Composite, AxB compact

Use (as first grid) the 5x5 carpet of the 3x3 magic square and (as second grid) the 3x3 carpet of the 5x5 magic square to construct a 15x15 magic square. Take [number -/- 1] x 25 from the first grid and add 1x number from the same cell of the second grid.

25 x [number -/- 1] from 5x5 carpet of a 3x3 magic square

2	9	4	2	9	4	2	9	4	2	9	4	2	9	4
7	5	3	7	5	3	7	5	3	7	5	3	7	5	3
6	1	8	6	1	8	6	1	8	6	1	8	6	1	8
2	9	4	2	9	4	2	9	4	2	9	4	2	9	4
7	5	3	7	5	3	7	5	3	7	5	3	7	5	3
6	1	8	6	1	8	6	1	8	6	1	8	6	1	8
2	9	4	2	9	4	2	9	4	2	9	4	2	9	4
7	5	3	7	5	3	7	5	3	7	5	3	7	5	3
6	1	8	6	1	8	6	1	8	6	1	8	6	1	8
2	9	4	2	9	4	2	9	4	2	9	4	2	9	4
7	5	3	7	5	3	7	5	3	7	5	3	7	5	3
6	1	8	6	1	8	6	1	8	6	1	8	6	1	8
2	9	4	2	9	4	2	9	4	2	9	4	2	9	4
7	5	3	7	5	3	7	5	3	7	5	3	7	5	3
6	1	8	6	1	8	6	1	8	6	1	8	6	1	8

+ 1x number from 3x3 carpet of a 5x5 (pan)magic square

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

= 15x15 magic square

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

Each random chosen 3x5 or 5x3 rectangle gives the magic sum of 1695.

I have used the method composite, AxB compact to construct

12x12, 15x15, 20x20, 21x21, 24x24, 28x28, 30x30a and 30x30b

15x15, Composite, AxB compact.xls

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15x15 magic square