A New Procedure for Magic Squares (Part III)

Consecutive Internal 13x13 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 second leftmost column and entered across every other cell. Consecutive numbers are then added to the next rows boustrophedonically. When n > 7 at least one row turns into a regular left to right order, increasing by one for each n. In addition every other number in the center row (starting with the second cell) will take on its complement. For example for a 13x13 square the second number 82 becomes 90 and 90 becomes 82 (see the 13x13 example below). The final square is 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. This method is being repeated for the 13x13 to show that a second row in non-boustrophedonic order is needed for the next higher 4n +1 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 13x13 Magic Square

1. Construct the 13x13 square (as was shown for the 9x9 square in Part II) first down and then up as shown in squares 1 and 2.
2. Rows 2 and 4 follow regular reading order as opposed to the rest of the rows which follow boustrophedonic order (like a snake). The last row shows the difference of the sum of the grey row from a sum of a typical 13x13 magic square, viz., S = 1105. The light grey entries, 1643 and and 1659, correspond to the diagonal sums.



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

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2 1643 137 1 136 2 135 3 134 4 133 5 132 6 131 959 146 7 143 8 142 9 141 10 140 11 139 12 138 13 919 192 150 14 149 15 148 16 147 17 146 18 145 19 144 1128 -23 20 156 21 155 22 154 23 153 24 152 25 151 26 1082 23 163 27 162 28 161 29 160 30 159 31 158 32 157 1297 -192 39 164 38 165 37 166 36 167 35 168 34 169 33 1251 -146 79 90 81 88 83 86 85 84 87 82 89 80 91 1105 0 40 92 41 93 42 94 43 95 44 96 45 97 46 868 237 104 52 103 51 102 50 101 49 100 48 99 47 98 1004 101 53 105 54 106 55 107 56 108 57 109 58 110 59 1037 68 117 65 116 64 115 63 114 62 113 61 112 60 111 1173 -68 66 118 67 119 68 120 69 121 70 122 71 123 72 1206 -101 130 78 129 77 128 76 127 75 126 74 125 73 124 1342 -237 1105 1105 1105 1105 1105 1105 1105 1105 1105 1105 1105 1105 1105 1659

⇒



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3. Since the columns are all equal to 1105 add or subtract the numbers in the last row from the center column values to generate square 3. At this point all sums are 1105 except for the diagonals. Also six duplicates have been generated.



3

													1643
137	1	136	2	135	3	280	4	133	5	132	6	131	1105
7	143	8	142	9	141	202	140	11	139	12	138	13	1105
150	14	149	15	148	16	124	17	146	18	145	19	144	1105
20	156	21	155	22	154	46	153	24	152	25	151	26	1105
163	27	162	28	161	29	-32	30	159	31	158	32	157	1105
39	164	38	165	37	166	-110	167	35	168	34	169	33	1105
79	90	81	88	83	86	85	84	87	82	89	80	91	1105
40	92	41	93	42	94	280	95	44	96	45	97	46	1105
104	52	103	51	102	50	202	49	100	48	99	47	98	1105
53	105	54	106	55	107	124	108	57	109	58	110	59	1105
117	65	116	64	115	63	46	62	113	61	112	60	111	1105
66	118	67	119	68	120	-32	121	70	122	71	123	72	1105
130	78	129	77	128	76	-110	75	126	74	125	73	124	1105
1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1659



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4. To convert all sums to a magic Sum we 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.




• We start by subtracting 1105 from each of the diagonals (1643,1659) to give 538 and 554, respectively and which will be used as what I call the "de la Hire constants".
  
  Addition of 538 and or554 to the diagonals and columns and rows gives 2197 a magic pre-sum.:
  
  The right diagonal: 2197 = 1643 + 554
  
  The left diagonal: 2197 = 1659 + 538
  
  Columns and rows: 2197 = 1105 +538 + 554



• Since six numbers from the center column must be modified the equations are modified such that the following conditions are obeyed:
  
  The right diagonal: 4851 = 1643 + 3(538) + 2(554)
  
  The left diagonal: 4851 = 1659 + 2(538) + 3(554)
  
  The rows and columns: 4851 = 1105 + 3(538) + 3(554).



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5. Generate the mask using the 538 and 554 factors adding these factors to the appropriate cells in square 3 to generate square 4. This will take a while especially if duplicates are generated.



Mask A 538 554 538 554 554 538 538 538 554 554 554 538 538 554 538 554 538 554 538 554 538 554 538 554 554 538 538 538 554 554 554 554 538 554 538 538 554 538 538 538 554 554 538 554 538 554 538 554 538 554 538 554 554 538 554 554 538 554 538 538 554 538 538 554 554 538 554 554 538 538 538 554 538 554 554 538 538 554

+     Square 3

⇒

6. Square 4 has a magic sum equal to 4851, i.e., S = 4851 = ½(n³ + 505n). The cell entries in color correspond to the color s of the factors from the Mask A.

This completes this section on a new Consecutive 13x13 Internally Added Mask-Generated Squares (Part III). To return to homepage.