Hi, people!

I propose to classify 18-clue symmetric patterns by their

ordered distributions of clues over boxes or simply "distributions". For example, pattern

- Code: Select all
`+-----+-----+-----+`

|x . .|. . .|. . .|

|x . .|. . .|. . .|

|x x x|. . .|. . .|

+-----+-----+-----+

|x x x|x x x|x x x|

|x x x|x x x|x x x|

|x x x|x x x|x x x|

+-----+-----+-----+

|x x x|x x x|x x x|

|x x x|x x x|x x x|

|x x x|x x x|x x x|

+-----+-----+-----+

has distribution 005999999 (each digit represents number of clues in some box; list must be sorted).

All distributions for 188268 possible valid patterns from the last version of

eleven's list (see link for downloading in my previous post) are published below.

- Code: Select all
`18-clue possible symmetric distributions (eleven's list)`

N Distribution Patterns

--------------------------

1 000022338 21

2 000022446 96

3 000022455 32

4 000033336 110

5 000033444 93

6 000122229 20

7 000122337 102

8 000122445 156

9 000133335 308

10 000222228 44

11 000222237 140

12 000222246 335

13 000222255 174

14 000222336 312

15 000222444 156

16 000223335 334

17 000223344 199

18 000233334 313

19 001111338 24

20 001111446 117

21 001111455 39

22 001112229 23

23 001112238 43

24 001112247 162

25 001112256 297

26 001112337 242

27 001112445 390

28 001113336 603

29 001113345 798

30 001113444 319

31 001122228 103

32 001122237 340

33 001122246 894

34 001122255 464

35 001122336 1689

36 001122345 1230

37 001122444 911

38 001123335 976

39 001123344 637

40 001133334 797

41 001222227 254

42 001222236 609

43 001222245 756

44 001222335 1246

45 001223334 1000

46 002222226 696

47 002222235 888

48 002222244 551

49 002222334 1610

50 011111229 9

51 011111337 116

52 011111355 129

53 011111445 301

54 011112228 54

55 011112237 180

56 011112246 378

57 011112255 349

58 011112336 522

59 011112444 903

60 011113335 1015

61 011113344 1286

62 011122227 356

63 011122335 3539

64 011122344 2573

65 011133333 2496

66 011222226 1350

67 011222235 2587

68 011222244 3512

69 011222334 8629 observed: 1

70 011223333 6865

71 012222225 1454

72 012222333 6584

73 022222224 3158 observed: 2

74 022222233 5937 observed: 1

75 111111228 44

76 111111336 414

77 111111444 756

78 111112227 592

79 111112236 1550

80 111112245 2485

81 111112335 3215

82 111112344 3112

83 111113334 5335

84 111122226 2378

85 111122235 4990

86 111122244 4782

87 111122334 15285 observed: 4

88 111123333 5663 observed: 1

89 111222225 6077

90 111222234 9976 observed: 2

91 111222333 12008 observed: 32

92 112222224 9653 observed: 3

93 112222233 15826 observed: 34

94 122222223 6632 observed: 24

95 222222222 1560 observed: 12

Total number of different distributions : 95

Total number of patterns in eleven's list: 188268

Total number of observed patterns : 116

So, this is my idea of classification - I propose to divide overall exhaustive search to separate small procedures of searching through alone distributions. Therefore overall search can be parallelised and we can see intermediate results of searching (for example, 352 patterns only should be searched through to prove that there are no valid symmetric 18-clue puzzles having 4 empty boxes).

It is worth noting that there exist 251 distributions for 18-clue puzzles (valid or invalid, symmetric or asymmetric).

Serg

[Edited: I updated number of observed patterns - 108 sym. essentially different patterns are known now (02.06.2012).]

[Edited: I updated number of observed patterns - 109 sym. essentially different patterns are known now (03.06.2012). Number of observed patterns for distribution 111222333 was updated.]

[Edited: I updated number of observed patterns - 110 sym. essentially different patterns are known now (04.06.2012). Number of observed patterns for distribution 122222223 was updated.]

[Edited: I updated number of observed patterns - 112 sym. essentially different patterns are known now (24.06.2012). Number of observed patterns for distributions 122222223 and 222222222 were updated (#111 has 222222222 distr., #112 has 122222223 distr.).]

[Updated number of observed patterns - 113 sym. essentially different patterns are known now (28.06.2012). Pattern #113 has new distribution: 011222334.]

[Updated number of observed patterns - 114 sym. essentially different patterns are known now (9.07.2012). Pattern #114 has distribution: 112222233.]

[Updated number of observed patterns - 115 sym. essentially different patterns are known now (23.07.2012). Pattern #115 has distribution: 112222233.]

[Updated number of observed patterns - 116 sym. essentially different patterns are known now (10.08.2012). Pattern #116 has new distribution: 111123333.]