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There are 3 similar types of squares: inlaid squares, bordered squares and concentric squares. See
for the exact descriptions on the website of Harvey Heinz: www.magic-squares.net/glossary.htm
A bordered square is an (impure) magic square inside a bigger (pure) magic square. A bordered square
consist of the center magic square with a singular border around the center square. The center square
consist of the middle numbers of the magic square and the border contains the lowest numbers and their
complements (the highest numbers). See below methods of construction to produce even and odd bordered
squares, a reference to a website with a method to produce concentric magic squares, reference to a web-
page with a method to produce inlaid squares and the presentation of two historic magic squares.
Even bordered squares
The smallest even bordered square is an (impure) 4x4 square inside a (pure) 6x6 square. A pure 6x6
square consist of all the digits from 1 up to 36.
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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17
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18
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19
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20
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21
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22
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23
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24
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25
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26
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27
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28
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29
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30
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31
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32
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33
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34
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35
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36
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-10
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-9
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-8
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-7
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-6
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-5
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-4
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-3
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-2
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-1
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Use the middle (yellow marked) 16 digits to produce the 4x4 inlaid square. Take for example the first basic
panmagic 4x4 square (see page panmagic 4x4) and add 10 to each digit.
Pure 4x4 +10 = (impure) inlaid 4x4
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1
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8
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13
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12
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11
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18
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23
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22
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15
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10
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3
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6
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25
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20
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13
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16
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4
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5
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16
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9
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14
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15
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26
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19
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14
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11
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2
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7
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24
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21
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12
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17
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To produce the border there are the following three boundary conditions:
● On the opposite of a possitive digit in the row, the column or the diagonal must be filled in the same
negative digit (= translation of the digits from 27 up to 36);
● In the top row, the bottom row, the left column and the right column must be filled in each time 6
different (3 positive and 3 negative) digits;
● The digits of the top row, the bottom row, the left column and the right column must each time total
to 0.
Produce the top row and the right column and fill in (for example) the digit 1 at the left corner on the top
row. Because the digit at the left corner on the top row is 1, at the right corner on the bottom row the
digit -1 must be filled in. The digit (?) to be filled in at the right corner on the top row is in the top row as
well as in the right column.
In the top row and the right column must be filled in 4x 3 digits from (+/-) 1 up to 10. We need 1 up to 10
plus two digits (out of 1 up to 10) double. The sum of 1 up to 10 is 55. If we choose 1 and 8 double, than
the sum of each 3 digits must be: [55 + 1 + 8] / 4 = 16. Solve the puzzle!
1+5+10
1+6+ 9
1+7+ 8
2+4+10
2+5+ 9
2+6+ 8
3+4+ 9
3+5+ 8
3+6+ 7
4+5+ 7
When you have solved the puzzle, the digits can be filled in as follows:
Fill in 4 possibilities Fill in opposite digits
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1
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6
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9
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-3
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-5
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-8
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1
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6
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9
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-3
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-5
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-8
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2
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-2
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2
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4
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-4
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4
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10
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-10
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10
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-7
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7
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-7
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-1
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8
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-6
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-9
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3
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5
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-1
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The final result is:
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1
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6
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9
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34
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32
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29
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35
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11
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18
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23
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22
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2
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33
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25
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20
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13
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16
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4
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27
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14
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15
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26
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19
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10
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7
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24
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21
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12
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17
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30
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8
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31
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28
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3
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5
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36
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You can use this method of construction for all even bordered magic squares (6x6 inside 8x8, 8x8 inside 10x10,
10x10 inside 12x12, …).
Odd bordered squares
The smallest bordered square is an (impure) 3x3 square inside the (pure) 5x5 square. It is possible to use the 9
(sequencing) middle digits out of 1 up to 25 to produce the 3x3 center square; see the result below.
3x3 inlay inside 5x5 square
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1
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22
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20
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19
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3
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2
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10
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17
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12
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24
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18
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15
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13
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11
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8
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21
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14
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9
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16
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5
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23
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4
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6
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7
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25
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An 5x5 inlay inside an 7x7 square, can be produced as follows:
Produce the column pattern and the row pattern of the 5x5 inlaid by using the method of construction on page
Pan magic 5x5 square. Take from the digits 0 up to 6 the five middle digits, 1 up to 5.
Column pattern 5x5 inlaid square Row pattern 5x5 inlaid square
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1
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2
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3
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4
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5
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1
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2
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3
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4
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5
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4
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5
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1
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2
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3
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3
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4
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5
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1
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2
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2
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3
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4
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5
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1
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5
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1
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2
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3
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4
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5
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1
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2
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3
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4
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2
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3
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4
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5
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1
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3
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4
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5
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1
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2
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4
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5
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1
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2
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3
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The 5x5 inlaid square contains all combinations of the digits 1 up to 5. The border must contain all combinations
of the difgits 0 and/or 6. Establish that the sum of two opposite digits are allways 6 and the sum of every row and
every column is allways 21. Notify that 2x the middle digit from 0 up to 6 (= 3) must be placed in opposite corners
and the lowest and highest digits (= 0 and 6) must be placed in the same row or column:
Column pattern 7x7 border
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3
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0
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3
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1
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2
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4
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5
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6
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0
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0
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6
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0
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6
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6
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0
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6
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0
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0
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6
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0
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6
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6
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0
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6
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0
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0
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6
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0
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6
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6
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3
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6
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5
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4
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2
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1
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0
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3
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Place the digits of the row pattern and establish that you get all combinations of the digits 0 and/or 6 between the
column pattern and the row pattern of the border (because the magic square must contain all the digits from 1 up
to 49):
Row pattern 7x7 border
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0
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6
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0
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0
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6
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6
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3
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1
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5
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2
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4
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5
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1
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4
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2
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6
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0
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3
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0
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6
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6
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0
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0
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6
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Take 7x a digit from the column pattern and add 1x a digit from the same cell of the row pattern and add 1 (to get
the digits 1 up to 49 instead of 0 up to 48).
7x digit from column pattern + 1x digit from row pattern
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3
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1
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2
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4
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5
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6
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0
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0
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6
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0
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0
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6
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6
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3
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0
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1
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2
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3
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4
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5
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6
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1
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1
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2
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3
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4
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5
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5
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6
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4
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5
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1
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2
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3
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0
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2
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3
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4
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5
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1
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2
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4
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0
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2
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3
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4
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5
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1
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6
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5
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5
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1
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2
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3
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4
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1
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6
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5
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1
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2
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3
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4
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0
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4
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2
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3
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4
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5
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1
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2
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0
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3
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4
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5
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1
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2
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6
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6
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4
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5
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1
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2
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3
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0
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6
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5
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4
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2
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1
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0
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3
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3
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0
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6
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6
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0
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0
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6
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+1 = 5x5 inlay inside 7x7 square
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21
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13
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14
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28
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41
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48
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3
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22
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14
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15
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29
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42
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49
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4
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1
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8
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16
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24
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32
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40
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47
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2
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9
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17
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25
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33
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41
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48
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44
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31
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39
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12
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15
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23
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4
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45
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32
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40
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13
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16
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24
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5
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5
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19
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22
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30
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38
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11
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43
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6
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20
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23
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31
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39
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12
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44
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46
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37
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10
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18
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26
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29
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2
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47
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38
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11
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19
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27
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30
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3
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6
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25
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33
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36
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9
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17
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42
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7
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26
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34
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37
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10
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18
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43
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45
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35
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34
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20
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7
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0
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27
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46
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36
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35
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21
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8
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1
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28
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You can use this method of construction also for a 7x7 inlaid inside a 9x9 square.
Webpage about the construction of an inlaid square
Find information about the construction of an inlaid square on this website:
www.perfectmagicsquares.com/inlaid square.html
Anxi bordered 6x6 square
Paul Michelet discovered that the 6x6 Anxi magic square from the 13th or 14th century is
not an ordinary 6x6 magic square, but a 6x6 bordered magic square with a 4x4 panmagic
square inside it.
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28
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4
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3
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31
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35
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10
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36
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18
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21
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24
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11
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1
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7
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23
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12
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17
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22
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30
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8
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13
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26
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19
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16
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29
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5
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20
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15
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14
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25
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32
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27
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33
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34
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6
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2
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9
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Read more about the Anxi magic square on:
http://hua.umf.maine.edu/China/Xian/Shaanxi_History/pages/284_History_Museum.html
Al-Antaakii concentric 15x15 magic square with extra feature
Paul Michelet drew my attention to the al-Antaakii concentric 15x15 magic square with an extra
magic feature dated from 987 (!!!); see youtube movie from Gresham College:
http://www.youtube.com/watch?v=YUsSaGnbcCw
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62
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2
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222
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220
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8
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10
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214
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213
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212
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16
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18
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206
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204
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24
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64
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126
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78
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26
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198
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196
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32
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11
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189
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207
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34
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190
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188
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40
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80
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100
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128
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122
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94
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42
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182
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7
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35
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173
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183
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203
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180
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48
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96
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104
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98
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50
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124
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118
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110
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3
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31
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51
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165
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167
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179
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199
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112
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108
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102
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176
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52
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70
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120
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201
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75
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159
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155
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153
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83
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87
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79
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25
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106
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156
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174
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54
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72
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205
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181
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141
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95
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135
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133
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103
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99
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85
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45
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21
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154
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172
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170
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209
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185
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169
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145
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125
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111
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121
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107
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101
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81
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57
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41
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17
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56
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211
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187
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171
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163
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149
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129
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109
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113
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117
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97
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77
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63
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55
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39
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15
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168
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9
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33
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49
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69
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89
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119
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105
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115
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137
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157
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177
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193
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217
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58
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60
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82
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5
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29
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65
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127
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91
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93
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123
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131
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161
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197
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221
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144
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166
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66
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142
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90
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1
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147
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67
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71
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73
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143
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139
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151
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225
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136
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84
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160
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158
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140
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92
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114
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223
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195
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175
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61
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59
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47
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27
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116
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134
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86
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68
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152
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88
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130
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184
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44
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219
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191
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53
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43
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23
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46
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178
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132
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138
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74
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76
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146
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200
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28
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30
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194
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215
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37
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19
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192
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36
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38
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186
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148
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150
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162
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224
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4
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6
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218
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216
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12
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13
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14
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210
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208
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20
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22
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202
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164
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N.B.: Extra magic feature is that the white diamond contains only the odd digits!!!
Read more about Al-Antaakii on http://en.wikipedia.org/wiki/Yahya_of_Antioch
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