) as described in previous section – and a 3^{th} order Semi Magic Corner Square (top/left corner, only one diagonal summing to 75).
22 
4 
49 
48 
44 
5 
3 
46 
28 
1 
2 
6 
45 
47 
7 
43 
25 
30 
41 
12 
17 
36 
14 
13 
19 
24 
32 
37 
8 
42 
31 
39 
9 
20 
26 
35 
15 
16 
27 
33 
38 
11 
21 
29 
40 
10 
18 
23 
34 
It can be proven that:

The application of a Simple Magic 3^{th} order square will require a(17) = a(9) = 25;

The application of a Semi Magic 3^{th} order square as shown, will require a(17) = 25 independent from the type of 5^{th} order Magic Square.
Consequently the 5^{th} order square can’t be Symmetric, Ultra Magic, Concentric or Eccentric as described in previous sections.
7.7.2 Analysis
As for the 5^{th} order Corner Square only (Pan) Magic Squares can be used, the full set of equations will be used for the generation of Magic Squares with overlapping sub squares (rather than using the equations of the supplementary rows and columns as a starting point as in Section 7.5.2).
a(1) 
a(2) 
a(3) 
a(4) 
a(5) 
a(6) 
a(7) 
a(8) 
a(9) 
a(10) 
a(11) 
a(12) 
a(13) 
a(14) 
a(15) 
a(16) 
a(17) 
a(18) 
a(19) 
a(20) 
a(21) 
a(22) 
a(23) 
a(24) 
a(25) 
a(26) 
a(27) 
a(28) 
a(29) 
a(30) 
a(31) 
a(32) 
a(33) 
a(34) 
a(35) 
a(36) 
a(37) 
a(38) 
a(39) 
a(40) 
a(41) 
a(42) 
a(43) 
a(44) 
a(45) 
a(46) 
a(47) 
a(48) 
a(49) 
The defining equations for the Simple Magic Square of the 7^{th} order are:
a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) + a( 7) = 175
a( 8) + a( 9) + a(10) + a(11) + a(12) + a(13) + a(14) = 175
a(15) + a(16) + a(17) + a(18) + a(19) + a(20) + a(21) = 175
a(22) + a(23) + a(24) + a(25) + a(26) + a(27) + a(28) = 175
a(29) + a(30) + a(31) + a(32) + a(33) + a(34) + a(35) = 175
a(36) + a(37) + a(38) + a(39) + a(40) + a(41) + a(42) = 175
a(43) + a(44) + a(45) + a(46) + a(47) + a(48) + a(49) = 175
a( 1) + a( 8) + a(15) + a(22) + a(29) + a(36) + a(43) = 175
a( 2) + a( 9) + a(16) + a(23) + a(30) + a(37) + a(44) = 175
a( 3) + a(10) + a(17) + a(24) + a(31) + a(38) + a(45) = 175
a( 4) + a(11) + a(18) + a(25) + a(32) + a(39) + a(46) = 175
a( 5) + a(12) + a(19) + a(26) + a(33) + a(40) + a(47) = 175
a( 6) + a(13) + a(20) + a(27) + a(34) + a(41) + a(48) = 175
a( 7) + a(14) + a(21) + a(28) + a(35) + a(42) + a(49) = 175
a( 1) + a( 9) + a(17) + a(25) + a(33) + a(41) + a(49) = 175
a( 7) + a(13) + a(19) + a(25) + a(31) + a(37) + a(43) = 175
The defining equations for the Pan Magic Corner Square of the 5^{th} order are:
a(17) + a(18) + a(19) + a(20) + a(21) = 125
a(24) + a(25) + a(26) + a(27) + a(28) = 125
a(31) + a(32) + a(33) + a(34) + a(35) = 125
a(38) + a(39) + a(40) + a(41) + a(42) = 125
a(45) + a(46) + a(47) + a(48) + a(49) = 125
a(17) + a(24) + a(31) + a(38) + a(45) = 125
a(18) + a(25) + a(32) + a(39) + a(46) = 125
a(19) + a(26) + a(33) + a(40) + a(47) = 125
a(20) + a(27) + a(34) + a(41) + a(48) = 125
a(21) + a(28) + a(35) + a(42) + a(49) = 125
a(17) + a(25) + a(33) + a(41) + a(49) = 125
a(18) + a(26) + a(34) + a(42) + a(45) = 125
a(19) + a(27) + a(35) + a(38) + a(46) = 125
a(20) + a(28) + a(31) + a(39) + a(47) = 125
a(21) + a(24) + a(32) + a(40) + a(48) = 125
a(21) + a(27) + a(33) + a(39) + a(45) = 125
a(17) + a(28) + a(34) + a(40) + a(46) = 125
a(18) + a(24) + a(35) + a(41) + a(47) = 125
a(19) + a(25) + a(31) + a(42) + a(48) = 125
a(20) + a(26) + a(32) + a(38) + a(49) = 125
The defining equations for the Semi Magic Square of the 3^{th} order are:
a( 1) + a( 2) + a( 3) = 75
a( 8) + a( 9) + a(10) = 75
a(15) + a(16) + a(17) = 75
a( 1) + a( 8) + a(15) = 75
a( 2) + a( 9) + a(16) = 75
a( 3) + a(10) + a(17) = 75
a( 1) + a( 9) + a(17) = 75
The resulting number of equations can be written in matrix representation as:
→ →
A * a = s
which can be reduced, by means of row and column manipulations, and results in following set of linear equations:
a(45) = 125  a(46)  a(47)  a(48)  a(49)
a(43) = 50  a(44)
a(39) = 150  a(40)  a(41)  a(42)  a(47)  a(48)
a(38) = 25 + a(47) + a(48)
a(36) = 50  a(37)
a(35) = 25  a(41)  a(42) + a(45) + a(46)
a(34) = 125  a(40)  a(41)  a(42)  a(48)
a(33) = 25 + a(42) + a(48)
a(32) = a(41) + a(42)  a(46)
a(31) = a(32) + a(40)  a(42)  a(45) + a(46)
a(29) = 50  a(30)
a(28) = 100  a(31)  a(34) + a(41)  a(45)  a(46)
a(27) =  a(34) + a(47) + a(49)
a(26) =  a(28) + a(41) + a(48)
a(25) = 125  a(41)  a(42)  a(48)  a(49)
a(24) = 125  a(40)  a(41)  a(47)  a(48)
a(23) = 100  a(30)  a(37)  a(44)
a(22) = 50  a(23)
a(21) =  a(32) + a(41) + a(47)
a(20) =  a(31) + a(40) + a(46)
a(19) = a(20)  a(40) + a(41)  a(46) + a(49)
a(18) =  a(35) + a(40) + a(48)
a(17) = 25
a(15) = 50  a(16)
a(13) = a(14)  a(37) + a(42)  a(43) + a(48)
a(11) = 100  a(12)  a(13)  a(14)
a( 9) = 50  0.5 * a(10)  0.5 * a(16)
a( 8) = 75  a( 9)  a(10)
a( 7) = 50  a(14)
a( 6) = 50  a(13)
a( 5) = 50  a(12)
a( 4) = 100  a( 5)  a(6)  a(7)
a( 3) = 50  a(10)
a( 2) = 50  a( 8)
a( 1) = 50  a( 9)
With an optimized guessing routine (MgcSqr7g), based on the equations above, following cases where considered:

Case 1: All independant variables constant except a(10) and a(16), which resulted in 8 Magic Squares within 0,63 seconds (ref. Attachment 7.6.1);

Case 2: The 5^{th} order Pan Magic  and the 3^{th} order Semi Magic Corner Squares constant, which resulted in 64 Magic Squares within 7,5 seconds (ref. Attachment 7.6.2);

Case 3: The border variables, including the 3^{th} order Semi Magic Corner Square constant, which resulted in 72 Magic Squares within 5 hours (ref. Attachment 7.6.3);

Case 4: Only the 5^{th} order Pan Magic Square constant, which resulted in 512 Magic Squares within 43,5 seconds (ref. Attachment 7.6.4).
It should be noted that the distinct  partly consecutive  integers for the 5^{th} order Pan Magic Square have been selected from the integers 1 through 49 as shown below (shaded):
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
Based on these 25 integers, 1152 Pan Magic Squares of the 5^{th} order with corner element 25 can be generated, which was realised with an appropriate guessing routine MgcSqr5a3 within 45 seconds and are shown in Attachment 7.6.5 which includes the 72 Pan Magic Squares found under Case 3 described above.
The number of 7^{th} order Simple Magic Squares which can be constructed based on one 5^{th} order Pan Magic Square depends from the sum s2 of the key variables a(19), a(25) and a(31) and is shown below.
Any of the 16 values of s2 corresponds with 72 Pan Magic Squares of the 5^{th} order (16 * 72 = 1152).
The total number of 7^{th} order Simple Magic Squares with overlapping Sub Squares is 72 * (8 * 512 + 4 * 256) = 368640.
Attachment 7.6.6 shows the first Simple Magic Square of the 7^{th} order found for each of the applicable Pan Magic Squares of the 5^{th} order.
7.7.3 Associated Magic Squares
Order 7 Associated Magic Squares with Overlapping Sub Squares should be composed out of:

Two each 4^{th} order Magic Corner Squares A and D (s4 = 100), with the center element in common, and

Two each 3^{th} order Semi Magic Corner Squares B and C (s3 = 75).
as shown below.
a1 
a2 
a3 
a4 
b1 
b2 
b3 
a5 
a6 
a7 
a8 
b4 
b5 
b6 
a9 
a10 
a11 
a12 
b7 
b8 
b9 
a13 
a14 
a15 
a16/d1 
d2 
d3 
d4 
c1 
c2 
c3 
d5 
d6 
d7 
d8 
c4 
c5 
c6 
d9 
d10 
d11 
d12 
c7 
c8 
c9 
d13 
d14 
d15 
d16 
Associated Magic Squares of order 7 can be constructed based on:

One Complementary Pair of order 4 Magic Anti Symmetric Corner Squares and

One Complementary Pair of order 3 Semi Magic Anti Symmetric Corner Squares (6 Magic Lines).
A (Semi) Magic Anti Symmetric Square of order n is a (Semi) Magic Square for which:
a_{i} + a_{j} ≠ 2 * s_{n} / n for any i and j (i,j = 1 ... n^{2}; i ≠ j)
The Magic Corner Square A is defined by following linear equations:
a(16) = s4/4
a(13) = s4  a(14)  a(15)  a(16)
a( 9) = s4  a(10)  a(11)  a(12)
a( 7) = a( 8)  a(10) + a(12)  a(13) + a(16)
a( 6) = s4  a( 8)  a(11)  a(12) + a(13)  a(16)
a( 5) =  a( 8) + a(10) + a(11)
a( 4) = s4  a( 7)  a(10)  a(13)
a( 3) = s4  a( 8) + a( 9) + 2*a(10) + 2*a(13) + a(14)
a( 2) = a( 8)  a( 9)  2*a(10) + a(15) + 2*a(16)
a( 1) = a( 8) + a(12)  a(13)
The Magic Corner Square C is defined by following linear equations:
c(7) = s3  c(8)  c(9)
c(4) = s3  c(5)  c(6)
c(3) =  c(6) + c(7) + c(8)
c(2) = s3  c(5)  c(8)
c(1) = c(5) + c(6)  c(7)
With an optimized guessing routine (MgcSqr7g2), based on the equations above, following cases where considered:

Case 1:
The total number of suitable 4^{th} order Anti Symmetric Magic Corner Squares A
has been determined (28800) and is broken down below for i = a(15) = 1 ... 24, 26 ... 49:
i 
n(i) 
i 
n(i) 
i 
n(i) 
i 
n(i) 
i 
n(i) 
i 
n(i) 
1  616 
2  712 
3  684 
4  668 
5  656 
6  712 
7  744 
8  632 
9  612 
10  576 
11  748 
12  560 
13  440 
14  540 
15  596 
16  520 
17  560 
18  624 
19  576 
20  456 
21  532 
22  588 
23  508 
24  540 
26  540 
27  508 
28  588 
29  532 
30  456 
31  576 
32  624 
33  560 
34  520 
35  596 
36  540 
37  440 
38  560 
39  748 
40  576 
41  612 
42  632 
43  744 
44  712 
45  656 
46  668 
47  684 
48  712 
49  616 

Case 2:
With the first occuring Magic Corner Square A constant, 144 Associated Magic Squares could be generated within 6,5 seconds (ref. Attachment 7.6.7)
The resulting number of Associated Magic Squares appeared to be 4156416 (= 144 * 28800 + 16 * 4 * 144).
Attachment 7.6.8 shows the first 7^{th} order Associated Magic Squares with Overlapping Sub Squares, found for some of the possible 4^{th} order Magic Corner Squares (48 ea).
7.7.4 Spreadsheet Solutions
The linear equations deducted in previous sections, have been applied in following Excel Spread Sheets:

CnstrSngl7d1,
Magic Squares of order 7, Overlapping Sub Squares (1)

CnstrSngl7d2,
Magic Squares of order 7, Overlapping Sub Squares (2)
Only the red figures have to be “guessed” to construct one of the applicable Magic Squares of the 7^{th} order
(wrong solutions are obvious).
