A New Procedure for Magic Squares (Part II)

Consecutive Boustrophedonic 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 first leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. The numbers are added to the last cell on the last row/column. in reverse order starting at the next to the last cell in the last row/column. The square which is not magic is modified into a form which can be converted into a magic one 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 follow the sum equation shown in the New block Loubère Method. :

Construction of a 9x9 Magic Square

1. Construct the 9x9 Square 1 where 5 = 4n + 1 by adding numbers in a consecutive manner starting at row 1 cell 1, then swing around at the end (boustrophedonically) until the final cell is reached (square 1).
2. On reaching the end cell reverse the process and continue adding numbers consecutively and boustrophedonically (Square 2).

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

⇒

2 208 1 81 2 80 3 79 4 78 5 333 36 73 9 74 8 75 7 76 6 77 405 -36 10 72 11 71 12 70 13 69 14 342 27 64 18 65 17 66 16 67 15 68 396 -27 19 63 20 62 21 61 22 60 23 351 18 55 27 56 26 57 25 58 24 59 387 -18 28 54 29 53 30 52 31 51 32 360 9 46 36 47 35 48 34 49 33 50 378 -9 37 45 38 44 39 43 40 42 41 369 0 333 405 342 396 351 387 360 378 369 189 36 -36 27 -27 18 -18 9 -9 0

⇒

3. Since all sums of all the columns or rows are not equal to 65 add or subtract the numbers in the last row from those numbers identical in sum in the last columns. At this point six duplicates have been generated (Square 3).

3

									208
37	81	2	80	3	79	4	78	5	369
73	-27	74	8	75	7	76	6	77	369
10	72	38	71	12	70	13	69	14	369
64	18	65	-10	66	16	67	15	68	369
19	63	20	62	39	61	22	60	23	369
55	27	56	26	57	7	58	24	59	369
28	54	29	53	30	52	40	51	32	369
46	36	47	35	48	34	49	24	50	369
37	45	38	44	39	43	40	42	41	369
369	369	369	369	369	369	369	369	369	189

4. 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 the magic sum.

This completes this section on a new Consecutive Boustrophedonic Mask-Generated Squares (Part II). To return to homepage.