Reversible magic square

Paul Michelet constructed a 9x9 reversible panmagic (and 3x3 compact & symmetric) square.

In the square are the numbers 11/19, 21/29, 31/39, 41/49, 51/59, 61/69, 71/79, 81/89 and 91/99.

If you reverse the digits of all numbers in the 9x9 panmagic square (i.e. 19 reversed is 91) you get another 9x9 panmagic (and 3x3 compact & symmetric) square.

9x9 reversible panmagic (and 3x3 compact & symmetric) square

88	81	86	18	11	16	68	61	66
38	31	36	58	51	56	78	71	76
48	41	46	98	91	96	28	21	26
83	85	87	13	15	17	63	65	67
33	35	37	53	55	57	73	75	77
43	45	47	93	95	97	23	25	27
84	89	82	14	19	12	64	69	62
34	39	32	54	59	52	74	79	72
44	49	42	94	99	92	24	29	22

Reversed 9x9 panmagic (and 3x3 compact & symmetric) square

88	18	68	81	11	61	86	16	66
83	13	63	85	15	65	87	17	67
84	14	64	89	19	69	82	12	62
38	58	78	31	51	71	36	56	76
33	53	73	35	55	75	37	57	77
34	54	74	39	59	79	32	52	72
48	98	28	41	91	21	46	96	26
43	93	23	45	95	25	47	97	27
44	94	24	49	99	29	42	92	22

Reversible 9x9 magic square.xls

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It is possible to construct a reversible 9x9 magic square, which is not only panmagic, 3x3 compact and symmetric, but also 1/3 of each row and 1/3 of each column gives 1/3 of the magic sum.

Take 1x number from first grid

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

+ 10x number from second grid

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

= Ultra magic 9x9 reversible square

11	85	69	16	87	62	18	83	64
58	33	74	51	35	79	56	37	72
96	47	22	98	43	24	91	45	29
61	15	89	66	17	82	68	13	84
78	53	34	71	55	39	76	57	32
26	97	42	28	93	44	21	95	49
81	65	19	86	67	12	88	63	14
38	73	54	31	75	59	36	77	52
46	27	92	48	23	94	41	25	99

If you reverse the digits of each number, you get:

Reversed ultra magic 9x9 square

11	58	96	61	78	26	81	38	46
85	33	47	15	53	97	65	73	27
69	74	22	89	34	42	19	54	92
16	51	98	66	71	28	86	31	48
87	35	43	17	55	93	67	75	23
62	79	24	82	39	44	12	59	94
18	56	91	68	76	21	88	36	41
83	37	45	13	57	95	63	77	25
64	72	29	84	32	49	14	52	99

Ultra magic reversible 9x9 square.xls

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