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How to produce a 9x9 pan magic square
I don’t know how many pure (pan)magic 9x9 squares exist.
A 9x9 square is an odd square, but is also a multiple of 3 (= can be devided by 3).
Can a 9x9 panmagic square be produced by using the same method of construction as the method to
produce a 5x5 panmagic square (which can be used to produce all possible panmagic 5x5 and pan-
magic 7x7 squares)? The answer is yes and no. If you choose as first row the digits 0-1-2-3-4-5-6-7-8
you get as result only a semimagic 9x9 square. If you choose as first row the digits 0-2-1-5-4-3-7-6-8
you get as result a correct panmagic 9x9 square.
Notify first that the row 0-2-1-5-4-3-7-6-8 leads to a correct result, because 0+5+7, 2+4+6 and 1+3+8 is
12, that is 1/3 of (0+1+2+3+4+5+6+7+8=) 36.
Notify second that if you choose as first row the digits 0-2-1-5-4-3-7-6-8 you can, instead of the 2nd square
that is produced by shifting the first row to the right, take as 2nd square the 1st square rotated by a quarter
to the right!
There is a method to produce a panmagic 9x9 square plus an extra magic feature:
Fill in the first row of the 1st square: 0, 1, 2, 3, 4, 5, 6, 7 and 8. Produce the second and third row of the 1st
square by moving the digits each time 3 places to the left.
1st square, first three rows
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0
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1
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2
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3
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5
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6
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8
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0
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7
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8
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0
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8
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0
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7
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8
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0
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7
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8
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0
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6
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7
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8
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The first three rows of the 1st square consist of three 3x3 sub-squares. Produce the second three rows of the
1st square by switching the sequence of the three columns of the three 3x3 sub-squares to 2-3-1. Produce the
second three rows of the 1st square by switching the sequence of the three columns of the three 3x3 sub-
squares to 3-1-2.
1st square
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0
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1
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2
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8
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7
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8
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0
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2
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6
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7
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8
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0
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1
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2
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3
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5
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1
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2
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0
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4
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7
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8
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6
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4
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5
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3
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7
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8
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6
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1
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2
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0
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7
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8
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6
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1
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2
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0
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4
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3
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2
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0
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1
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5
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8
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7
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3
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8
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7
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0
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8
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7
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2
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0
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1
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5
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4
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Produce the 2nd square by rotating the 1st square (a quarter turn to the right). Take a digit from the 1st square
multiplied by 9 and add (1x) the digit from the same cell of the 2nd square.
9x digit +1x digit = panmagic 9x9 square
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0
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1
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2
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3
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5
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6
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7
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8
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8
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5
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2
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7
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4
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1
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6
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3
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0
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8
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14
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20
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34
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40
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46
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60
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66
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72
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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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0
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1
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2
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6
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3
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0
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8
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5
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2
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7
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4
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1
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33
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39
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45
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62
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68
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74
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7
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13
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19
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6
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7
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8
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0
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1
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2
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3
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7
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4
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1
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6
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3
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0
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8
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5
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2
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61
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67
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73
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6
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12
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18
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35
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41
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47
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1
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2
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0
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4
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5
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3
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7
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8
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6
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2
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8
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5
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1
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7
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4
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0
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6
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3
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11
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26
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5
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37
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52
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31
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63
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78
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57
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4
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5
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3
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7
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8
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6
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1
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2
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0
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0
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6
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3
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2
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8
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5
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1
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7
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4
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36
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51
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30
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65
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80
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59
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10
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25
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4
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7
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8
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6
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1
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2
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0
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1
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7
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4
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0
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6
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3
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2
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8
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5
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64
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79
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58
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9
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24
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3
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38
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53
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32
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2
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0
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1
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5
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3
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4
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8
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6
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7
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5
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8
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1
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7
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3
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0
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6
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23
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2
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17
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49
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28
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43
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75
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54
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69
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5
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3
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4
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8
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6
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7
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2
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0
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1
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3
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0
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6
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5
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2
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8
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1
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7
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48
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27
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42
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77
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56
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71
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22
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1
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16
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8
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6
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7
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2
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0
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1
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4
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1
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7
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3
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0
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6
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2
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8
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76
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55
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70
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21
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0
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15
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50
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29
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44
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In the square consists of the digits 0 up to 80. By adding 1 to each digit you can produce a square which con-
sists of the digits 1 up to 81.
Extra magic feature is that the sum of the digits of each 3x3 sub-square is the magic sum of 360.
You can use this method to produce squares which have the size of odd number squared (= 9x9, 25x25, 49x49,
81x81, 121x121, ...). For example, to produce a 25x25 square use the digits 0 up to 24 as first row of the 1st
square. Produce the second up to fifth row of the 1st square by moving the digits each time 5 places to the left.
Produce the second up to fifth five rows (= 5x5 sub-squares) of the 1st square by switching the sequence of the
five columns of the five 5x5 sub-squares to 2-3-4-5-1, 3-4-5-1-2, 4-5-1-2-3 respectively 5-1-2-3-4.
There are not many possibilities (just a little alternative combinations of digits) to produce a 9x9 panmagic square.
Click for a method to produce many more 9x9 panmagic squares on [NEXT>>
Ultra magic 9x9 square:
| 1x digit from row grid |
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| 0 |
4 |
8 |
5 |
6 |
1 |
7 |
2 |
3 |
| 7 |
2 |
3 |
0 |
4 |
8 |
5 |
6 |
1 |
| 5 |
6 |
1 |
7 |
2 |
3 |
0 |
4 |
8 |
| 0 |
4 |
8 |
5 |
6 |
1 |
7 |
2 |
3 |
| 7 |
2 |
3 |
0 |
4 |
8 |
5 |
6 |
1 |
| 5 |
6 |
1 |
7 |
2 |
3 |
0 |
4 |
8 |
| 0 |
4 |
8 |
5 |
6 |
1 |
7 |
2 |
3 |
| 7 |
2 |
3 |
0 |
4 |
8 |
5 |
6 |
1 |
| 5 |
6 |
1 |
7 |
2 |
3 |
0 |
4 |
8 |
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| + 9x digit from column grid +1 |
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| 0 |
7 |
5 |
0 |
7 |
5 |
0 |
7 |
5 |
| 4 |
2 |
6 |
4 |
2 |
6 |
4 |
2 |
6 |
| 8 |
3 |
1 |
8 |
3 |
1 |
8 |
3 |
1 |
| 5 |
0 |
7 |
5 |
0 |
7 |
5 |
0 |
7 |
| 6 |
4 |
2 |
6 |
4 |
2 |
6 |
4 |
2 |
| 1 |
8 |
3 |
1 |
8 |
3 |
1 |
8 |
3 |
| 7 |
5 |
0 |
7 |
5 |
0 |
7 |
5 |
0 |
| 2 |
6 |
4 |
2 |
6 |
4 |
2 |
6 |
4 |
| 3 |
1 |
8 |
3 |
1 |
8 |
3 |
1 |
8 |
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| = Ultra magic 9x9 square |
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| 1 |
68 |
54 |
6 |
70 |
47 |
8 |
66 |
49 |
| 44 |
21 |
58 |
37 |
23 |
63 |
42 |
25 |
56 |
| 78 |
34 |
11 |
80 |
30 |
13 |
73 |
32 |
18 |
| 46 |
5 |
72 |
51 |
7 |
65 |
53 |
3 |
67 |
| 62 |
39 |
22 |
55 |
41 |
27 |
60 |
43 |
20 |
| 15 |
79 |
29 |
17 |
75 |
31 |
10 |
77 |
36 |
| 64 |
50 |
9 |
69 |
52 |
2 |
71 |
48 |
4 |
| 26 |
57 |
40 |
19 |
59 |
45 |
24 |
61 |
38 |
| 33 |
16 |
74 |
35 |
12 |
76 |
28 |
14 |
81 |
N.B.: The square is panmagic, symmetric, 3x3 compact and each 1/3 row and each 1/3 column gives
1/3 of the magic sum.
See also: 3x extra magic 9x9 square or Panmagic 15x15 square
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