Explanation most perfect (Franklin pan)magic 8x8 square

The most perfect (Franklin pan)magic 8x8 square consist of 4 proportional 4x4 panmagic squares.

4x4 panmagic square 4x4 sub-square of 8x8

1	8	13	12	1	54	12	63
15	10	3	6	16	59	5	50
4	5	16	9	53	2	64	11
14	11	2	7	60	15	49	6

In both squares the sum of two numbers of a colour always totals to the lowest plus the highest number of the magic square (1+16=17 respectively 1+64=65). With each time two colours you can get all eight (pan)diagonals (see 4x4 magic square, explanation).

Look below at the patterns of the panmagic 4x4 square and the most perfect 8x8 square.

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

9 + 25 = 34 16 + 18 = 34

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

55 + 75 + 59 + 71 = 130 + 130 = 260 17 + 113 + 49 + 91 = 130 + 130 = 260

Because of the structure the sum of the numbers of each 1/2 row/column/(pan)diagonal and of each 2x2 sub-square is allways (half of the magic sum: 1/2 x 260 =) 130.

There are the 3 following swap possibilities:

[1th] You can swap row 1&3 and/or row 2&4 and/or row 5&7 and/or row 6&8 and/or column 1&3 and/or column 2&4 and/or column 5&7 and/or column 6&8.

[2nd] You can swap the upper half with the down half and/or the right half with the left half.

[3rd] You can swap row 1&2 and row 3&4 and row 5&6 and row 7&8 and/or column 1&2 and column 3&4 and column 5&6 and column 7&8.

If you combine the 3 swap possibilities you can get each number out of 1 up to 64 in the top left corner. Try it!!!

From Willem Barink we learn that a small part (1/64) of the most perfect (8x8) magic squares has an extra magic feature. See the following most perfect magic 8x8 square:

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

33 + 97 = 130 61 + 69 = 130

The extra feature is that in each row and each column (not only ) adding the numbers from position (1 up to 4 and 5 up to 8, but also from) 3 up to 6 gives the magic sum of 130.

Most perfect on this website is different from most perfect on other websites. In my opinion the 8x8 Franklin panmagic square is most perfect. But according to Kathleen Ollerenshaw, the famous mathematician, the 8x8 complete magic square is most perfect. See below how you can transform a 8x8 Franklin panmagic square into a 8x8 complete magic square (by swapping rows and columns systematically).

	*Franklin panmagic 8x8 (= according to me most perfect) square*

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


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


*Complete (= K. Ollerenshaw's most perfect) magic 8x8 square*

			98	162	98	162	98	162	98	162
			162	98	162	98	162	98	162	98
		128									128
66	194		1	32	5	28	38	59	34	63		260	260
194	66		46	51	42	55	9	24	13	20		260	260
66	194		11	22	15	18	48	49	44	53		260	260
194	66		40	57	36	61	3	30	7	26		260	260
66	194		27	6	31	2	64	33	60	37		260	260
194	66		56	41	52	45	19	14	23	10		260	260
66	194		17	16	21	12	54	43	50	47		260	260
194	66		62	35	58	39	25	8	29	4
		132									132
			130	130	130	130	130	130	130
			130	130	130	130	130	130	130
			130	130	130	130	130	130	130
			130	130	130	130	130	130	130
			130	130	130	130	130	130	130
			130	130	130	130	130	130	130
			130	130	130	130	130	130	130

Addition of the numbers in half rows/columns/diagonals of the complete magic square don't give half of the magic sum. That is why I think the Franklin panmagic 8x8 square is most perfect and the complete magic square is not.

You can transform all most perfect magic squares which are a multiple of 4 from 8x8 to infinite (= 8x8, 12x12, 16x16, 20x20, ...) into complete magic squares. In all downloads of most perfect squares on this website you find the transformation from most perfect into complete.

most perfect 8x8 magic square.xls

Microsoft Excel werkblad 507.5 KB

Download

8x8 magic square