Shift method (2)

It is possible to use the shiftmethod to get a more tight structure of the panmagic 15x15 square.

You have to puzzle to get the first row. Construct a 3x5 = 5x3 matrix:

Matrix 3x5 = Matrix 5x3

0	9	12	21	0	9	12	21
1	14	6	21	1	14	6	21
11	2	8	21	11	2	8	21
13	3	5	21	13	3	5	21
10	7	4	21	10	7	4	21

35	35	35		35	35	35

The magic sum of 0 up to 14 is 105. In the matrix the sum of each column is (5/15 x 105 =) 35 and the sum of each row is (3/15 x 105 =) 21 is. Put the numbers in the first row:

First row according to 3x5 matrix

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

First row according to 5x3 matrix

0

8

7

1

5

9

11

4

14

13

12

2

10

6

3

Construct row 2 up to 15 by shifting the first row each time 4 places to the left.

0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3
5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1
14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4
10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2
8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0
9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5
13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14
6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10
7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8
11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9
12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13
3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6
1	5	9	11	4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3
4	14	13	12	2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3
2	10	6	3	0	8	7	1	5	9	11	4	14	13	12	2	10	6	3

We have constructed the first grid. The second grid is a reflection (rotated by a quarter and mirrored) of the first grid. Take 15x number +1 from first grid and add 1x number from second grid to get a 15x15 panmagic square.

Take 15x number + 1

0	8	7	1	5	9	11	4	14	13	12	2	10	6	3
5	9	11	4	14	13	12	2	10	6	3	0	8	7	1
14	13	12	2	10	6	3	0	8	7	1	5	9	11	4
10	6	3	0	8	7	1	5	9	11	4	14	13	12	2
8	7	1	5	9	11	4	14	13	12	2	10	6	3	0
9	11	4	14	13	12	2	10	6	3	0	8	7	1	5
13	12	2	10	6	3	0	8	7	1	5	9	11	4	14
6	3	0	8	7	1	5	9	11	4	14	13	12	2	10
7	1	5	9	11	4	14	13	12	2	10	6	3	0	8
11	4	14	13	12	2	10	6	3	0	8	7	1	5	9
12	2	10	6	3	0	8	7	1	5	9	11	4	14	13
3	0	8	7	1	5	9	11	4	14	13	12	2	10	6
1	5	9	11	4	14	13	12	2	10	6	3	0	8	7
4	14	13	12	2	10	6	3	0	8	7	1	5	9	11
2	10	6	3	0	8	7	1	5	9	11	4	14	13	12

+ 1x number

3	1	4	2	0	5	14	10	8	9	13	6	7	11	12
6	7	11	12	3	1	4	2	0	5	14	10	8	9	13
10	8	9	13	6	7	11	12	3	1	4	2	0	5	14
2	0	5	14	10	8	9	13	6	7	11	12	3	1	4
12	3	1	4	2	0	5	14	10	8	9	13	6	7	11
13	6	7	11	12	3	1	4	2	0	5	14	10	8	9
14	10	8	9	13	6	7	11	12	3	1	4	2	0	5
4	2	0	5	14	10	8	9	13	6	7	11	12	3	1
11	12	3	1	4	2	0	5	14	10	8	9	13	6	7
9	13	6	7	11	12	3	1	4	2	0	5	14	10	8
5	14	10	8	9	13	6	7	11	12	3	1	4	2	0
1	4	2	0	5	14	10	8	9	13	6	7	11	12	3
7	11	12	3	1	4	2	0	5	14	10	8	9	13	6
8	9	13	6	7	11	12	3	1	4	2	0	5	14	10
0	5	14	10	8	9	13	6	7	11	12	3	1	4	2

= 15x15 panmagic square

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

Use the shift method to construct magic squares of odd order from 5x5 to infinity.

See 5x5, 7x7, 9x9 (1), 9x9 (2), 11x11, 13x13, 15x15 (1), 15x15 (2), 17x17, 19x19, 21x21 (1), 21x21 (2), 23x23, 25x25, 27x27 (1), 27x27 (2), 29x29 and 31x31

15x15, shift method (2).xls

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15x15 magic square