A New Procedure for Magic Squares

A Loubère Type Method - The Slant Break

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 added by the staircase method starting at (one down, one right) to the center cell. The Loubère method is applied (the staircase method). When a break is encountered the move is (2 down, 2 right), i.e., a 2 down slant. The result is the generation of a Loubère type square different from the standard Loubère.

In addition, those numbers divisible by three will not be magic but will give triads of sums, viz. for a 9 x9 this triad is (360,369,379). To convert this square to magic we can use the mask method as shown in Cross and Mask Methods. The sums of these squares is also shown to follow the new sum equation as was shown in the New block Loubère Method:

Construction of a 5x5 Slant Break Magic Square

1. Construct Square 1 by adding consecutive numbers at position (one down, one right) to the center cell. On reaching 5 break (2 down,2 right) and continue with 6.
2. Fill in the rest of the cell in like manner.

Construction of a slant break 9x9 Magic Square

1. Construct Square 1 by adding consecutive numbers at position (one down, one right) to the center cell. On reaching 5 break (2 down,2 right) and continue with 10.
2. Fill in the rest of the cell in like manner (Squares 2 and 3).
3. At this point not all rows sum to 369.

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

⇒

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

⇒

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

4. Adjust the values in the center row by adding and subtracting the values in the last columns to generate 4. At this point fourduplicates (in light orange) have been generated.
5. Generate a mask using n² as a factor to be added to the appropriate cell of square 5. Since two duplicates are present in columns 4 and 6, two n² factors must be present per row, column and diagonal. In this case to the center cell is added 2n².

4 369 21 6 72 48 24 18 75 60 45 369 5 71 47 32 17 74 59 44 20 369 70 46 31 16 82 58 43 19 4 369 54 30 15 81 48 42 27 3 69 369 29 14 80 56 41 26 2 68 53 369 13 79 55 40 34 1 67 52 28 369 78 63 39 24 0 66 51 36 12 369 62 38 23 8 65 50 35 11 77 369 37 22 7 64 58 34 10 76 61 369 369 369 369 369 369 369 369 369 369 369

+

Mask B 81 81 81 81 81 81 81 81 162 81 81 81 81 81 81 81 81

⇒

6. Addition of these factors to square 4 gives square 85 with S = 531 = ½(n³ + 37n) and the modification of 17 numbers.

5

									531
21	6	72	129	24	18	75	141	45	531
86	71	128	32	17	74	59	44	20	531
70	46	31	16	82	139	43	100	4	531
135	30	15	81	48	42	108	3	69	531
29	14	80	56	203	26	2	68	53	531
13	79	136	40	34	1	67	52	109	531
78	144	39	105	0	66	51	36	12	531
62	38	23	8	65	50	116	11	158	531
37	103	7	64	58	115	10	76	61	531
531	531	531	531	531	531	531	531	531	531

This completes this section on a new slant break Loubère Type 5x5 and 9x9 Mask-Generated Methods. To return to homepage.

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Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com