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What do I need to know, before I read about most perfect magic squares?
Which size (order) have most perfect magic squares?
Most perfect squares are multiples of 4 (4x4, 8x8, 12x12, 16x16, 20x20, … magic squares).
Which special features have most perfect magic squares?
● Most perfect magic squares are panmagic and the sum of the digits of each 2x2 sub-square is 4 / n
(n = the size/order of the square) x the magic sum.
● A most perfect magic square, which is a multiple of 8 (= 8x8, 16x16, 24x24, 32x32, 40x40, … magic
square), has all features of a Franklin panmagic square. It means (simply spoken), that the total of the
digits of ½ rows/columns/diagonals is ½ of the magic sum.
● For each multiple of 8 from 16x16 and up, the most perfect magic square is even more magic than a
Franklin panmagic square. For example the 16x16 most perfect square: (not only the total of ½ but also)
the total of ¼ rows/columns/diagonals gives ¼ of the magic sum.
N.B.: There are as many most perfect magic squares as complete magic squares and that is no coincidence.
You can transform a most perfect 8x8 magic square by swapping row 3 up to 4 with row 5 up to 6 and
column 3 up to 4 with column 5 up to 6. It is possible to transform each most perfect magic square (8x8,
12x12, 16x16, 20x20, ..., nxn) by swapping row 3 up to 1/2n with row 1/2n+1 up to n-2 and column 3 up to
1/2n with column 1/2n+1 up to n-2. It is a pity that Kathleen Ollerenshaw (I admire her brilliant work) has
called the complete magic square 'most perfect', while the complete magic square is less magic (= has less
magic features) than the real most perfect magic square. On this website I present only the real most perfect
magic square. But it is possible to swap each most perfect magic square on this website into a complete magic
square, if you prefer complete magic squares.
What is the structure of a most perfect magic square?
Do you know the structure of the most perfect magic square, than you understand why the most perfect
magic square has its special features.
The most perfect magic square consists of one or more proportional panmagic 4x4 (sub-)squares. The
meaning of ‘proportional’ becomes clear as a panmagic 4x4 square is compared with (one of the four
4x4 sub-squares of) a most perfect 8x8 magic square.
panmagic 4x4 square 4x4 sub-square of 8x8
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1
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8
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13
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12
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1
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54
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12
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63
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15
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10
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3
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6
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16
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59
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5
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50
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4
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5
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16
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9
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53
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2
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64
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11
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14
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11
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2
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7
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60
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15
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49
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6
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In both squares the sum of two digits of a colour always equals to the lowest plus the highest digit of the
magic square (1+16=17 respectively 1+64=65). With each time two colours you can get all eight (pan)dia-
gonals (see page panmagic 4x4 square, explanation).
Look below at the patterns of the panmagic 4x4 square and the most perfect 8x8 square.
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1
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8
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13
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12
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1
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8
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13
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12
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15
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10
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3
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6
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15
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10
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3
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6
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4
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5
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16
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9
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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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11
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2
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7
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14
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11
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2
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7
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9 + 25 = 34 16 + 18 = 34
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1
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54
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12
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63
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3
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56
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10
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61
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1
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54
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12
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63
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3
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56
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10
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61
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16
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59
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5
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50
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14
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57
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7
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52
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16
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59
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5
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50
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14
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57
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7
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52
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53
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2
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64
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11
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55
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4
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62
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9
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53
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2
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64
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11
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55
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4
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62
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9
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60
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15
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49
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6
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58
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13
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51
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8
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60
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15
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49
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6
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58
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13
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51
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8
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17
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38
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28
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47
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19
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40
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26
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45
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17
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38
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28
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47
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19
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40
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26
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45
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32
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43
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21
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34
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30
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41
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23
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36
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32
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43
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21
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34
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30
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41
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23
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36
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37
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18
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48
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27
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39
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20
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46
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25
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37
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18
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48
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27
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39
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20
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46
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25
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44
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31
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33
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22
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42
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29
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35
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24
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44
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31
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33
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22
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42
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29
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35
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24
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55 + 75 + 59 + 71 = 130 + 130 = 260 17 + 113 + 49 + 91 = 130 + 130 = 260
Now you understand why the sum of the digits of each ½ row/column/diagonal and of each 2x2
sub-square is allways (half of the magic sum: ½ x 260 =) 130.
There are the 3 following swap possibilities:
[1th] You can swap row 1&3 and/or row 2&4 and or row 5&7 and/or row 6&8 and/or column 1&3
and/or column 2&4 and/or column 5&7 and/or column 6&8.
[2nd] You can swap the upper half with the down half and/or the right half with the left half.
[3rd] You can swap row 1&2 and row 3&4 and row 5&6 and row 7&8 and/or column 1&2 and
column 3&4 and column 5&6 and column 7&8.
If you combine the 3 swap possibilities you can get each digit out of 1 up to 64 in the top left corner.
Try it!!!
From Willem Barink we learn that a small part of the most perfect magic squares has an extra magic
feature. See the following most perfect magic 8x8 square:
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1
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32
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43
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54
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9
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24
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35
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62
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1
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32
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43
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54
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9
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24
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35
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62
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60
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37
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18
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15
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52
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45
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26
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7
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60
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37
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18
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15
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52
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45
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26
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7
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22
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11
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64
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33
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30
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3
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56
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41
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22
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11
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64
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33
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30
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3
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56
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41
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47
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50
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5
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28
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39
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58
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13
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20
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47
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50
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5
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28
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39
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58
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13
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20
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17
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16
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59
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38
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25
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8
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51
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46
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17
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16
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59
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38
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25
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8
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51
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46
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44
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53
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2
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31
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36
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61
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10
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23
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44
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53
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2
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31
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36
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61
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10
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23
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6
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27
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48
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49
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14
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19
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40
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57
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6
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27
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48
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49
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14
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19
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40
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57
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63
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34
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21
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12
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55
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42
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29
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4
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63
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34
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21
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12
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55
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42
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29
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4
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33 + 97 = 130 61 + 69 = 130
The extra magic feature is that in each row and each column (not only ) adding the digits from
position (1 up to 4 and 5 up to 8, but also from) 3 up to 6 gives the magic sum of 130.
How to produce most perfect magic squares?
To produce most perfect magic square there are a eight methods of construction. All methods are
on this website:
You can use the transformation method to produce all most perfect magic squares (= 4x4, 8x8, 12x12,
16x16, …).
You can use the basic key method, the quadrant method and Sudoku method 2 to produce
all most perfect magic squares from 8x8 and up (= 8x8, 12x12, 16x16, 20x20, …).
You can use basic patttern method @ 1 and Sudoku method 3 to produce most perfect magic squares
8x8, 16x16, 32x32, 64x64, …
You can use the binary method to produce most perfect 4x4 or 8x8 magic squares
You can use basic pattern method 3 to produce most perfect 16x16 magic squares.
@ With the adapted basic pattern method it is possible to produce most perfect 8x8, 12x12, 16x16 20x20,
24x24, 28x28 and 32x32 magic squares with the extra magic feature X (see above).
See a complete classification of all 368640 most perfect (Franklin pan)magic 8x8 squares on:
'Analysis Franklin panmagic 8x8 (2)'
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