Franklin’s Construction Method
Part 3

Franklin's original 8x8

A








B
52
61
4
13
20
29
36
45

6
7
0
1
2
3
4
5

3
4
3
4
3
4
3
4
14
3
62
51
46
35
30
19

1
0
7
6
5
4
3
2

5
2
5
2
5
2
5
2
53
60
5
12
21
28
37
44

6
7
0
1
2
3
4
5

4
3
4
3
4
3
4
3
11
6
59
54
43
38
27
22

1
0
7
6
5
4
3
2

2
5
2
5
2
5
2
5
55
58
7
10
23
26
39
42

6
7
0
1
2
3
4
5

6
1
6
1
6
1
6
1
9
8
57
56
41
40
25
24

1
0
7
6
5
4
3
2

0
7
0
7
0
7
0
7
50
63
2
15
18
31
34
47

6
7
0
1
2
3
4
5

1
6
1
6
1
6
1
6
16
1
64
49
48
33
32
17

1
0
7
6
5
4
3
2

7
0
7
0
7
0
7
0
                       
Franklin's magic circle

A








B
62
73
14
25
30
41
46
57

6
7
0
1
2
3
4
5

2
5
2
5
2
5
2
5
24
15
72
63
56
47
40
31

1
0
7
6
5
4
3
2

4
3
4
3
4
3
4
3
71
64
23
16
39
32
55
48

7
6
1
0
3
2
5
4

3
4
3
4
3
4
3
4
17
22
65
70
49
54
33
38

0
1
6
7
4
5
2
3

5
2
5
2
5
2
5
2
69
66
21
18
37
34
53
50

7
6
1
0
3
2
5
4

1
6
1
6
1
6
1
6
19
20
67
68
51
52
35
36

0
1
6
7
4
5
2
3

7
0
7
0
7
0
7
0
60
75
12
27
28
43
44
59

6
7
0
1
2
3
4
5

0
7
0
7
0
7
0
7
26
13
74
61
58
45
42
29

1
0
7
6
5
4
3
2

6
1
6
1
6
1
6
1
                       
Franklin's unpublished

A








B
17
47
30
36
21
43
26
40

1
0
7
6
5
4
3
2

7
0
7
0
7
0
7
0
32
34
19
45
28
38
23
41

6
7
0
1
2
3
4
5

1
6
1
6
1
6
1
6
33
31
46
20
37
27
42
24

0
1
6
7
4
5
2
3

2
5
2
5
2
5
2
5
48
18
35
29
44
22
39
25

7
6
1
0
3
2
5
4

4
3
4
3
4
3
4
3
49
15
62
4
53
11
58
8

1
0
7
6
5
4
3
2

3
4
3
4
3
4
3
4
64
2
51
13
60
6
55
9

6
7
0
1
2
3
4
5

5
2
5
2
5
2
5
2
1
63
14
52
5
59
10
56

0
1
6
7
4
5
2
3

6
1
6
1
6
1
6
1
16
50
3
61
12
54
7
57

7
6
1
0
3
2
5
4

0
7
0
7
0
7
0
7

Row Types

6
7
0
1
2
3
4
5










7
6
1
0
3
2
5
4




























1
0
7
6
5
4
3
2










0
1
6
7
4
5
2
3

 

A comparison of Franklin’s three 8x8s

Franklin produced three 8x8s-original, magic circle, and an unpublished square found among his papers.

His magic circle can be transcribed into an eight by eight grid. Spokes are columns, concentric circles are rows, and eccentric circles are bent diagonals. Horizontal bent diagonals start in columns one, three, five, and seven and have row totals. In each column they can start in rows one through five and be in the circle. Starting in rows six, seven, and eight they run off the square and can’t be used. That leaves five for each of the four starting columns for a total of twenty, the number of eccentric circles in his magic circle.

Unpublished A and B are rotated 90 degrees clockwise for better comparisons.

In the three squares A there are four types of rows. Types one and two are in original, 67012345 and 10765432 and are simply consecutive sequences. The other two types are those two with pairs of numbers rotated, 76103254 and 01674523.

In magic circle half diagonals now have a pair totaling one fewer than complimentary and a pair one greater. These offset and cure the diagonal total problem in square A. However that causes square B to be flawed. In magic circle square B column one complimentary pairs must be placed in rows one and four, two and three, five and eight, and six and seven to get a match up of all the numbers in squares A and B. This destroys diagonal totals in square B. Franklin had traded a problem in square A for the same problem in square B.

Unpublished square A simply repositions the pairs of rows. In unpublished square B he placed complimentary pairs in rows one and eight, two and seven, three and six, and four and five. That matched up all the numbers in A and B but didn’t help the diagonal totals.

All rows in squares A have one of each number in the number set. In squares B the same applies to all columns.

Magic circle doesn't have vertical bent diagonals. Unpublished corrects that.

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