Inlaid 14x14 magic square (with inlays 4x4, 6x6 and 12x12)

I was inspired by the inlaid squares on the website of Harvey Heinz: www.magic-squares.net/magicsquare.htm#Orders 3, 5, 7, 9 Inlaid and the inlaid squares of John Hendricks: www.magic-squares.net/hendricks.htm

First we construct a 12x12 inlay, which consist of four pan-4x4 in 6x6 magic squares in five steps:

[1] To get the four 4x4 panmagic inlay squares we take a 8x8 most perfect (Franklin pan)magic square, add 40 to each number and split up the 8x8 square in four 4x4 (inlay) squares.

Most perfect 8x8 square + 40 = four 4x4 inlay squares

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

[2] To construct the four borders we need (4 x 20 =) 80 numbers. Take the numbers 1 up to 40 and 105 up to 144. Translate the numbers 105 up to 144 into -/- 1 up to -/- 40.

[3] Put in each side of the border 3 positive and 3 negative numbers and take care that the sum of the 6 numbers is exactly 0. In the four x four corners you need 16 numbers extra = 8 positive and 8 negative numbers double. The average number is ([the lowest number + the highest number] divided by 2: [1+40]/2 =) 20,5. Take as sum of the 8 double numbers (8 x 20,5 = ) 164. Take as sum of 3 numbers (3 x 20,5 =) 61,5 = 61 (8x) or 62 (8x). I puzzeled and got the following table:

+	15	20	26	61	16	21	25	62	17	22	23	62	18	19	24	61	164
+	7	28	26	61	5	32	25	62	8	31	23	62	1	36	24	61
-/-	15	9	37	61	16	6	40	62	17	10	35	62	18	4	39	61
-/-	13	14	34	61	3	29	30	62	2	27	33	62	11	12	38	61

[4] Use the table to construct the 4 borders and translate the negative numbers into -/- 1 up to -/- 40 into 105 up to 144).

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


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


15	20	132	131	111	26	16	21	142	116	115	25	17	22	143	118	112	23	18	19	134	133	107	24
117					28	113					32	114					31	109					36
138					7	140					5	137					8	144					1
37					108	40					105	35					110	39					106
9					136	6					139	10					135	4					141
119	125	13	14	34	130	120	124	3	29	30	129	122	123	2	27	33	128	121	126	11	12	38	127

[5] Combine the borders and the 4x4 inlay squares.

12x12 magic square = four 6x6 magic squares with pan-4x4 inlay

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

The magic sum of each 4x4 panmagic inlay square is 290.The magic sum of each 6x6 magic square is 435. The magic sum of the 12x12 magic square is 870.

Because the 12x12 magic square consists of four proportional 6x6 magic squares, each 1/2 row/column/diagonal gives 1/2 of the magic sum.

And now we construct the 14x14 inlaid square:

[1] Add 26 to each number of the 12x12 magic square;

[2] Use 52 numbers (= 1 up to 26 and 171 up to 196) to construct the 14x14 border.

The sum of the numbers 1 up to 26 is 351. Add 33 to the sum of 351 and you get 384 = 4x96. To get 33 take the numbers 16 and 17 double. I constructed the following table:

16	17	1	26	2	25	9	96
16	4	24	5	22	6	19	96
17	3	23	7	21	10	15	96
8	11	12	13	14	18	20	96

Use the table to construct the 14x14 border (the numbers 171 up to 196 have been translated into -/- 1 up to -/- 26):

16	1	26	2	25	9	-8	-11	-12	-13	-14	-18	-20	17
													3
													23
													7
													21
													10
													15
													-4
													-24
													-5
													-22
													-6
													-19
													-16


16	1	26	2	25	9	-8	-11	-12	-13	-14	-18	-20	17
-3													3
-23													23
-7													7
-21													21
-10													10
-15													15
4													-4
24													-24
5													-5
22													-22
6													-6
19													-19
-17	-1	-26	-2	-25	-9	8	11	12	13	14	18	20	-16


16	1	26	2	25	9	189	186	185	184	183	179	177	17
194													3
174													23
190													7
176													21
187													10
182													15
4													193
24													173
5													192
22													175
6													191
19													178
180	196	171	195	172	188	8	11	12	13	14	18	20	181

Combine the 14x14 border and the 12x12 inlay to complete the 14x14 magic inlaid square.

Magic 14x14 inlaid square (with inlays 4x4, 6x6 and 12x12)

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

Who told you that only simple 14x14 (= double odd) magic squares exist!!!

14x14, Inlaid (2).xls

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14x14 magic square