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Introduction
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 a method of construction to produce even bordered
squares, an example of an odd bordered square and a reference to a website with (more complicated)
inlaid squares (= more squares inside a bigger magic square).
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 inlaid 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 inlaid 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 inlaid 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
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