A New Procedure for Magic Squares (Part I)

Zig Zag Consecutive 5x5 and 9x9 Mask-Generated Squares

A Discussion of the New Method

Magic squares such as the Loubère have a center cell which must always contain the middle number of a series of consecutive numbers, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n² + 1). The properties of these regular or associated Loubère squares are:

1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n² + 1, i.e., or twice the number in the center cell and are complementary to each other.

In this method the numbers on the square are placed consecutively starting from the leftmost column and zigged zagged up and down between two rows as was done in the Zig-Zag and Mask generated 9x9 square method Zig Zag 9x9 Cross and Mask-Generated Methods (Part II) using odd numbers to start and even numbers to finish.

In addition, it will also be shown that the sums of these squares follow the new sum equation as was shown in the New block Loubère Method:

Construction of a 5x5 Magic Square

1. Construct Square 1 by adding consecutive numbers in a zig zag manner to the cells. Don't fill in the center row but proceed to the first cell the last row (the number 6).
2. On reaching 10 reverse the pattern by adding consecutive numbers, filling the center row then proceeding from 15 to 16 along the yellow path, and filling in the last two rows. On reaching 20 proceed to 21 and fill up the top two rows in a zig zag manner(Squares 2 and 3).

1 1 3 5 2 4    7 9 6 8 10

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2 1 3 5 2 4 11 12 13 14 15 7 9 6 8 10

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3 35 1 22 3 24 5 55 10 21 2 23 4 25 75 -10 11 12 13 14 15 65 0 16 7 18 9 20 70 -5 6 17 8 19 10 60 5 55 60 65 70 75 35 10 5 0 -5 -10

⇒



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3. Where the grey sums on the penultimate right hand column intersect the grey sums in the next to the last row adjust the values in these cells by adding and subtracting the values in the last row and columns to generate 4. At this point four duplicates (in green) have been generated.
4. Generate a mask whereby the sums of the columns and rows are each 50 and the right and left diagonals are 80 and 75, respectively. This assures that when each of these values is added to the corresponding cell in square 4 (as in the de la Hire method) that all sums will equal 115. In addition the corresponding equation for the sum of this square is S = ½(n³ + 21n).

Construction of a 9x9 Magic Square

1. Construct Square 1 by adding consecutive numbers numbers in a zig zag manner to the cells as was done to the 5x5 square above (square 1).
2. After reaching number 36 jump to the center row and fill it consecutively (Square 2).
3. After reaching 45 jump to the last row and zig zag fill up to 54 (Square 3).



1 1 3 5 7 9 2 4 6 8 10 12 14 16 18 11 13 15 17     20 22 24 26 19 21 23 25 27 29 31 33 35 28 30 32 34 36

⇒

2 1 3 5 7 9 2 4 6 8 10 12 14 16 18 11 13 15 17 37 38 39 40 41 42 43 44 45 20 22 24 26 19 21 23 25 27 29 31 33 35 28 30 32 34 36

⇒

3 1 3 5 7 9 2 4 6 8 10 12 14 16 18 11 13 15 17 37 38 39 40 41 42 43 44 45 20 22 24 26 19 21 23 25 27 46 29 48 31 50 33 52 35 54 28 47 30 49 32 51 34 53 36



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4. From 54 go across to 55 and fill up to 63 (Square 4), followed by 64, 72 and 73 following the zig zag path. (Square 5 and 6).



4 1 3 5 7 9 2 4 6 8 10 12 14 16 18 11 13 15 17 37 38 39 40 41 42 43 44 45 55 20 57 22 59 24 61 26 63 19 56 21 58 23 60 25 62 27 46 29 48 31 50 33 52 35 54 28 47 30 49 32 51 34 53 36

⇒

5 1 3 5 7 9 2 4 6 8 10 65 12 67 14 69 16 71 18 64 11 66 13 68 15 70 17 72 37 38 39 40 41 42 43 44 45 55 20 57 22 59 24 61 26 63 19 56 21 58 23 60 25 62 27 46 29 48 31 50 33 52 35 54 28 47 30 49 32 51 34 53 36

⇒



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5. At this point not all columns, rows or diagonals sum to 369.


6. Where the grey sums on the penultimate right hand column intersect the grey sums in the next to the last row adjust the values in these cells by adding and subtracting the values in the last row and columns to generate 7. At this point five duplicates (in light orange) have been generated.

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6 189 1 74 3 76 5 78 7 80 9 333 36 73 2 75 4 77 6 79 8 81 405 -36 10 65 12 67 14 69 16 71 18 342 27 64 11 66 13 68 15 70 17 72 396 -27 37 38 39 40 41 42 43 44 45 369 0 55 20 57 22 59 24 61 26 63 387 -18 19 56 21 58 23 60 25 62 27 351 18 46 29 48 31 50 33 52 35 54 378 -9 28 47 30 49 32 51 34 53 36 360 9 333 342 351 360 369 378 387 396 405 189 36 27 18 9 0 -9 -18 -27 -36

⇒

7 207 37 74 3 76 5 78 7 80 9 369 73 2 75 4 77 6 79 8 45 369 10 92 12 67 14 69 16 71 18 369 64 11 66 13 68 15 70 -10 72 369 37 38 39 40 41 42 43 44 45 369 55 20 57 22 59 24 43 26 63 369 19 56 39 58 23 60 25 62 27 369 46 29 48 31 50 24 52 35 54 369 28 47 30 58 32 51 34 53 36 369 369 369 369 369 369 369 369 369 369 225

+




7. A possible mask for use with square 7 adds either 81 or 99 (each a multiple of 9) of mask B to the appropriate cells of square 7. The number 81 was picked so as to use numbers >= to n². 91 was pulled out a hat in order that the sum of 81 + 99 with 369 added to 549. The diagonals were similarly adjusted using these two numbers.
8. Addition of the right factors gives square 8 with S = 549 = ½(n³ + 41n) and the modification of 18 numbers.

Mask B 81 99 81 99 99 81 99 81 81 99 81 99 99 81 99 81 99 81

⇒

8 549 118 74 3 76 5 177 7 80 9 549 73 2 75 4 77 6 79 89 144 549 10 92 12 67 113 69 97 71 18 549 64 110 66 94 68 15 70 -10 72 549 37 137 39 40 41 42 43 143 45 549 55 20 57 22 59 105 142 26 63 549 19 56 138 58 104 60 25 62 27 549 145 29 147 31 50 24 52 35 54 549 28 47 30 157 32 51 34 53 117 549 549 549 549 549 549 549 549 549 549 549



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9. Alternatively Mask C may be produced which uses a more logical construction.




• We start by subtracting each of the diagonals(207,225) from square 7 from 369 to give 162 and 144, respectively and which will be used as what I call the "de la Hire constants" much as 81 and 99 were used above.
  
  Addition of the sum of these two numbers, 162 + 144 = 406 to 369 gives 775 a magic pre-sum.


• If we subtract the sum 775 from the two diagonals we obtain the following two sums:
  
  775 = 207 + 568 and 775 = 225 + 560.


• At this point it may be seen that 568 or 569 cannot be broken down into exact multiples of 144 and 162.


• Only by converting the pre-sum 775 into 819 (by the addition of 44) can we obtain a solution such that the following conditions are obeyed:
  
  The right diagonal: 819 = 207 + 612 = 207 + 2(144) + 2(162)
  
  The left diagonal: 819 = 225 + 594 = 225 + 3(144) + 162
  
  The rows and columns: 819 = 369 + 2(144) + 162.


• Generate the mask using the 144 and 162 factors adding these factors to the appropriate cells in square 7 to generate square 9. Sometimes duplicates are formed so go back and tweak the mask a little (move some of the cell numbers around) until no duplicates are generated. (Note that the numbers about the diagonals are somewhat symmetrical about the center cell.)


• Square 9 has a magic sum equal to 819, i.e., S = 819 = ½(n³ + 101n).

This completes this section on a new zig zag consecutive 5x5 and 9x9 Mask-Generated Methods (Part I). The next section deals with new zig zag 7x7 Mask-Generated Methods (Part II). To return to homepage.