New Method for Loubère, Méziriac and Méziriac Type Magic Squares (Part IB)

The Siamese method which includes both the Loubère and Méziriac magic squares have the property that the center cell must always contain the middle number of the series of numbers used, 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). In addition, the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S. Both squares also require an upward stepwise addition of consecutive numbers, i.e., 1,2,3...

Summary of Part IA

The previous page gave an account of a new method to generate 7×7 and 9×9 Méziriac type squares, a method previously unreported since Loubère and Méziriac concentrated only on those squares where the initial digit 1 were located on the periphery of the square (i.e., the cell at the center of row one) or on the cell adjacent to the central cell. There was no attempt to look into those squares where the digit 1 resided on cells other than these two positions. It was shown on that page that the 7×7 square was the first to contain three cells to the right of the central cell and that one could also place the digit 1 in the second cell to generate a new type of Méziriac square.

It was also shown that certain squares, particularly those where the size of the number n used to generate the square was divisible by three, produced less than the required number of squares. Thus, where the starting digit 1 is placed to the right of the central cell towards C determines what right hand shift to use when moving from one diagonal end to the start of the next diagonal. It was mentioned that there are four positions in which the digit 1 can be placed. An E with an arrow meant that a right cell shift of the value in the cell is required towards the periphery of the square. On the other hand, if it points inwards the movement is towards C:

Right Cell Shifts
C	2	4	6	8	10	12	14	16	18	20	22	24	26	...	E →

In addition, the caption for each square included both a letter and the two arrows showing the shift to the right and the accompanying shift to the left, e.g., K (6→ 5←) where the sum of the two numbers is n.

Two Possible Squares Derived from 11×11 and 13×13

Examination of the 11×11 (K) and the 13×13 (L) Méziriac type squares where the initial digit 1 is place three cells to the right of the central cell are shown below. Again the position of this cell implies that a shift of 6 cells to the right is to be used when moving from one diagonal to the next. Although the right hand shifts of these squares are 6 one may also visualize them as left hand shifts of 5 for (K) and 7 for (L) where the sum of the

left hand shift + right hand shift = n

K (6→ 5←)
79	103	6	30	54	67	91	115	18	42	66
102	5	29	53	77	90	114	17	41	65	78
4	28	52	76	89	113	16	40	64	88	101
27	51	75	99	112	15	39	63	87	100	3
50	74	98	111	14	38	62	86	110	2	26
73	97	121	13	37	61	85	109	1	25	49
96	120	12	36	60	84	108	11	24	48	72
119	22	35	59	83	107	10	23	47	71	95
21	34	58	82	106	9	33	46	70	94	118
44	57	81	105	8	32	45	69	93	117	20
56	80	104	7	31	55	68	92	116	19	43

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

So the upshot is that the full complement of m is possible for these squares when n is prime and it doesn't matter whether n is of type 4N+1 or 4N+3.

A Subset of Three 15×15 Squares

The next square, the 15×15 (M), shows (1) that filling in the right hand diagonal, (2) then placing the starting digit 1 six cells to the right of the central cell, (3) doing a stepwise construction, will eventually result in a cell number being superimposed over another. The same result is produced when placing the digit 1 thre cells to the right of the central cell (not shown). In both cases the 30 superimposes over 111 even though the former is a 12 cell right shift and the latter is a 6 cell right shift.

M (12→ 3←)
		24			8			217						120
	23			7			218						119
22			6			215						118
		5			214						117			21
	4			213						116			20
3			212						115			19
		211						114			18			2
	225						113			17			1
224						112			16			15
					111			30			14			233
				110			29			13			222
			109			28			12			221
		108			27			11			220
	107			26			10			219
106			25			9			218

Since n = 3×5 then those right shifts divisible by 5 will not produce a square. Thus, placing the digit 1 five cells to the right of the central cell will, according to the shift table below, result in a right cell shift of 10 cells to the right, whereby the number 15 superimposes over the cell value of 112. The upshot is that one more square is not constructible as seen in partial square Mp.

15×15 Right Cell Shifts
C	2	4	6	8	10	12	14	E →

Mp (10→ 5←)
				115					3
			114					2
		113					1
	112					15
111					14

Although, the square (N) with the digit 1 in position 4 cells from the center is magic (below), the square with the digit in 2 cells to the east of the central cell is not magic (square not shown here but the identical square rotated by 180^o degrees along the main diagonal is shown at the end of Part IV). Thus according to the 15×15 right shift table above, only those right shifts of 2, 8 and 14 are possible while the 4 is non magic the 6, 10, and 12 being impossible.

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

Finally, for the larger squares, we can mention that when n = 25, i.e., p₁ × p₂= 25 and p₁ = p₂ then those cells in the shift table divisible by five will be impossible to construct. For example, placing the digit 1 into the 5th cell from center will result in a right shift of 10, superimposing the digit 51 onto number 311 of the main diagonal. Similarly, for n = 3 × 7, placing the digit 1 in cell number 7 will result in a right shift of 14 and superimpose the digit 21 onto 220 of the main diagonal. The superimpositions, thereby, reduce the number of Méziriac type magic squares.

This completes this section (Part IB). To go to next section Part II. To return to Part IA. To return to homepage.

		24			8			217						120
	23			7			218						119
22			6			215						118
		5			214						117			21
	4			213						116			20
3			212						115			19
		211						114			18			2
	225						113			17			1
224						112			16			15
					111			30			14			233
				110			29			13			222
			109			28			12			221
		108			27			11			220
	107			26			10			219
106			25			9			218

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

		24			8			217						120
	23			7			218						119
22			6			215						118
		5			214						117			21
	4			213						116			20
3			212						115			19
		211						114			18			2
	225						113			17			1
224						112			16			15
					111			30			14			233
				110			29			13			222
			109			28			12			221
		108			27			11			220
	107			26			10			219
106			25			9			218

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

		24			8			217						120
	23			7			218						119
22			6			215						118
		5			214						117			21
	4			213						116			20
3			212						115			19
		211						114			18			2
	225						113			17			1
224						112			16			15
					111			30			14			233
				110			29			13			222
			109			28			12			221
		108			27			11			220
	107			26			10			219
106			25			9			218

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