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What do I need to know about most perfect squares?
What do I need to know, before I read about most perfect squares?
Which size (order) have most perfect squares?
Most perfect squares are multiples of 4 (4x4, 8x8, 12x12, 16x16, 20x20, … magic squares).
Which special features have most perfect squares?
● Most perfect 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 fea-
tures 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 is ¼ of the magic sum.
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 ‘propor-
tional’ 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)diagonals (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.
How to produce most perfect magic squares?
To produce most perfect magic square there are a several methods of construction. Most of this methods are on this
website:
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