Franklin Squares
Part 4
My Construction Method  

					Order 12											1/3 rows
				A	1	2	3	4	5	6		odd	even			must equal 22
				1	0	10	9	3	4	5	22	13		dia
				2	11	1	2	8	7	6	22		15	28		diagonal total
																must equal 33
					Order 12
				B	1	2	3	4	5	6		odd	even
				1	0	9	10	3	7	5	22	17		dia
				2	11	2	1	8	4	6	22		16	33
				C	row one square A
				1	0	9	10	3	7	5	6	4	8	1	2	11
				D	column one square B
				1	0	9	10	3	7	5	11	2	1	8	4	6
				E	column one square B
				1	0	9	10	3	7	5	4	6	11	2	1	8
					12x12 A													12x12 B
					0	9	10	3	7	5	6	4	8	1	2	11		0	11	0	11	0	11	0	11	0	11	0	11
					11	2	1	8	4	6	5	7	3	10	9	0		9	2	9	2	9	2	9	2	9	2	9	2
					0	9	10	3	7	5	6	4	8	1	2	11		10	1	10	1	10	1	10	1	10	1	10	1
					11	2	1	8	4	6	5	7	3	10	9	0		3	8	3	8	3	8	3	8	3	8	3	8
					0	9	10	3	7	5	6	4	8	1	2	11		7	4	7	4	7	4	7	4	7	4	7	4
					11	2	1	8	4	6	5	7	3	10	9	0		5	6	5	6	5	6	5	6	5	6	5	6
					0	9	10	3	7	5	6	4	8	1	2	11		4	7	4	7	4	7	4	7	4	7	4	7
					11	2	1	8	4	6	5	7	3	10	9	0		6	5	6	5	6	5	6	5	6	5	6	5
					0	9	10	3	7	5	6	4	8	1	2	11		11	0	11	0	11	0	11	0	11	0	11	0
					11	2	1	8	4	6	5	7	3	10	9	0		2	9	2	9	2	9	2	9	2	9	2	9
					0	9	10	3	7	5	6	4	8	1	2	11		1	10	1	10	1	10	1	10	1	10	1	10
					11	2	1	8	4	6	5	7	3	10	9	0		8	3	8	3	8	3	8	3	8	3	8	3
					magic square
					1	120	121	48	85	72	73	60	97	24	25	144
					142	27	22	99	58	75	70	87	46	123	118	3
					11	110	131	38	95	62	83	50	107	14	35	134
					136	33	16	105	52	81	64	93	40	129	112	9
					8	113	128	41	92	65	80	53	104	17	32	137
					138	31	18	103	54	79	66	91	42	127	114	7
					5	116	125	44	89	68	77	56	101	20	29	140
					139	30	19	102	55	78	67	90	43	126	115	6
					12	109	132	37	96	61	84	49	108	13	36	133
					135	34	15	106	51	82	63	94	39	130	111	10
					2	119	122	47	86	71	74	59	98	23	26	143
					141	28	21	100	57	76	69	88	45	124	117	4
					Order 20										1/5 rows must equal 38
					1	2	3	4	5	6	7	8	9	10			diagonal total must equal 95
				1	0	18	17	3	4	14	13	7	8	9	38	38	42
				2	19	1	2	16	15	5	6	12	11	10	38	38		44	86
					Order 20
					1	2	3	4	5	6	7	8	9	10
				3	3	18	17	0	4	14	13	7	11	9	38	38	48
				4	16	1	2	19	15	5	6	12	8	10	38	38		47	95
					Row one square A
					3	18	17	0	4	14	13	7	11	9	10	8	12	6	5	15	19	2	1	16
					Column one square B
					3	18	17	0	4	14	13	7	11	9	8	10	16	1	2	19	15	5	6	12

Orders that are odd multiples of four, 12, 20, 28,…, present problems. In order twelve row total is 66. Half row total is 33. In row one numbers in odd columns are in the diagonal and the compliments of the numbers in even columns are in the diagonal. If odd columns plus even columns equal half row total and odd columns plus compliments of even columns also equal half row total then odd columns=even columns=forth row totals. For all orders that are odd multiples of four half row total will always be odd. For this reason I believe squares of these orders can’t have half rows and/or columns, full diagonals, and bent diagonals with proper totals in the same square.

I’ve given up half rows and half columns with half row totals and added third rows and third columns with third row totals. Any row or column can be broken into three equal segments and each segment will equal one third of row total.

Set up the number set in row A1 and row A2 as before. Adjust the pairs as in rows B1 and B2. In row one of square A row B1 is the left half and row B2 in reverse order is the right half as in C1. This places complimentary pairs symmetrically opposite in the row. In column one of square B complimentary pairs must be both in odd rows or both in even rows. Row D1 shows B1 as the left half and B2 in order as the right half with complimentary pairs in rows one and seven, two and eight, three and nine, and so forth. This places complimentary pairs properly, both elements in odd rows and both elements in even rows, but ruins third column totals. Moving the four and six from rows eleven and twelve to rows seven and eight solve the problem. Row E1 is column one of square B.

Multiply square A by twelve and add square B plus one to get the magic square. It has all the features as the square with symmetry.

The work up for order twenty is shown. In order twenty, rows and columns must be broken into five equal segments with 1/5th row totals.

Return to My Contruction Method Part 3