A New Procedure for Magic Squares (Part IB)

Equation Generated Equation Generated Centered Sequential Mask-Generated Squares

A Discussion of the New Method

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In this method the numbers on a 4n + 1 square are placed consecutively starting from the center cell in the top row and entered until every other cell is filled. (This is a modification of the 9x9 square constructed previously ). Consecutive numbers are then added to the next rows using a (1,down,1,right) knight move as shown below in the examples. However, there is one exception to the rule:

Additions to the square going up are performed by filling in the center row followed by the empty cells below the center row. The cells above the filled centered row are filled according to the general sequence formula shown below, i.e., 2 cells to the right, 4 cells to the right and 3 cells to the left as shown in the sequence table derived from the sequence equation. Again this is modified from the 9x9 method where the row adjacent to the center row was initially filled in a reverse manner instead of to the right as in this new method.

After converting the squares into semi-magic ones the square are converted into magic ones by the use of a mask. This mask generates numbers which are added to certain cells in the square to produce a final square composed of numbers which may not be in serial order. For example, negative numbers or numbers greater than n² may be present in the square.

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 9x9 Magic Square

1. Construct the 9x9 Square 1 where 9 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 5 and filling every other the cell (square 1).
2. Upon reaching the numeral 5 perform a knight break enter 6 at cell 4 and continue filling cells in a normal fashion.
3. After skipping over the center row add 19 to the eight cell in the 6th row and continue adding numerals in normal fashion in the last row of the square.
4. From 36 go to the center row and fill in the row consecutively (square 2). Fill in the rest of the square, going backwards this time (square 3), starting at 45 → 46. At 63 go to 64 adjacent to the numeral 2 on the first row.



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

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

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5. Fill in the rest of row 1 in a backwards manner, then fill in the rest of the square using the sequence table to generate the moves. For example, to go to the second row move 2 cells (knight), followed by filling the row to the right. Continue filling the next rows using first, 4 moves right then 3 moves left.
6. Since only the columns are equal to the magic sum while the row and diagonal sums are not, add or subtract the numbers in the last column to those of the center column. At this point four duplicates have been generated in pink (Square 4).



3 193 4 67 5 66 1 65 2 64 3 277 92 72 9 68 6 69 7 70 8 71 380 -11 14 76 10 73 11 74 12 75 13 358 11 78 15 79 16 80 17 81 18 77 461 -92 37 38 39 40 41 42 43 44 45 369 0 47 20 48 21 49 22 50 19 46 322 47 25 52 26 53 27 54 23 51 24 335 34 57 30 58 31 59 28 55 29 56 403 -34 35 62 36 63 32 60 33 61 34 416 -47 369 369 369 369 369 369 369 369 369 188

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4 193 4 67 5 66 93 65 2 64 3 369 72 9 68 6 58 7 70 8 71 369 14 76 10 73 22 74 12 75 13 369 78 15 79 16 -12 17 81 18 77 369 37 38 39 40 41 42 43 44 45 369 47 20 48 21 96 22 50 19 46 369 25 52 26 53 61 54 23 51 24 369 57 30 58 31 25 28 55 29 56 369 35 62 36 63 -15 60 33 61 34 369 369 369 369 369 369 369 369 369 369 188

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7. 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 3 (as in the de la Hire method) that all sums will equal.

This completes this section on a new Equation Generated Centered Sequential Mask-Generated Squares (Part IB). The next section deals with Equation Generated Centered Sequential 13x13 Mask-Generated Squares (Part IC). To return to homepage.