How to make perfect magic squares & cubes

14x14 magic square

	How to make perfect magic squares & cubes
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14x14 magic square

How to produce a 14x14 magic square?

Use the method of Strachey to produce a 14x14 magic square. As extra I show you a 14x14 magic
square with a 12x12 Bree/Ollerenshaw magic square as inlay.

Method of Strachey

You need 2x2 the same 7x7 magic square

There are a several methods to produce a 7x7 magic square:

7x7, method 1

Use the diagonal method of the Dutch Professor van der Blij:

						1
					8		2
				15		9		3
			22		16		10		4
		29		23		17		11		5
	36		30		24		18		12		6
43		37		31		25		19		13		7
	44		38		32		26		20		14
		45		39		33		27		21
			46		40		34		28
				47		41		35
					48		42
						49


			22	47	16	41	10	35	4
			5	23	48	17	42	11	29
			30	6	24	49	18	36	12
			13	31	7	25	43	19	37
			38	14	32	1	26	44	20
			21	39	8	33	2	27	45
			46	15	40	9	34	3	28

7x7, method 2
Use the method of De la Loubère (also know as the Siamese method):

- Put digit 1 exactly in the middle of the top row;
- Put the next digits diagonal up right in the cells until there is a cell that has already been filled in
(for example the 8 cannot been put in the cell diagonal up right, because the 7 is already in that
cell);
- Put the next digit below the previous digit (for example put the 8 below the 7);
- Start a new diagonal untill you meet a cell that has already been filled in, et cetera.

30	39	48	1	10	19	28
38	47	7	9	18	27	29
46	6	8	17	26	35	37
5	14	16	25	34	36	45
13	15	24	33	42	44	4
21	23	32	41	43	3	12
22	31	40	49	2	11	20

7x7, method 3
Use the method of the (Dutch) Wikipedia:

- Put digit 1 exactly in the middle of the top row;

- Put the digits 2 up to 7 each time one place to the right and two places down (= chess horse
movement);

- Put digit 8 below 7.

			1
5
				2
	6
					3
		7
		8				4

- Put the digits 9 up to 14 each time one place to the right and two places down (= chess horse
movement);

- Put digit 15 below 14;

- …

- Put digit 43 below 42;

- Put the digits 44 up to 49 each time one place to the right and two places down (= chess horse
movement).

46	31	16	1	42	27	12
5	39	24	9	43	35	20
13	47	32	17	2	36	28
21	6	40	25	10	44	29
22	14	48	33	18	3	37
30	15	7	41	26	11	45
38	23	8	49	34	19	4

N.B.: It is also possible to use a panmagic 7x7 square (see method on page ‘panmagic 5x5 square’).

Produce the second, third and fourth 7x7 magic squares by adding (7 x 7 = ) 49, (2 x 49 = ) 98
respectively (3 x 49 = ) 147 to all digits of the first 7x7 magic square. Put the first square in the top
left corner, put the second square in the down right corner, put the third square in the top right cor-
ner and put the fourth square in the down left corner.

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

The columns give already the magic sum. To get the right sum in the rows and the diagonals,
you need to swap digits. Split up the 7x7 square in the top left corner and the 7x7 square in the
down left corner both in four parts (see blue digits). Swap the ‘parts’ top left and down left of
the 7x7 square in the top left corner with the ‘parts’ top left and down left of the 7x7 square in
the down left corner. Swap the digits of the (one place to the right shifted) border between the
two parts of the 7x7 square in the top left corner with the digits of the (one place to the right
shifted) border between the two parts of the 7x7 square in the down left corner. Swap finally
all digits of the top half of the last columns with the digits of the down half of the last columns.
N.B.: Because we swapped digits from the first 3 columns, we need to swap digits from the last
(3 – 1 = ) 2 columns. See the final result below.

14x14 magic square

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

14x14 Magic square with 12x12 Bree/Ollerenshaw inlay

Produce a 14x14 magic square with 12x12 inlay. Use the method of Bree/Ollerenshaw to
produce the 12x12 inlay. First put the digits 1 up to 144 in sequence and secondly mix the
digits of the four quarters (see colours).

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


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

N.B.: This 12x12 square is panmagic and each random chosen 2x2 sub-square gives 1/3 of the
magic sum.

To produce the border you need the digits 1 up to 26 (and 171 up to 196); to produce the 12x12
inlay you need to add 26 to each digit. To produce the border, you produce a 14x14 concentric
magic square on website http://users.eastlink.ca/~sharrywhite/BorderedMagicSquares.html. Use only
the border. Put the 12x12 inlay and the border together, and you get the following 14x14 magic square with
12x12 inlay:

14x14 magic square (with 12x12 inlay)

14	8	188	10	186	12	196	7	182	16	180	18	178	184
195	27	160	29	162	31	164	38	169	36	167	34	165	2
3