Franklin Squares
Part 3
My Construction Method

	A	1	2	3	4	5	6	7	8
	1	0	1	2	3	4	5	6	7
	2	15	14	13	12	11	10	9	8
	B	1	2	3	4	5	6	7	8		odd	even
	3	0	14	13	3	11	5	6	8	60	30	30
	4	15	1	2	12	4	10	9	7	60
16x16 A
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
	0	14	13	3	11	5	6	8	7	9	10	4	12	2	1	15
	15	1	2	12	4	10	9	7	8	6	5	11	3	13	14	0
magic square
	1	240	209	64	177	96	97	144	113	160	161	80	193	48	17	256
	242	31	34	207	66	175	146	127	130	111	82	191	50	223	226	15
	16	225	224	49	192	81	112	129	128	145	176	65	208	33	32	241
	255	18	47	194	79	162	159	114	143	98	95	178	63	210	239	2
	3	238	211	62	179	94	99	142	115	158	163	78	195	46	19	254
	244	29	36	205	68	173	148	125	132	109	84	189	52	221	228	13
	14	227	222	51	190	83	110	131	126	147	174	67	206	35	30	243
	253	20	45	196	77	164	157	116	141	100	93	180	61	212	237	4
	5	236	213	60	181	92	101	140	117	156	165	76	197	44	21	252
	246	27	38	203	70	171	150	123	134	107	86	187	54	219	230	11
	12	229	220	53	188	85	108	133	124	149	172	69	204	37	28	245
	251	22	43	198	75	166	155	118	139	102	91	182	59	214	235	6
	7	234	215	58	183	90	103	138	119	154	167	74	199	42	23	250
	248	25	40	201	72	169	152	121	136	105	88	185	56	217	232	9
	10	231	218	55	186	87	106	135	122	151	170	71	202	39	26	247
	249	24	41	200	73	168	153	120	137	104	89	184	57	216	233	8
C	1	2	3	4	5	6	7	8	9	10	11	12
1	0	1	2	3	4	5	6	7	8	9	10	11
2	23	22	21	20	19	18	17	16	15	14	13	12
D	1	2	3	4	5	6	7	8	9	10	11	12		odd	even
3	0	22	21	3	4	18	17	7	8	14	13	11	138	63	75
4	23	1	2	20	19	5	6	16	15	9	10	12	138
E	1	2	3	4	5	6	7	8	9	10	11	12		odd	even
5	0	22	21	3	4	18	17	7	14	8	13	11	138	69	69
6	23	1	2	20	19	5	6	16	9	15	10	12	138

The other feature of interest in square B was complimentary symmetry. If square A can be constructed with that feature it will be in the combined magic square. To accomplish this complimentary pairs must be placed symmetrically opposite the center line. Enter the number set in two lines as shown and block it off into blocks of four pairs. In the first block invert the middle two pairs. In the second block invert the outside two pairs. In the top row the numbers in the odd columns are in the half diagonal. The numbers in the even columns have their compliments in the diagonal. The numbers in the odd columns must total 30 as do the numbers in the even columns to have both half rows and half diagonals with the proper totals. This construction accomplishes that. Place the top row, B3, in the left half of row one in square A and the bottom row, B4, in the right half in reverse order.

This construction also produces a pandiagonal square with all the Franklin features including wraparound of the bent diagonals.

Complete the square with complimentary construction. Multiply this square by 16 and add square B of the previous square plus one.

This construction is general for orders that are multiples of sixteen. They have an even number of blocks of four pairs which is necessary-pairs of blocks offset each other. Orders that are odd multiples of eight have an odd number of blocks and must be adjusted to get the right totals. The workup for order 24 is shown. Note that the center two pairs are inverted in all blocks and the adjustment is made by reversing columns nine and ten. This is only one of many ways to do this. I don’t have a general construction.

Return to My Contruction Method Part 2

Continue to My Contruction Method Part 4