Symmetric transformation (Liki)

Paulus Gerdes introduced the Liki magic square (see http://plus.maths.org/content/new-designs-africa). He showed that it is possible to transform a square with consecutive numbers into a magic square by swapping half of the numbers symmetrically. You can use this method to construct magic squares which are a multiple of 4 (= 4x4, 8x8, 12x12, 16x16, ... magic square).

Paulus Gerdes constructed the following symmetric 8x8 magic square:

8x8 square with consecutive numbers

		232	240	248	256	264	272	280	288
	260									260
36		1	2	3	4	5	6	7	8
100		9	10	11	12	13	14	15	16
164		17	18	19	20	21	22	23	24
228		25	26	27	28	29	30	31	32
292		33	34	35	36	37	38	39	40
356		41	42	43	44	45	46	47	48
420		49	50	51	52	53	54	55	56
484		57	58	59	60	61	62	63	64

Symmetric 8x8 magic square

		260	260	260	260	260	260	260	260
	260									260
260		1	63	3	61	60	6	58	8
260		56	55	11	12	13	14	50	49
260		17	18	46	45	44	43	23	24
260		40	26	38	28	29	35	31	33
260		32	34	30	36	37	27	39	25
260		41	42	22	21	20	19	47	48
260		16	15	51	52	53	54	10	9
260		57	7	59	5	4	62	2	64

It is a beautiful result, but it is possible to get a more magic result.

			52	56	60	64	68	72	76	80
			180	184	188	192	196	200	204	208
		260									260
10	26		1	2	3	4	5	6	7	8
42	58		9	10	11	12	13	14	15	16
74	90		17	18	19	20	21	22	23	24
106	122		25	26	27	28	29	30	31	32
138	154		33	34	35	36	37	38	39	40
170	186		41	42	43	44	45	46	47	48
202	218		49	50	51	52	53	54	55	56
234	250		57	58	59	60	61	62	63	64

			130	130	130	130	130	130	130	130
			130	130	130	130	130	130	130	130
		260									260
130	130		1	63	62	4	5	59	58	8
130	130		56	10	11	53	52	14	15	49
130	130		48	18	19	45	44	22	23	41
130	130		25	39	38	28	29	35	34	32
130	130		33	31	30	36	37	27	26	40
130	130		24	42	43	21	20	46	47	17
130	130		16	50	51	13	12	54	55	9
130	130		57	7	6	60	61	3	2	64

Each 1/2 row/column gives 1/2 of the magic sum.

By refining the grids you can construct an ultramagic 8x8 square.

1x number from first grid + 1x [number -/- 1] from second grid =

1	2	2	1	2	1	1	2	1	1	2	2	3	3	4	4
3	4	4	3	4	3	3	4	1	1	2	2	3	3	4	4
3	4	4	3	4	3	3	4	5	5	6	6	7	7	8	8
1	2	2	1	2	1	1	2	5	5	6	6	7	7	8	8
3	4	4	3	4	3	3	4	9	9	10	10	11	11	12	12
1	2	2	1	2	1	1	2	9	9	10	10	11	11	12	12
1	2	2	1	2	1	1	2	13	13	14	14	15	15	16	16
3	4	4	3	4	3	3	4	13	13	14	14	15	15	16	16

symmetric transformation = Ultra magic 8x8 square

1	2	6	5	10	9	13	14	1	63	6	60	10	56	13	51
3	4	8	7	12	11	15	16	62	4	57	7	53	11	50	16
19	20	24	23	28	27	31	32	19	45	24	42	28	38	31	33
17	18	22	21	26	25	29	30	48	18	43	21	39	25	36	30
35	36	40	39	44	43	47	48	35	29	40	26	44	22	47	17
33	34	38	37	42	41	45	46	32	34	27	37	23	41	20	46
49	50	54	53	58	57	61	62	49	15	54	12	58	8	61	3
51	52	56	55	60	59	63	64	14	52	9	55	5	59	2	64

This 8x8 magic square is panmagic, 2x2 compact, symmetric and each 1/2 row/column gives 1/2 of the magic sum.

Use this method to construct magic squares of order is multiple of 4 from 4x4 to infinity. See 4x4, 8x8, 12x12, 16x16, 20x20, 24x24, 28x28, 32x32

8x8, Symmetric transformation (Liki).xls

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8x8 magic square