A New Procedure for Magic Squares (Part III)

9x9 and 13x13 Cross Squares

A Discussion of the New Method

This follows as a continuation of new 7x7 Mask generated Squares (Part II). This page, however, will treat only the cross squares of two 4n + 1 squares as was shown 5x5 cross squares. These squares have non-yellow squares whose sum is equal to ½(n² + 1) ½(n - 1)² .

Construction of 9x9 and 13x13 Cross Magic Squares

Method I: Reading from left to right

Produce square 1 with the sum and row columns (in gray).
As shown below this square is not magic because all the columns and rows don't sum to 369. The last column shows numerically how far this sum is from 369.
Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 369 and then...
Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 369 to generate 3, containing yellow duplicate cells only in the cross.
The four white squares have sums of 41x16 = 656.

1
									369
1	80	3	78	5	76	7	74	9	333	36
72	11	70	13	68	15	66	17	64	396	-27
19	62	21	60	23	58	25	56	27	351	18
54	29	52	31	50	33	48	35	46	378	-9
37	44	39	42	41	40	43	38	45	369	0
36	47	34	49	32	51	30	53	28	360	9
55	26	57	24	59	22	61	20	63	387	-18
18	65	16	67	14	69	12	71	10	342	27
73	8	75	6	77	4	79	2	81	405	-36
365	372	367	370	369	368	371	366	373	369
-4	3	-2	1	0	-1	2	-3	4

⇒

2
									369
1	168	3	78	41	76	7	74	9	369
72	11	70	13	41	15	66	17	64	369
19	62	21	60	41	58	25	56	27	369
54	29	52	31	41	33	48	35	46	369
41	41	41	41	41	41	41	41	41	369
36	47	34	49	41	51	30	53	28	369
55	26	57	24	41	22	61	20	63	369
18	65	16	67	41	69	12	71	10	369
73	8	75	6	41	4	79	2	81	369
369	369	369	369	369	369	369	369	369	369

Produce square 3 with the sum and row columns (in gray).
As shown below this square is not magic because all the columns and rows don't sum to 1105. The last column shows numerically how far this sum is from 1105.
Adjust the center column by adding and subtracting numbers from the selected cells so that the sums become 1105 and then...
Adjust the center row by adding and subtracting numbers from the selected cells so that the sums become 1105 to generate 3, containing yellow duplicate cells only in the cross.
The four white squares have sums of 85x36 = 3060 according to the sum equation from above.

3
													1105
1	168	3	166	5	164	7	162	9	160	11	158	13	1027	-78
156	15	154	17	152	19	150	21	148	23	146	25	144	1170	65
27	142	29	140	31	138	33	136	35	134	37	132	39	1053	-52
130	41	128	43	126	45	124	47	122	49	120	51	118	1144	39
53	116	55	114	57	112	59	110	61	108	63	106	65	1079	-26
104	67	102	69	100	71	98	73	96	75	94	77	92	1118	13
79	90	81	88	83	86	85	84	87	82	89	80	91	1105	0
78	93	76	95	74	97	72	99	70	101	68	103	66	1092	-13
105	64	107	62	109	60	111	58	113	56	115	54	117	1131	26
52	119	50	121	48	123	46	125	44	127	42	129	40	1066	-39
131	38	133	36	135	34	137	32	139	30	141	28	143	1157	52
26	145	24	147	22	149	20	151	18	153	16	155	14	1040	-65
157	12	159	10	161	8	163	6	165	4	167	2	169	1183	78
1099	1110	1101	1108	1103	1106	1105	1104	1107	1102	1109	1100	1111	1105
-6	5	-4	3	-2	1	0	-1	2	-3	4	-5	6

⇒

4
													1105
1	168	3	166	5	164	85	162	9	160	11	158	13	1105
156	15	154	17	152	19	85	21	148	23	146	25	144	1105
27	142	29	140	31	138	85	136	35	134	37	132	39	1105
130	41	128	43	126	45	85	47	122	49	120	51	118	1105
53	116	55	114	57	112	85	110	61	108	63	106	65	1105
104	67	102	69	100	71	85	73	96	75	94	77	92	1105
85	85	85	85	85	85	85	85	85	85	85	85	85	1105
78	93	76	95	74	97	85	99	70	101	68	103	66	1105
105	64	107	62	109	60	85	58	113	56	115	54	117	1105
52	119	50	121	48	123	85	125	44	127	42	129	40	1105
131	38	133	36	135	34	85	32	139	30	141	28	143	1105
26	145	24	147	22	149	85	151	18	153	16	155	14	1105
157	12	159	10	161	8	85	6	165	4	167	2	169	1105
1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105	1105

This completes this section on the new 9x9 and 13x13 Cross Squares Methods (Part III).
The next section deals with new 7x7 and 11x11 Cross Squares Methods (Part III). To return to homepage.