Chess - Programming - de Bruijn Sequence - De Bruijn Sequence on a Chess Board

A directed De Bruijn Graph of B(2, 6) sequences with Little-Endian Rank-File Mapping board coordinates (a1 = 0, b1 = 1, h8 = 63).

• For topology reasons, almost each node (except a1 and h8) of the graph is de-concentrated and appears twice in the form of two reversed binary trees.
• The leaf outputs join the respective reversed tree.
• Between c6 and f3 is a direct cycle, since 42 is 2*21 and 21 is (2*42 + 1) % 64, with both six-bit pattern reversed - 010101 (21) versus 101010 (42).
• The challenge is to traverse the graph in any way to visit each of the 64 nodes aka squares exactly once.

+----------------->---------------a1--------------<-----------------+
|                                 |                                 |
|                                 b1                                |
|                 +--------------/  \-------------+                 |
|                c1                               d1                |
^         +-----/  \-----+                 +-----/  \-----+         |
|        e1              f1               g1              h1        |
|     +-/  \-+        +-/  \-+         +-/  \-+        +-/  \-+     |
|    a2      b2      c2      d2       e2      f2      g2      h2    |
|  a3  b3  c3  d3  e3 |f3| g3  h3   a4  b4  c4  d4  e4  f4  g4  h4  |
+-a5b5c5d5e5f5g5h5a6b6c6d6e6f6g6h6 a7b7c7d7e7f7g7h7a8b8c8d8e8f8  |  |
     ^ ^ ^ ^ ^ ^ ^ ^ ^   ^ ^ ^ ^ ^  ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^  |  |
                                                                 |  |
+----------------->---------------h8--------------<--------------+  ^
|                                  |                                |
|                                 g8                                |
|                 +--------------/  \-------------+                 |
|                f8                               e8                |
^         +-----/  \-----+                 +-----/  \-----+         |
|        d8              c8               b8              a8        |
|     +-/  \-+        +-/  \-+         +-/  \-+        +-/  \-+     |
|    h7      g7      f7      e7       d7      c7      b7      a7    |
|  h6  g6  f6  e6  d6 |c6| b6  a6   h5  g5  f5  e5  d5  c5  b5  a5--+
+-h4g4f4e4d4c4b4a4h3g3f3e3d3c3b3a3 h2g2f2e2d2c2b2a2h1g1f1e1d1c1
     ^ ^ ^ ^ ^ ^ ^ ^ ^   ^ ^ ^ ^ ^  ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^

De Bruijn sequences can be described by two parameters:

• k: The number of entities in the alphabet e.g. {0,1,2,3,4,5,6,7,8,9} for k=10
• n: the order (length of sub-sequence required) e.g. n=4 for a four digit long PIN.

B(2, 6) implies 2^6 or 64 bit sequences.

• There are 2^26 or 67,108,864 odd 64-bit sequences with 64 overlapping unique six-bit sub-sequences, for instance 0x022fdd63cc95386d.

 i  0000001000101111110111010110001111001100100101010011100001101101|00000 s[i]
 0  000000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  0
 1   000001                                                                 1
 2  . 000010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2
 3     000100                                                               4
 4  . . 001000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8
 5       010001                                                            17
 6  . . . 100010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
 7         000101                                                           5
 8  . . . . 001011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
 9           010111                                                        23
10  . . . . . 101111 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
11             011111                                                      31
12  . . . . . . 111111 . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13               111110                                                    62
14  . . . . . . . 111101 . . . . . . . . . . . . . . . . . . . . . . . . . 61
15                 111011                                                  59
16  . . . . . . . . 110111 . . . . . . . . . . . . . . . . . . . . . . . . 55
17                   101110                                                46
18  . . . . . . . . . 011101 . . . . . . . . . . . . . . . . . . . . . . . 29
19                     111010                                              58
20  . . . . . . . . . . 110101 . . . . . . . . . . . . . . . . . . . . . . 53
21                       101011                                            43
22  . . . . . . . . . . . 010110 . . . . . . . . . . . . . . . . . . . . . 22
23                         101100                                          44
24  . . . . . . . . . . . . 011000 . . . . . . . . . . . . . . . . . . . . 24
25                           110001                                        49
26  . . . . . . . . . . . . . 100011 . . . . . . . . . . . . . . . . . . . 35
27                             000111                                       7
28  . . . . . . . . . . . . . . 001111 . . . . . . . . . . . . . . . . . . 15
29                               011110                                    30
30  . . . . . . . . . . . . . . . 111100 . . . . . . . . . . . . . . . . . 60
31                                 111001                                  57
32  . . . . . . . . . . . . . . . . 110011 . . . . . . . . . . . . . . . . 51
33                                   100110                                38
34  . . . . . . . . . . . . . . . . . 001100 . . . . . . . . . . . . . . . 12
35                                     011001                              25
36  . . . . . . . . . . . . . . . . . . 110010 . . . . . . . . . . . . . . 50
37                                       100100                            36
38  . . . . . . . . . . . . . . . . . . . 001001 . . . . . . . . . . . . .  9
39                                         010010                          18
40  . . . . . . . . . . . . . . . . . . . . 100101 . . . . . . . . . . . . 37
41                                           001010                        10
42  . . . . . . . . . . . . . . . . . . . . . 010101 . . . . . . . . . . . 21
43                                             101010                      42
44  . . . . . . . . . . . . . . . . . . . . . . 010100 . . . . . . . . . . 20
45                                               101001                    41
46  . . . . . . . . . . . . . . . . . . . . . . . 010011 . . . . . . . . . 19
47                                                 100111                  39
48  . . . . . . . . . . . . . . . . . . . . . . . . 001110 . . . . . . . . 14
49                                                   011100                28
50  . . . . . . . . . . . . . . . . . . . . . . . . . 111000 . . . . . . . 56
51                                                     110000              48
52  . . . . . . . . . . . . . . . . . . . . . . . . . . 100001 . . . . . . 33
53                                                       000011             3
54  . . . . . . . . . . . . . . . . . . . . . . . . . . . 000110 . . . . .  6
55                                                         001101          13
56  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 011011 . . . . 27
57                                                           110110        54
58  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101101 . . . 45
59                                                             01101|0     26
60  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101|00  . 52
61                                                               101|000   40
62  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01|0000  16
63                                                                 1|00000 32