Patterns in Chess Square Enumerations

In my bit manipulation post awhile back I posed a problem involving "closed form" calculation of chess endgame table indices. Back when I originally solved the problem, it was a big help to create some visualizations of some of the patterns embedded in the square numbering scheme. I had mentioned that I thought the patterns were interesting, so I thought I'd come back and elaborate on that statement.

One way of visualizing some of the patterns involved is to partition the squares according to the equivalence classes defined by the binary bits of the index. Bits 0-2 are the three low-order bits and bits 3-5 are the three high-order bits. Here are the equivalence classes defined by each of the bits of the natural square numbering.

Bit 0

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Bit 3

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Bit 1

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Bit 4

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Bit 2

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Bit 5

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Here we can clearly see the natural row and column oriented behavior of the index. Nothing earth shattering. If we take octal digits instead of binary digits to define our equivalence class, we get the following more colorful partition.

Low Digit

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

High Digit

70	71	72	73	74	75	76	77
60	61	62	63	64	65	66	67
50	51	52	53	54	55	56	57
40	41	42	43	44	45	46	47
30	31	32	33	34	35	36	37
20	21	22	23	24	25	26	27
10	11	12	13	14	15	16	17
00	01	02	03	04	05	06	07

Now on to the interesting stuff. Here are the same six equivalence classes of the less obvious square numbering for which I posed my puzzle.

Bit 0

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

Bit 3

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

Bit 1

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

Bit 4

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

Bit 2

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

Bit 5

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

And for the octal digits...

Low Digit

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

High Digit

64	04	14	24	26	16	06	66
05	65	34	44	46	36	67	07
15	35	74	54	56	76	37	17
25	45	55	75	77	57	47	27
21	41	51	71	73	53	43	23
11	31	70	50	52	72	33	13
01	61	30	40	42	32	63	03
60	00	10	20	22	12	02	62

Ahhhhhh....I don't know about you, but I think that's a beautiful square numbering. There's also plenty of group theory going on. What do you see here?

The Haskell code used to generate these tables is here.