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The 5x5 magic square is square, because it has as many rows (from left to right = horizontal) as columns
(from top to bottom = vertical).
The 5x5 magic square consists of 5 rows which multiplied by 5 columns is 25 cells.
The 5x5 magic square must contain 25 different digits. A pure magic 5x5 square contains the digits 1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 and 25.
The magic square is magic, because the sum of the digits of each row, each column and both diagonals al-
ways give the same result. The sum can be calculated as follows, the (odd) size of the square multiplied by
the middle digit: 5 x 13 = 65.
What is a panmagic 5x5 square?
Panmagic (5x5) squares have an extra magic feature. Not only the sum of 5 digits from each row, each
column and both diagonals - but also the sum of 5 digits from each pandiagonal - totals to the magic sum
of 65. Pandiagonals don’t start (unlike ordinary diagonals) in the corner, and pandiagonals are broken dia-
gonals; see below (the ordinary diagonal is yellow and the pandiagonals are red, blue, pink and green):
from right to left from left to right
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1
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7
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13
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19
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25
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1
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7
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13
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19
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25
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14
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20
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21
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2
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8
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14
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20
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21
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2
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8
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22
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3
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9
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15
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16
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22
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3
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9
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15
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16
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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18
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24
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5
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6
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12
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18
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24
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5
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6
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12
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Specialty of the panmagic square is, that you can produce a 2x2 carpet of the same square:
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1
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7
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13
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19
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25
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1
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7
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13
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19
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25
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|
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1
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7
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13
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19
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25
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1
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7
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13
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19
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25
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14
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20
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21
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2
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8
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14
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20
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21
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2
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8
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14
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20
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21
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2
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8
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14
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20
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21
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2
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8
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22
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3
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9
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15
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16
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22
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3
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9
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15
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16
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|
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22
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3
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9
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15
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16
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22
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3
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9
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15
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16
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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18
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24
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5
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6
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12
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18
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24
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5
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6
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12
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|
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18
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24
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5
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6
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12
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18
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24
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5
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6
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12
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1
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7
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13
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19
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25
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1
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7
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13
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19
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25
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1
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7
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13
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19
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25
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1
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7
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13
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19
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25
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14
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20
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21
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2
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8
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14
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20
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21
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2
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8
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|
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14
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20
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21
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2
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8
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14
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20
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21
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2
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8
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22
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3
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9
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15
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16
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22
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3
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9
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15
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16
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22
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3
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9
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15
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16
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22
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3
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9
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15
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16
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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10
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11
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17
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23
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4
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18
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24
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5
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6
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12
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18
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24
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5
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6
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12
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18
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24
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5
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6
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12
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18
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24
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5
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6
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12
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Each 5x5 subsquare on the 2x2 carpet is panmagic, because of the parallel running diagonals on the carpet!!!
What is the secret behind the panmagic 5x5 square?
Imagine: A boat sales on a lake of 5 x 5 = 25 square miles. A radar detects the boat and determines in which
square mile (= cell) the boat is located. The position of the boat can be expressed in a row coordinate (0, 1, 2,
3, or 4) and a column coördinate (0, 1, 2, 3, or 4).
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All combinations of
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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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row- and column coordinates
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0
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(0,0)
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(1,0)
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(2,0)
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(3,0)
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(4,0)
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1
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(0,1)
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('1,1)
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(2,1)
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(3,1)
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(4,1)
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2
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(0,2)
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(1,2)
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(2,2)
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(3,2)
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(4,2)
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3
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(0,3)
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(1,3)
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(2,3)
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(3,3)
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(4,3)
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4
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(0,4)
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(1,4)
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(2,4)
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(3,4)
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(4,4)
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If you add 5x the column coordinate to the row coordinate, you can number the cells from 0 up to 24.
row coordinate + 5x column coordinate = digits 0 up to 24
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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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10
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0
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0
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0
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0
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0
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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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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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10
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1
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1
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1
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1
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1
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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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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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10
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2
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2
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2
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2
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2
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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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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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10
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3
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3
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3
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3
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3
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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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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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10
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4
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4
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4
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4
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4
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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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10
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10
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10
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10
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10
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If you use the digits 0 up to 24 (in stead of 1 up to 25), the magic sum is (the size of the odd square
multiplied by the middle digit: 5 x 12 = ) 60.
You can also calculate the magic sum as follows: add 5x the sum of the column coordinates (0+1+2+
3+4=10) to the sum of the row coordinates (0+1+2+3+4=10): 5x10 + 10 = 60.
If you ensure that all the column coordinates from 0 up to 4 and all the row coordinates from 0 up to
4 are in each row/column/(pan)diagonal, than you will produce everywhere the magic sum of 60 (see
calculation mentioned above). Notify that the connection between the column coordinates and the row
coordinates must lead to unique column/row-combinations (see table mentioned above), so you can find
all the digits 0 up to 24 in the square.
How to produce a 5x5 panmagic square?
A trick to make a 5x5 panmagic square is the knight movement method:
Knight movement 1-5 place 6th digit and repeat Indent marked digits
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1
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1
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7
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13
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19
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25
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1
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7
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13
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19
|
25
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2
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20
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21
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2
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8
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14
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14
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20
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21
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2
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8
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3
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9
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15
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16
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22
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3
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22
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3
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9
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15
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16
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4
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23
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4
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10
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11
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17
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10
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11
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17
|
23
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4
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5
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6
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12
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18
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24
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5
|
6
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18
|
24
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5
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6
|
12
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For a trick to make all panmagic 5x5 squares, see page panmagic 5x5 square
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