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Pan Magic Squares 4 x 4, Interactive Solution based on Binary Matrices
Introduction
Another well known construction method for Pan Magic Squares is based on the binary representation of numbers (0 = 0000,
1 = 0001, 2 = 0010, 3 = 0011 ... 15 = 1111).
Any Magic Square of the 4th order with the numbers 1 ... 16 can be written as
B1 +
2 * B2 +
4 * B3 +
8 * B4 + [1],
where the matrices B1,
B2,
B3 and
B4
- further referred to as Binaries - contain only the numbers 0 and 1.
The 8 for Pan Magic Squares applicable Binaries are shown below in the bottom part of the form.
A 4th order Pan Magic Square can be constructed by means of the Matrix Operation PM =
B1 +
2 * B2 +
4 * B3 +
8 * B4 + [1].
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Selected:
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Pan Magic:
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Binaries:
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Procedure:
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Select the matrices
B1,
B2,
B3 or
B4
with the selection button left.
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Select for each of the matrices one Binary from the pairs
H1a/H1b,
H2a/H2b,
V1a/V1b and
V2a/V2b
by clicking the corresponding image shown in the bottom part of the form.
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A selection can be cancelled by clicking subject image in the top part of the form, after which another Binary can be selected.
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Press the button ‘Show Square’ to calculate and visualise the resulting Pan Magic Square.
All 384 Pan Magic Squares can be determined by means of this method as the 24 = 16 possible combinations and 4! = 24 permutations of the Binaries will result in 16 * 24 = 384 different Pan Magic Squares.
All combinations, permutations and resulting Pan Magic Squares of the 4th order
are shown in Attachment 4b.
Have Fun!
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