How to produce extra magic 9x9 squares?
On the previous webpage I present a method to produce a
panmagic 9x9 square. On this webpage I present
three different methods to produce 9x9 magic squares, which are not panmagic, but the squares have other
extra magic features.
Method 1
We use 9 magic 3x3 squares to build up the 9x9 magic square. We choose a 3x3 magic square and add each
time 9 to all digits. Now we get 9 magic 3x3 squares with all the digits from 1 up to 81. We put the 9 magic 3x3
squares in order by using an (other or the same) magic 3x3 square.
9 x magic 3x3 square (by adding each time 9 to all digits)
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52
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61
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70
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51
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60
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62
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69
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71
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78
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73
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80
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Put the 3x3 squares in order by using a 3x3 magic square
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2
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9
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9
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1
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1
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1
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8
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Magic 9x9 square consisting of 9 magic 3x3 squares
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13
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74
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81
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76
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79
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78
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73
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80
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56
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63
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38
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27
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61
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43
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60
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55
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42
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69
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64
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71
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Method 2
We build up the 9x9 magic square by using (again) 9 magic 3x3 squares. Only this time the 9 magic 3x3 squares
are proportional semi magic squares. Proportional means that all 9 magic 3x3 squares have the same magic sum
of (1/3 x 369 = ) 123.
We use the row and column patterns of the magic 3x3 square (see method 3 on page
3x extra magic 15x15 square).
As column coordinates we use
not the digits 0 up to 2 but 0 up to (9 x 3 -/- 1 = ) 26 and we spread the column
coordinates proportional over the 9 semi magic 3x3 squares.
1 x row coordinate + 3x column coordinate + 1 = semi magic 3x3 square
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7
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11
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65
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36
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Put the 9 magic 3x3 squares together.
9x9 magic square consisting of 9 proportional semi magic 3x3 squares
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40
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81
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2
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52
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57
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14
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28
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69
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26
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3
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41
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79
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15
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53
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55
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27
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67
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80
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1
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42
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56
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68
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25
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65
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Notify that each 1/3 row and each 1/3 column gives 1/3 of the magic sum (1/3 of 369 = 123) and both diagonals
give the magic sum of of 369.
Method 3
We build up the 9x9 magic square by using 1x a digit from the grid with 9x the same 3x3 magic square and 9x a
digit from the same cell of the grid with 9x the shifted versions of the 3x3 magic square.
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+1x digit from 9x the same 3x3 m.s.
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+9x [digit-1] from 9x shifted 3x3
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= extra magic 9x9 square
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80
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10
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33
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71
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1
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51
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62
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19
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42
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3
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50
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70
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21
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41
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61
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12
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32
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79
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40
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63
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20
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31
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81
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11
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49
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72
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2
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53
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64
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6
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44
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55
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24
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35
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73
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15
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57
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23
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43
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75
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14
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34
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66
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5
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52
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13
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36
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74
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4
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54
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65
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22
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45
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56
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26
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37
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60
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17
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28
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78
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8
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46
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69
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30
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77
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16
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48
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68
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7
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39
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59
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25
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67
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9
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47
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58
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27
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38
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76
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18
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29
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Notify that each 1/3 row and each 1/3 column gives 1/3 of the magic sum (1/3 of 369 = 123) and both diagonals
give the magic sum of of 369.
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