A New Procedure for Magic Squares (Part ID)

Centered Sequential Zig Zag 7x7 Mask-Generated Squares

A Discussion of the New Method

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In this method the numbers on a 4n + 3 square are placed consecutively starting from the center cell in the top row and entered in a zig zag manner. Additions to the square are done by leaving the center row unfilled and continuing the zig zag pattern below the center row. (The addition across the center row is different from previous additions which began on the first cell. The addition of numbers in this case may come elsewhere)). The center row is then filled in consecutively followed by the filling in of the bottom and top rows again in a zig zag manner. Breaking involves moving down 2 cells to the next row and continuing addition of numbers to the right.

After the square is filled, the square is converted into a semi-magic one by addition or subtraction i,e. by the differences of numbers in the last row. The square is then converted by means of a numerical mask into a magic square. Moreover, this new square may have negative numbers or numbers greater than n².

In addition, it will also be shown that the sums of these squares follows either of the two sum equations shown in the New block Loubère Method and Consecutive 5x5 Mask Generated squares:

Construction of a 7x7 Magic Square

1. Construct the 7x7 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 4 and filling every other cell (square 1). Note that on reaching the final cell in the row the addition is continued at the other end of the row.
2. Upon reaching 7 go down one cell to 8 and add numbers same manner eventually skipping over the center row.
3. At 14 perform move down two cells (a down break of 2) and continue the addition to the cells in a zigzag fashion.
4. From 21 go to the center row and fill in the row consecutively (square 2).

1 6 1 3 5 7 2 4 13 8 10 12   14 9 11 21 16 18 20 15 17 19

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2 6 1 3 5 7 2 4 13 8 10 12 22 23 24 25 26 27 28 14 9 11 21 16 18 20 15 17 19

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5. From 28 go to 29 and repeat the zig zag in a reverse manner.
6. From 35 break 2 up to 36 and add numbers zigzag across the center row but in a right hand fashion.
7. At 42 go a distance of four units up to 43 and continue adding in zigzag fashion to fill in square 4.
8. At this point all columns at this point sum to 175, while the row sums are to be adjusted (by + or - values) according to the last cell in square 4 in order to sum to 175.
9. Square 5 shows the result of the adjustment with the generation of 2 pink duplicates.

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

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4 225 46 6 48 1 43 3 45 192 -17 5 47 7 49 2 44 4 158 17 13 39 8 41 10 36 12 159 16 22 23 24 25 26 27 28 175 0 38 14 40 9 42 11 37 191 -16 21 31 16 33 18 35 20 174 1 30 15 32 17 34 19 29 176 -1 175 175 175 175 175 175 175 232

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5 225 46 6 48 -16 43 3 45 175 5 47 7 66 2 44 4 175 13 39 8 57 10 36 12 175 22 23 24 25 26 27 28 175 38 14 40 -7 42 11 37 175 21 31 16 34 18 35 20 175 30 15 32 16 34 19 29 175 175 175 175 175 175 175 175 230

10. Generate a mask whereby the sums of the columns and rows are constructed as in the box below. 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 be equal.

This completes this section on a new Centered Sequential Mask-Generated Squares (Part IC). The next section deals with Centered 11x11 Sequential Mask-Generated Zig Zag Squares (Part IE). To return to homepage.