A New Procedure for Magic Squares (Part IB)

Centered Sequential 9x9 Zig Zag Mask-Generated Squares

A Discussion of the New Method

In this method the numbers on a 4n + 1 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 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:

S = ½(n³ ± an)
S = ½(n³ ± an + b)

Construction of a 9x9 Magic Square

Method: Sequential Readout - use of a mask

Construct the 9x9 Square 1 by adding numbers in a consecutive zigzag manner starting at row 1 cell 5 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.
Upon reaching the numeral 9 a 2 down cell break is performed to numeral 10 and addition to the cells in a normal fashion.
To skip over the center row at numeral 18, a cell break of 4 down is performed and addition to the cells in a normal fashion.
From 36 go to the center row and fill in the row consecutively (square 2).

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

	27		20		22		24
26		19		21		23		25
	28		30		32		34
36		29		31		33		35

⇒

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

⇒

After filling in the center row the rest of the square is filled starting at 46 and continuing in zigzag fashion (square 3), Thence to 64 (second row) and filling and completing the square.
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 six duplicates have been generated in pink or the blue line (Square 4).

3
									178
6	68	8	70	1	72	3	65	5	298	71
67	7	69	9	71	2	64	4	66	359	10
16	78	18	80	11	73	13	75	15	379	-10
77	17	79	10	81	12	74	14	76	440	-71
37	38	39	40	41	42	43	44	45	369	0
47	27	49	20	51	22	53	24	46	339	30
26	48	19	50	21	52	23	54	25	318	51
57	28	59	30	61	32	63	34	56	420	-51
36	58	29	60	31	62	33	55	35	399	-30
369	369	369	369	369	369	369	369	369	196

⇒

4
									178
6	68	8	70	72	72	3	65	5	369
67	7	69	9	81	2	64	4	66	369
16	78	18	80	1	73	13	75	15	369
77	17	79	10	10	12	74	14	76	369
37	38	39	40	41	42	43	44	45	369
47	27	49	20	81	22	53	24	46	369
26	48	19	50	72	52	23	54	25	369
57	28	59	30	10	32	63	34	56	369
36	58	29	60	1	62	33	55	35	369
369	369	369	369	369	369	369	369	369	196

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 square4 (as in the de la Hire method) that all sums will equal.

We start by subtracting the diagonals(178,196) from 369 to give 173 and 191, respectively and which will be used as what I call the "de la Hire constants".
Addition of the sum of these two numbers, 173 + 191 = 364 to 369 gives 733 a magic pre-sum. However, these factors do not produce the necessary mask and new equations with the right number of factors must be recalculated.
The following equations are used such that the following conditions are obeyed:
The right diagonal: 1097 = 178 + 2(173) + 3(179)
The left diagonal: 1097 = 196 + 3(173) + 2(191)
The rows and columns: 1097 = 369 + 2(173) + 2(191)
Generate the mask using the 173 and 191 factors and adding these factors to the appropriate cells in square 4 to generate square 5.
Square 5 has a magic sum equal to 1097, i.e., S = 1097 = ½(n³ + 162n + 7).

Mask B
173					191	191		173
173		173		191			191
	191		173	191		173
191		173	191				173
	173	191			173			191
	364						173	191
		191	191	173		173
			173	173	191		191
191					173	191		173

⇒

5
									1097
179	68	8	70	72	263	194	65	178	1097
246	7	242	9	272	2	64	195	66	1097
16	269	18	253	192	73	186	75	15	1097
268	17	252	201	10	12	74	187	76	1097
37	211	230	40	41	215	43	44	236	1097
47	391	49	20	81	22	53	197	237	1097
26	48	210	241	245	52	196	54	25	1097
57	28	59	203	183	223	63	225	56	1097
227	58	29	60	1	235	224	55	208	1097
1097	1097	1097	1097	1097	1097	1097	1097	1097	1097

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