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Why does the basic key method of construction lead to a perfect 16x16 Franklin panmagic square?
Why does the basic key method lead to perfect Franklin panmagic squares that are multiples of 8, for example
the perfect 16x16 Franklin pan magic square.
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16 = 136.
(1st) sum = sum = 34 = 1/4 x 136
That is the condition which ensures that the sum of the digits of the quarter rows and the quarter columns is a
quarter of the magic sum.
(2nd) sum = sum = 34 = 1/4 x 136
That is the condition which ensures that the sum of the digits of the quarter diagonals (and the [parallel] [mir-
rored] bent diagonals) is a quarter of the magic sum.
(3rd) An odd number of digits must always be between reverse vertical combinations of digits (reverse vertical
combinations may not lie next to each other).
This is the condition that allows it to be used as a second (column) square, thus rotate the first (row) square
by a quarter turn to the left and all the digits from 1 to 256 can be found in the square.
How to use the basic key method of construction for the 12x12 square?
For squares that are odd multiples of 4, for example the 12x12 square, it is not possible to comply with all three
above-mentioned conditions. You must choose between the following:
[option a]
(1st) ) sum = sum = 26 = 1/3 x 78
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This is the condition which ensures that the sum of the digits of 1/3 rows and 1/3 columns are 1/3 of the magic
sum, but the sum of the digits of 1/2 rows and 1/2 columns are not ½ of the magic sum. Besides, the sum of the
digits of circles and other symmetric figures are a proportional part of the magic sum.
(2nd) sum = sum = 39 = 1/2 x 78
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This is the condition which ensures that the sum of the digits of ½ diagonals (and the [parallel] [mirrored] bent
diagonals) is ½ of the magic sum.
(3rd) This is the condition that allows it to be used as second (column) square, thus rotate the first (row) square
by a quarter turn to the left, and you will find all the digits from 1 to 144 can be found in the square.
Row pattern
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Column pattern
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12x12 square (option a)
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[option b]
(1st) sum = sum = 39 = 1/2 x 78
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This is the condition which ensures that the sum of the digits of ½ rows and ½ columns are ½ of the
magic sum. Besides, the sum of the digits of circles and other symmetric figures are not a proportional
part of the magic sum.
(2nd) The sum of the ½ twisting rows is not 39, but the sum of the whole twisting rows is 78. That is
the boundary condition which ensures that the sum of the digits of the whole diagonal is the magic sum,
but the sum of the digits of ½ diagonals (and the [parallel] [mirrored] bent diagonals) is not ½ of the
magic sum.
(3rd) Odd number of digits are between reverse vertical combinations rotating the first (row) square by
a quarter turn to the left enables you to use it as a second (column) square and all the digits from 1 to
144 can be found in the square.
Row pattern
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Colum pattern
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12x12 square (option b)
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140
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