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Illustration group 1
Magic squares of group 1 can be constructed by means of combining G-grids (i.e. all quadrants have
a G-structure) with G*-grids (i.e. all quadrants have a G*-structure).
First you construct the 8x8 row grid. Fill the upper left quadrant with the digits after G1, G2, G3, G4,
G5 or G6. In the example G1 has been chosen.
G1 (row grid), step 1
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Necessarily the same G-quadrant must be repeated in the down left corner (as is shown with the
purple digits).
G1 (row grid), step 2
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The right half of the row grid must be filled with the digits 2, 3, 4, and 5. In column 5 and 7 it is
only possible to continue with an alteration of 2-5, 5-2, 3-4, or 4-3, that is four options. In the
example 2-5 has been chosen.
G1 (row grid), step 3
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Now in column 6 and 8 it is only possible to continue with an alteration of 3-4 or 4-3, that is two
options. In the example 4-3 has been chosen.
G1 (row grid), step 4
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In total there are 6 (G1 to G6) x 4 (options of step 3) x 2 (options of step 4) = 48 possible row grids.
By reflecting the 48 row grids diagonally you can produce 48 column grids. In the G-group it is pos-
sible to match all 48 row grids with all 48 column grids. See below one of the 64 possible squares
of G1/G1*. Note that in the example the column grid is the reflection of the row grid.
1x digit from row grid + + 8x digit from column grid = most perfect 8x8 magic square
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1
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63
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8
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58
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3
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61
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6
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60
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1
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6
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6
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1
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56
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10
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49
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15
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54
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12
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51
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13
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57
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7
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64
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2
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59
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62
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16
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50
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9
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55
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14
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52
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11
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53
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2
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5
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17
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47
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24
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42
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19
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45
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22
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44
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4
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4
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4
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40
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26
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33
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31
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38
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28
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35
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29
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41
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23
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48
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18
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43
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21
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46
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20
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32
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25
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39
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30
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36
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27
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37
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The combination G/G* gives in total 48x48 = 2304 squares. The collection of squares belonging to
the combination G*/G (= swapping row and column grids) gives also 2304 squares. But: are they
new squares?
Something about swapping (c/r-switch)
In the above special example it is easy to see that the swapping gives the diagonal reflected speci-
men of the original square. But what about this (arbitrary) second example:
G1 G3*
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63
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48
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40
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Swapping the grids gives the following:
G3* G1
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56
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16
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40
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41
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32
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5
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62
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8
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59
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61
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60
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Despite the incompleteness of the final squares it can easily be established that the second square
is different from the first one! So, swapping gives new squares? Let us continue our investigation
and now reflect diagonally this second square, and see what happens:
G3 G1*
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0
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1
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62
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8
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59
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2
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61
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7
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60
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7
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1
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56
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0
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1
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7
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0
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7
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57
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1
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1
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6
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1
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6
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1
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6
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1
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6
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16
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5
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2
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2
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5
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17
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4
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3
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4
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3
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4
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3
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4
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3
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40
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0
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7
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2
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1
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6
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5
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2
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5
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2
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5
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5
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2
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41
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3
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4
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4
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4
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32
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And we establish that the swapping of G1/G3* gives the diagonal reflection of G3/G1*, and that
is a square belonging to the first 2304 squares already made.
So, the conclusion is that combinations of two grids that structurally are reflections
of each other (e.g. combination 1, 3a,3b, 4a, etc…) give no new squares when swap-
ping the grids.
Consequently, the amount of squares of combination 1, the G-group, is 2304.
The first square we made contains the extra magic property X. Make sure that this property can
only arise in case of the following 6 sequences of digits in the first row or column: 0-6-7-1-2-4-5-3,
0-6-7-1-4-2-3-5, 0-5-7-2-1-4-6-3, 0-5-7-2-4-1-3-6, 0-3-7-4-1-2-6-5 and 0-3-7-4-2-1-5-6. Conse-
quently the amount of squares in group 1 containing this property is 6 x 6 = 36.
Illustration group 2
Magic squares of group 2 can be produced by means of combining A-grids with B-grids.
First you construct the row grid. Fill the upper left quadrant after A1, A2 or A3. In the example A1
has been chosen.
A1 (row grid), step 1
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In the 5th row you can only continue with 0-7-6-1 or 1-6-7-0. With both options you can finish the
down left quadrant, continuing the A-structure (N.B.: that is a choice, you are going to construct
an A-grid!). In the example 0-7-6-1 has been chosen, which means repeating the first quadrant.
A1 (row grid), step 2
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0
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1
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7
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0
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