Statistics

Problem Statement for "UnfinishedTournamentEasy"

Problem Statement

A tournament is being played. There are n participants, numbered 0 through n-1. The tournament is a single round-robin tournament. In other words, each player is supposed to play each other player exactly once. There are no ties, each match will be won by one of the two players.

Some games may have already been played. You are given the information about the current state of the tournament in the String[] G. For each x and y, if x won the game against y, we have G[x][y] = 'W' and G[y][x] = 'L'. For the games that haven't been played yet we have G[x][y] = G[y][x] = '?'.

The winning rate of a player is the percentage of games they won. In particular, at the end of the tournament the winning rate of a player can be computed as the number of games they won divided by n-1.

The variance of a sequence is the average square distance of an element of the sequence from the mean of the sequence. Formally, suppose you have a sequence s[0], ..., s[n-1]. Then:
  • The mean of this sequence is mu(s) = ( s[0] + ... + s[n-1] ) / n.
  • The variance of this sequence is var(s) = ( (s[0]-mu(s))^2 + ... + (s[n-1]-mu(s))^2 ) / n.

Let r[0], ..., r[n-1] be the sequence of the winning rates of our n players at the end of the tournament. Fill in the results of the remaining matches in a way that maximizes the variance of r. Return the maximal possible value of var(r).

Definition

Class:
UnfinishedTournamentEasy
Method:
maximal
Parameters:
String[]
Returns:
double
Method signature:
double maximal(String[] G)
(be sure your method is public)

Notes

  • Your answer will be considered correct if its absolute or relative error does not exceed 10^(-9).

Constraints

  • n will be between 2 and 20, inclusive.
  • G will contain exactly n elements.
  • Each element in G will contain exactly n characters.
  • Each character in G will be one of {'W', 'L', '?', '-'}.
  • For each i, G[i][i] = '-'.
  • For each i != j: {G[i][j], G[j][i]} = {'W', 'L'} or G[i][j] = G[j][i] = '?'.

Examples

  1. {"-??", "?-?", "??-"}

    Returns: 0.16666666666666669

    One of the optimal solutions is the following tournament: -WW L-W LL- For this tournament the winning rates of the three players are {1, 0.5, 0}. Thus, the mean is 0.5 and the variance is ((1-0.5)^2 + (0.5-0.5)^2 + (0-0.5)^2) / 3 = 0.5 / 3 = 0.16667.

  2. {"-WL", "L-W", "WL-"}

    Returns: 0.0

    This time all matches are finished. The winning rates are {0.5, 0.5, 0.5} thus the variance is 0.

  3. {"-WWL", "L-??", "L?-?", "W??-"}

    Returns: 0.1388888888888889

  4. {"-WWL", "L-WW", "LL-?", "WL?-"}

    Returns: 0.08333333333333331

  5. {"-?", "?-"}

    Returns: 0.25

  6. {"-??W?WWWLL???LL?LW?L","?-WL?W??W?W???LL????","?L-?L?????W?WWWL??W?","LW?-???????L?????LWW","??W?-?L?LWL???LWL?W?","LL???-?L???L????WL??","L???W?-W???????WWLWW","L????WL-?LLWLW?LL???","WL??W???-?WW?????WLL","W???L??W?-W??W??LL??","?LL?W??WLL-L?L?WLWW?","???W?W?LL?W-?W??????","??L????W????-??W??W?","W?L????L?LWL?-W?W??L","WWL?W????????L-????L","?WW?L?LW??L?L??-??WW","W???WLLW?WW??L??-?LW","L??W?WW?LWL??????-LL","??LLL?L?W?L?L??LWW-W","W??L??L?W????WWLLWL-"}

    Returns: 0.058310249307479245

  7. {"-L?????W?WL???W?L???","W-?????????W?W??????","??-W??W????L????WL?W","??L-L?LWL??????W???L","???W-??????????LW?L?","?????-?????W???LL???","??LW??-?????LW???L??","L??L???-?L???LLW????","???W????-W?W????????","L??????WL-???????WL?","W?????????-???LWLL??","?LW??L??L??-?????WL?","??????W?????-???????","?L????LW?????-?L????","L??????W??W???-L???W","???LWW?L??L??WW-???L","W?L?LW????W?????-W??","??W???W??LWL????L-??","????W????W?W??????-?","??LW??????????LW???-"}

    Returns: 0.08296398891966761

  8. {"-?W???L???LL????WW?","?-???W???W?????????","L?-?WWL?L????????L?","???-W?????L????????","??LL-L????????L????","?LL?W-?L??WW????WLW","W?W???-???W??L???W?","?????W?-????????WLW","??W?????-W???L??LWL","?L??????L-WWW??W???","W??W?LL??L-?LLW??L?","W????L???L?-??LL???","?????????LW?-?W???L","??????W?W?W??-??LW?","????W?????LWL?-LL?W","?????????L?W??W-??W","L????L?LW????WW?-?W","L?W??WLWL?W??L???-?","?????L?LW???W?LLL?-"}

    Returns: 0.08057179987004548

  9. {"-LWLLLLWWWW?WL?L?W?","W-LWLLL??L?LL?L??W?","LW-WLWL?WLLW?W?LLLW","WLL-???LLWLLWWL?LL?","WWW?-LW?LL?LLWLWLLW","WWL?W-L?W?WLWLLL?L?","WWW?LW-??WLLL?L?LLL","L??W???-LWLWWL?W?LL","L?LWWL?W-WWW?W?WL??","LWWLW?LLL-?WWLLL?WW","L?WW?LWWL?-LLWWWW??","?WLWWWWLLLW-?W?WW?L","LW?LWLWL?LW?-?WW?WL","W?LLLW?WLWLL?-WLWLW","?W?WWWW??WL?LL-?LW?","W?W?LW?LLWLLLW?-?LL","??WWW?W?W?LL?LW?-?W","LLWWWWWW?L??LWLW?-?","??L?L?WW?L?WWL?WL?-"}

    Returns: 0.031189083820662766

  10. {"-??","?-?","??-"}

    Returns: 0.16666666666666669

  11. {"-WW?WWWWWWWWWW?WW","L-WWW?WWWWW?WWWWW","LL-?W?WWWWWWWW?WW","?L?-LWWWWWWWWWWW?","LLLW-W?WW?WW?WLW?","L??LL-WWWWW?WWWWW","LLLL?L-W?WWWW??WW","LLLLLLL-WWWWWWWW?","LLLLLL?L-LW?WW?WL","LLLL?LLLW-?WLW?WW","LLLLLLLLL?-WWWWWL","L?LLL?LL?LL-WWWWW","LLLL?LLLLWLL-WWWW","LLLLLL?LLLLLL-WLW","?L?LWL?L??LLLL-WL","LLLLLLLLLLLLLWL-W","LLL??LL?WLWLLLWL-"}

    Returns: 0.0827205882352941

  12. {"-WWL","L-L?","LW-W","W?L-"}

    Returns: 0.08333333333333331

  13. {"-L?L?L?L?L??LL???","W-LL?WLL?LLL?LLLL","?W-L???LL?L?LL??L","WWW-?LLL????L???L","????-L?LL??LLLL?L","WL?WW-?L?LL?WWL?L","?W?W??-L?LLLL?LL?","WWWWWWW-L?LLLLL??","??W?W??W-LLL??L?L","WW???WW?W-L??LLL?","?WW??WWWWW-LLL?LL","?W??W?WWW?W-LL?L?","W?WWWLWW??WW-LW??","WWW?WL?W?WWWW-LLL","?W??WWWWWW??LW-LL","?W????W??WWW?WW-?","?WWWWW??W?W??WW?-"}

    Returns: 0.0859375

  14. {"-WL","L-L","WW-"}

    Returns: 0.16666666666666669

  15. {"-WW?WWWWW","L-W?WWWWW","LL-WWWWWW","??L-W?W?W","LLLL-WWWW","LLL?L-WW?","LLLLLL-WW","LLL?LLL-W","LLLLL?LL-"}

    Returns: 0.10416666666666669

  16. {"-?????????","?-????????","??-??????L","???-??????","????-L????","????W-????","??????-???","???????-??","????????-?","??W??????-"}

    Returns: 0.10185185185185186

  17. {"-LL","W-L","WW-"}

    Returns: 0.16666666666666669

  18. {"-L???LL?????W","W-????LLL?L??","??-????????W?","???-L??L?WWL?","???W-?LW??L?W","W????-?L??LW?","WW??W?-W?WL?W","?W?WLWL-L?WWL","?W?????W-?L??","???L??L??-LLW","?W?LWWWLWW-LL","??LW?L?L?WW-W","L???L?LW?LWL-"}

    Returns: 0.07264957264957267

  19. {"-?","?-"}

    Returns: 0.25

  20. {"-LWWL?WWWWWWL","W-WWWWWWWWLLW","LL-WWLWWWWWWL","LLL-WWWWWWLLW","WLLL-W?WLWW?L","?LWLL-WWWWWWW","LLLL?L-WW?WWW","LLLLLLL-WWLL?","LLLLWLLL-WWLW","LLLLLL?LL-WWW","LWLWLLLWLL-WW","LWLW?LLWWLL-?","WLWLWLL?LLL?-"}

    Returns: 0.03739316239316237

  21. {"-WWWW","L-WWW","LL-WW","LLL-L","LLLW-"}

    Returns: 0.125

  22. {"-WLL?LLW","L-?L?W?L","W?-W?LLL","WWL-WWWL","???L-?WL","WLWL?-LW","W?WLLW-W","LWWWWLL-"}

    Returns: 0.040816326530612235

  23. {"-LLWLW","W-WWWW","WL-?W?","LL?-WW","WLLL-L","LL?LW-"}

    Returns: 0.09000000000000002

  24. {"-??????L?????L??L??","?-?????????W???????","??-L?????????L?????","??W-???????????????","????-???????L??????","?????-??W???????L??","??????-???W?????L??","W??????-??WLLL?WL??","?????L??-?L????????","?????????-???L?L???","??????LLW?-????L???","?L?????W???-??????L","????W??W????-??LL??","W?W????W?W???-?????","??????????????-?L??","???????L?WW?W??-???","W????WWW????W?W?-??","?????????????????-?","???????????W??????-"}

    Returns: 0.09226770630279402

  25. {"-?????????L?WL","?-?LLLL?L???LL","??-??WLL??LLLL","?W?-????L?L???","?W??-??????W??","?WL??-LL?LL???","?WW??W-?LL??L?","??W??W?-??????","?W?W??W?-L??L?","?????WW?W-W??L","W?WW?W???L-???","??W?L??????-L?","LWW???W?W??W-L","WWW??????W??W-"}

    Returns: 0.09615384615384615

  26. {"-??LWWL?WLW","?-?L??LWLLL","??-LLLWWL??","WWW-?WLLW??","L?W?-?WL??L","L?WL?-WW??L","WWLWLL-LWLL","?LLWWLW-W??","LWWL??LL-L?","WW????W?W-?","LW??WWW???-"}

    Returns: 0.06545454545454549

  27. {"-?LWWWL?WWWLWW","?-WWLL?WLLWWLL","WL-WWLWW?W??L?","LLL-LL?WLLW?W?","LWLW-LW?LLWLLW","LWWWW-LLLWLLWW","W?L?LW-LWLWLL?","?LLL?WW-LW?LWL","LW?WWWLW-LLW?W","LWLWWLWLW-WWLL","LL?LLWL?WL-WLW","WL??WWWWLLL-LL","LWWLWLWL?WWW-W","LW??LL?WLWLWL-"}

    Returns: 0.024302620456466584

  28. {"-WW??W????W??WW","L-??WWW?WWWWWW?","L?-W?WW?WLL?W?W","??L-W??W?WWL???","?L?L-?W?W????WW","LLL??-?W???WW??","?LL?L?-??WL??W?","???L?L?-?WW???W","?LL?L???-?WW?W?","?LWL??LL?-????W","LLWL??WLL?-?WW?","?L?W?L??L??-???","?LL??L????L?-WW","LL??L?L?L?L?L-?","L?L?L??L?L??L?-"}

    Returns: 0.08503401360544222

  29. {"-???????W?W","?-?WLLWW??W","??-??L???W?","?L?-L?WL?L?","?W?W-L????L","?WW?W-?????","?L?L??-?W?W","?L?W???-W?W","L?????LL-??","??LW?????-?","LL??W?LL??-"}

    Returns: 0.09272727272727277

  30. {"-??L","?-L?","?W-W","W?L-"}

    Returns: 0.1388888888888889

  31. {"-?LWW?WWWWL??","?-L?WWWWW?WW?","WW-WWWW?WWL?W","L?L-?WWW?LWLW","LLL?-L??WL?L?","?LLLW-?WW?WWW","LLLL??-?WW?WW","LL?L?L?-???W?","LLL?LLL?-WLWL","L?LWW?L?L-L?W","WLWL?L??WW-?W","?L?WWLLLL??-L","??LL?LL?WLLW-"}

    Returns: 0.07692307692307693

  32. {"-???L?????LW??","?-?LL????LW?WW","??-L????LL???W","?WW-?LL?WLW??L","WW??-?????L??L","???W?-LW?WLW??","???W?W-?LLL?WL","?????L?-LL?L??","??WL??WW-L?L??","?WWW?LWWW-L???","WL?LWWW??W-L??","L????L?WW?W-??","?L????L?????-?","?LLWW?W??????-"}

    Returns: 0.07079459002535932

  33. {"-LLLWLLWLWLLL?WL","W-LW?LLLLWWLLLLL","WW-LL?LWLWL??LWW","WLW-WLL??LL?LLWL","L?WL-WLW?LLWWWLW","WW?WL-LWLWWLL?WW","WWWWWW-LLL?WLWLW","LWL?LLW-WWWLLWWL","WWW??WWL-LLLLL?L","LLLWWLWLW-WL?LWL","WLWWWL?LWL-WLLLL","WW??LWLWWWL-LLWL","WW?WLWWWW?WW-WW?","?WWWL?LLWWWWL-WL","LWLLWLWL?LWLLL-?","WWLWLLLWWWWW?W?-"}

    Returns: 0.034444444444444444

  34. {"-WLW???W?WLL","L-LL??LWLLLL","WW-?LWWWLLLL","LW?-LLLL?WLL","??WW-WL?L??W","??LWL-LW?L??","?WLWWW-LWLLW","LLLW?LW-LL??","?WW?W?LW-LLL","LWWL?WWWW-LL","WWWW??W?WW-L","WWWWL?L?WWW-"}

    Returns: 0.06542699724517903

  35. {"-W?WL?LWW?LL?","L-W?L???LLL?W","?L-L??W???LL?","L?W-????L?L?W","WW??-??????WW","?????-LW?LWL?","W?L??W-LLLWWW","L????LW-?W?WL","LW?W??W?-W?W?","?W???WWLL-?WW","WWWW?LL???-??","W?W?LWLLLL?-?","?L?LL?LW?L??-"}

    Returns: 0.07051282051282054

  36. {"-L?WL?WLL","W-?L?W?W?","??-LWLWLW","LWW-LLLWW","W?LW-??LW","?LWW?-L?L","L?LW?W-W?","WLWLW?L-L","W?LLLW?W-"}

    Returns: 0.03125

  37. {"-L","W-"}

    Returns: 0.25

  38. {"-WWW","L-WW","LL-W","LLL-"}

    Returns: 0.1388888888888889

  39. {"-LWWLLLWLLWLLL","W-WLWWLWLWWWWW","LL-LWLW?WLLWLW","LWW-LWWLLWLWWW","WLLW-WWLWWWWLW","WLWLL-LLLLWWWW","WWLLLW-WWLLWLW","LL?WWWL-WWWLWL","WWLWLWLL-LWLLL","WLWLLWWLW-LWLW","LLWWLLWLLW-LWW","WLLLLLLWWLW-W?","WLWLWLWLWWLL-W","WLLLLLLWWLL?L-"}

    Returns: 0.020076077768385436

  40. {"-????????W??L?????","?-????????????????","??-L??????W???????","??W-W?????????????","???L-?????????????","?????-W?L??WW?????","?????L-??W????????","???????-?????????L","?????W??-?W???????","L?????L??-??W?W?L?","??L?????L?-???????","?????L?????-??W???","W????L???L??-?????","?????????????-????","?????????L?L??-???","???????????????-??","?????????W??????-W","???????W????????L-"}

    Returns: 0.09275278738946557

  41. {"-?WW?W????L??W?WW?","?-WWW????WW?LW??WW","LL-WWWW?W??WW?W???","LLL-WWW?WWW?W?WWW?","?LLL-WWWLWW?WW?WWW","L?LLL-WWWW?WWLW?WW","??LLLL-WW?W??WWW??","????LLL-WW?WWWW?WW","??LLWLLL-WW?W?WWW?","?L?LLL?LL-WW?WW?WW","WL?LL?L?LL-W?WW?LW","??L??L?L?LL-WWW?WW","?WLLLL?LL??L-WWL?W","LL??LWLL?LLLL-W?WW","??LL?LLLLLLLLL-?LW","L??LL?L?L???W??-WW","LL?LLL?LLLWL?LWL-W","?L??LL?L?LLLLLLLL-"}

    Returns: 0.0792964244521338

  42. {"-WW?WW","L-WLWW","LL-?LL","?W?-?W","LLW?-W","LLWLL-"}

    Returns: 0.1166666666666667

  43. {"-???W????????????","?-???WW??????????","??-W?????????W???","??L-???????W?WW??","L???-????????????","?L???-???W????W??","?L????-???W??????","???????-???????WW","????????-?????WWW","?????L???-???????","??????L???-??W???","???L???????-?????","????????????-????","??LL??????L??-???","???L?L??L?????-W?","???????LL?????L-?","???????LL???????-"}

    Returns: 0.09375

  44. {"-W??LLL?L?L????L????","L-?L?WLLL????W?LL??L","??-LW?LL???LL??L??LL","?WW-LLL?L??LLW??????","W?LW-L?L??L????LW?L?","WL?WW-?L?LW?W??LLLL?","WWWW??-?L??W????WWL?","?WW?WW?-L?LL?L??L???","WW?W??WW-L??????LLL?","?????W??W-L?L?WL??LW","W???WL?W?W-WLW?LLWLL","??WW??LW??L-LLL????L","??WW?L???WWW-?????W?","?L?L???W??LW?-W?LLLL","?????????L?W?L-?WL?L","WWW?WW???WW????-LWL?","?W??LWLWW?W??WLW-LW?","?????WL?W?L??WWLW-??","??W?WWW?WWW?LW?WL?-?","?WW??????LWW?WW????-"}

    Returns: 0.07409972299168976

  45. {"-???","?-?L","??-?","?W?-"}

    Returns: 0.1388888888888889

  46. {"-??W?W?W","?-W??W??","?L-?WW?W","L??-W???","??LL-??W","LLL??-??","??????-L","L?L?L?W-"}

    Returns: 0.10714285714285715

  47. {"-????L?LL?LL?","?-????L??????","??-??L???????","???-?L??L????","????-L???????","W?WWW-???????","?W????-?L????","W??????-??L??","W??W??W?-????","?????????-?L?","W??????W??-??","W????????W?-?","????????????-"}

    Returns: 0.09722222222222221

  48. {"-L?L?L??W","W-?????WW","??-WL??WL","W?L-???WL","??W?-W?LW","W???L-???","??????-??","?LLLW??-?","LLWWL???-"}

    Returns: 0.0798611111111111

  49. {"-L","W-"}

    Returns: 0.25

  50. {"-???????W???W?","?-????W?WWWL?L","??-?????W???L?","???-L??L??WWWW","???W-L????????","????W-?W??W???","?L????-?W??WW?","???W?L?-???L??","LLL???L?-???L?","?L???????-W?L?","?L?L?L???L-??L","?W?L??LW???-WW","L?WL??L?WW?L-L","?W?L??????WLW-"}

    Returns: 0.09277261200338127

  51. {"-??????W?","?-???????","??-?L????","???-??L??","??W?-????","?????-???","???W??-?L","L??????-?","??????W?-"}

    Returns: 0.10416666666666669

  52. {"-???????LL?","?-?????????","??-LL?????L","??W-???LLLL","??W?-L?L??L","????W-??L??","??????-?L??","???WW??-??L","W??W?WW?-??","W??W?????-L","??WWW??W?W-"}

    Returns: 0.10000000000000003

  53. {"-W?W?W????WW?WW","L-WW??W????W??W","?L-WW???W?W??WW","LLL-?WWWWWW????","??L?-LLW??WW???","L??LW-??WL?W?WW","?L?LW?-??WW?WW?","???LL??-??W???W","??LL?L??-??????","???L?WL??-W???W","L?LLL?LL?L-?WW?","LL??LL?????-?WW","??????L???L?-??","L?L??LL???LL?-W","LLL??L?L?L?L?L-"}

    Returns: 0.09523809523809523

  54. {"-??L??W???WW???WW","?-???????WW??????","??-???????????W?W","W??-??????WW?????","????-W???W????W??","????L-???W???????","L?????-???L??W?W?","???????-?WW?W?W??","????????-????????","?L??LL?L?-??W????","LL?L??WL??-?W????","L??L???????-??W??","???????L?LL?-????","??????L??????-W??","??L?L??L???L?L-W?","L?????L???????L-?","L?L?????????????-"}

    Returns: 0.09375

  55. {"-LW?WL?L?","W-LL?LLLW","LW-?LLLL?","?W?-LLWLW","L?WW-WLL?","WWWWL-WLW","?WWLWL-WL","WWWWWWL-L","?L?L?LWW-"}

    Returns: 0.048611111111111105

  56. {"-WWLL","L-WLL","LL-L?","WWW-W","WW?L-"}

    Returns: 0.125

  57. {"-?","?-"}

    Returns: 0.25

  58. {"-???LL?????WL?W??L?","?-???L?L?LL??LWL?WL","??-LL?L????????L??W","??W-?W??L???L?????L","W?W?-LL???L?LLW?LL?","WW?LW-LWL???L????WW","??W?WW-W??L?L?LL?LW","?W???LL-??LLLLWWLW?","???W?W??-W?W???LWW?","?W??????L-?L??W???W","?W??W?WW??-?WL?LL??","L??????WLW?-??LW??L","W??WWWWW??L?-???W?L","?W??W??W??W??-?WL?L","LL??L?WL?L?W??-?L?L","?WW???WLW?WL?L?-??L","????W??WL?W?LWW?-W?","WL??WLWLL???????L-?","?WLW?LL??L?WWWWW??-"}

    Returns: 0.06985055230669268

  59. {"-LW","W-W","LL-"}

    Returns: 0.16666666666666669

  60. {"-?","?-"}

    Returns: 0.25

  61. {"-??LW???W??W","?-?WL?W??WWW","??-W?W??W?WW","WLL-????WWW?","LW??-WW?????","??L?L-??????","?L??L?-?W??W","???????-W??L","L?LL??LL-?L?","?L?L?????-W?","?LLL????WL-W","LLL???LW??L-"}

    Returns: 0.09573002754820936

  62. {"-WWWWWLLWLWWWWWL","L-WWWWLLLLLLWLLW","LL-WWWWLL?WLWWWL","LLL-WWLWWLLWWLLW","LLLL-WLW?WWWLWLW","LLLLL-LLWW?LLWLW","WWLWWW-WWWWWWWLW","WWWLLWL-WLL?WWL?","LWWL?LLL-WLWWLWW","WW?WLLLWL-?WLWWW","LWLWL?LWW?-W?WWW","LWWLLWL?LLL-LWWW","LLLLWWLLLW?W-LWW","LWLWLLLLWLLLW-WW","LWLWWWWWLLLLLL-L","WLWLLLL?LLLLLLW-"}

    Returns: 0.02888888888888891

  63. {"-WLLLWWLWLW?LL","L-LWL?WWWLLWLL","WW-LLW?WWLLLWL","WLW-WWLLWWLLLL","WWWL-WLLLLWLWW","L?LLL-LWLLL?LW","LL?WWW-LLLLLLL","WLLWWLW-?WLWWL","LLLLWWW?-WLWWW","WWWLWWWLL-WLWW","LWWWLWWWWL-LLW","?LWWW?WLLWW-LW","WWLWLWWLLLWW-L","WWWWLLWWLLLLW-"}

    Returns: 0.022612003381234147

  64. {"-L?LLLLLL?LLLL?LLLWL","W-LLLLLL?L????LL??W?","?W-LLLLLL??L??LLL???","WWW-?LL??????L??L???","WWW?-WLL?LL?WLLL???L","WWWWL-LL?WL???LLLLL?","WWWWWW-?L??LL????L??","WWW?WW?-LLLLLW?LL?LL","W?W???WW-LLLLLL?LLLL","?W??WL?WW-W?LL?????L","W???WW?WWL-L??LL??LL","W?W???WWW?W-??LLLLLL","W???L?WWWW??-LW?WLWL","W??WW??LWW??W-LL?LLL","?WW?WW??W?WWLW-L?LLL","WWW?WW?W??WW?WW-L?L?","W?WW?W?WW??WL??W-L??","W????WW?W??WWWW?W-LL","LL???W?WW?WWLWWW?W-L","W???W??WWWWWWWW??WW-"}

    Returns: 0.07714681440443216

  65. {"-?WL??WL??","?-????????","L?-L????L?","W?W-?????L","????-???W?","?????-?L??","L?????-?W?","W????W?-??","??W?L?L?-?","???W?????-"}

    Returns: 0.10185185185185186

  66. {"-?L?","?-??","W?-?","???-"}

    Returns: 0.1388888888888889

  67. {"-L???LL?","W-??LLLL","??-L??LL","??W-??L?","?W??-??W","WW???-?L","WWWW??-L","?WW?LWW-"}

    Returns: 0.10714285714285715

  68. {"-??????","?-W????","?L-?W??","???-?W?","??L?-??","???L?-?","??????-"}

    Returns: 0.1111111111111111

  69. {"-????W???W?","?-??????W??","??-??WW????","???-?WW?L??","????-??????","L?LL?-????W","??LL??-????","???????-??L","?L?W????-??","L????????-?","?????L?W??-"}

    Returns: 0.10000000000000003

  70. {"-?LLWLLL","?-?L??L?","W?-?LLLL","WW?-?LWL","L?W?-?L?","W?WW?-LL","WWWLWW-L","W?WW?WW-"}

    Returns: 0.09693877551020408

  71. {"-?LWL?L?WW?LWWLL?","?-WWWWW?WW?WL??WW","WL-LWL????LWW?W?W","LLW-LW?WWW??LWWWL","WLLW-LW?W?L?WWWLW","?LWLW-LLWWL????L?","WL??LW-?WWWW?LW?W","???L?W?-??WWWW?WL","LL?LLLL?-?LWW?LL?","LL?L?LL??-W?LL?W?","??W?WWLLWL-?WWW??","WLL???LLL??-WL?WL","LWLWL??LLWLL-WWW?","L??LL?WL?WLWL-WWW","W?LLL?L?W?L?LL-LL","WL?LWW?LWL?LLLW-?","?LLWL?LW???W?LW?-"}

    Returns: 0.05514705882352944

  72. {"-?L?????","?-?L??LL","W?-??L??","?W?-?LLL","????-?LL","??WW?-??","?W?WW?-?","?W?WW??-"}

    Returns: 0.10714285714285715

  73. {"-WLLLWLLLLLWWLLWWW","L-??LLLWL??W?LLLLL","W?-LW?WLLLLLWLLL?L","W?W-LLLLWLL?LLWLW?","WWLW-LLLLWLLL?LLLW","LW?WW-LLLLLLLLL?LL","WWLWWW-LLWLL?LLWL?","WLWWWWW-WLWL?WLWWL","WWWLWWWL-LL??LLLLW","W?WWLWLWW-LLLLL?LL","W?WWWWWLWW-LWLWLLL","LLW?WWWW?WW-WLLWLW","L?LWWW???WLL-LW?LW","WWWW?WWLWWWWW-?LLL","WWWLWWWWWWLWL?-WLL","LWWWW?LLW?WL?WL-LL","LW?LWWWLWWWWWWWW-?","LWW?LW?WLWWLLWWW?-"}

    Returns: 0.038927335640138394

  74. {"-?","?-"}

    Returns: 0.25

  75. {"-??","?-?","??-"}

    Returns: 0.16666666666666669

  76. {"-LLLLW???L","W-L?W?????","WW-L???W?W","W?W-?WLL??","WL??-?LLLL","L??L?-L??W","???WWW-?LW","??LWW??-??","????W?W?-?","W?L?WLL??-"}

    Returns: 0.09444444444444444

  77. {"-LWLLLLW?LLLLWL","W-?LLL?LLWLLLL?","L?-L??LL????LLL","WWW-WLLLLWL?LL?","WW?L-LLLL?LLLLL","WW?WW-LLL?W??LL","W?WWWW-LLL?LLWL","LWWWWWW-LLLLLLL","?W?WWWWW-LLLLLW","WL?L??WWW-L?LLL","WW?WWL?WWW-L?LL","WW??W?WWW?W-LLL","WWWWW?WWWW?W-LL","LWWWWWLWWWWWW-?","W?W?WWWWLWWWW?-"}

    Returns: 0.06938775510204082

  78. {"-??LL?L???????","?-L???L?LLL?LL","?W-???L?????L?","W??-LL?LL?LL?L","W??W-?L?L????L","???W?-L?LL??LL","WWW?WW-?LLL?L?","???W???-L????L","?W?WWWWW-LLL?L","?W???WW?W-?L??","?W?W??W?W?-LL?","???W????WWW-?L","?WW??WW???W?-L","?W?WWW?WW??WW-"}

    Returns: 0.09615384615384615

  79. {"-L","W-"}

    Returns: 0.25

  80. {"-LLLLL","W-?WLW","W?-LLL","WLW-LL","WWWW-L","WLWWW-"}

    Returns: 0.10333333333333333

  81. {"-LLLWLLLLLLWLLL","W-LLLLLLLLLLLLL","WW-LLLLLLLLLLLL","WWW-LLLLLLLLLLL","LWWW-LLLLLLLLLW","WWWWW-LLLLLLLLL","WWWWWW-LLLLLLLL","WWWWWWW-LLLLLLL","WWWWWWWW-LLLLLL","WWWWWWWWW-LLLLL","WWWWWWWWWW-LLLL","LWWWWWWWWWW-LLL","WWWWWWWWWWWW-LL","WWWWWWWWWWWWW-L","WWWWLWWWWWWWWW-"}

    Returns: 0.08027210884353747

  82. {"-WLL","L-LL","WW-L","WWW-"}

    Returns: 0.1388888888888889

  83. {"-?W??","?-WW?","LL-?W","?L?-?","??L?-"}

    Returns: 0.125

  84. {"-???","?-??","??-?","???-"}

    Returns: 0.1388888888888889

  85. {"-WWWWWLW?LWLLWWWWWW","L-WWLLLLWWWLLLLLWLW","LL-WWWWWWLLWWL?LWWW","LLL-WWWLWW?LWL?WW?W","LWLL-WLWWWWLWLWWLLW","LWLLL-?WLLLLLL??LW?","WWLLW?-WLWWLWLW?LWW","LWLWLLL-LWWWLWLWLLW","?LLLLWWW-W?LWL??WWW","WLWLLWLLL-LLLLWLLWW","LLW?LWLL?W-LWLWWWW?","WWLWWWWLWWW-WWLWLWW","WWLLLWLWLWLL-WLLWW?","LWWWWWWLWWWLL-W?WLL","LW??L?LW?LLWWL-LLWW","LWWLL??L?WLLW?W-L?L","LLLLWWWWLWLWLLWW-LW","LWL?WLLWLLLLLWL?W-?","LLLLL?LLLL?L?WLWL?-"}

    Returns: 0.03183885640025991

  86. {"-???W??","?-?????","??-????","???-??W","L???-??","?????-L","???L?W-"}

    Returns: 0.1111111111111111

  87. {"-LL?LLLLLL?LL","W-LL????LLL?L","WW-?L?L?????L","?W?-W?LL?LLL?","W?WL-LL?LWL?L","W???W-WLL??LL","W?WWWL-LL??LL","W??W?WW-L?L?L","WW??WWWW-LL??","WW?WL???W-?LL","?W?WW??WW?-?L","W??W?WW??W?-L","WWW?WWWW?WWW-"}

    Returns: 0.09188034188034189

  88. {"-?W????W????W????","?-??W?????????W??","L?-???W?W??W?WW??","???-?W????W??W???","?L??-??W??????W??","???L?-??W??????W?","??L???-W?W?????WW","L???L?L-????W??W?","??L??L??-??????W?","??????L??-???????","???L??????-???W??","??L????????-????W","L??????L????-?W??","??LL?????????-???","?LL?L?????L?L?-?W","?????LLLL??????-?","??????L????L??L?-"}

    Returns: 0.09375

  89. {"-?LL","?-LL","WW-W","WWL-"}

    Returns: 0.1388888888888889

  90. {"-??????????","?-???WW??W?","??-????W???","???-W??????","???L-??W???","?L???-????L","?L????-??W?","??L?L??-???","????????-??","?L????L??-?","?????W????-"}

    Returns: 0.10000000000000003

  91. {"-WWL?", "L-L??", "LW-?W", "W??-L", "??LW-"}

    Returns: 0.07500000000000001

  92. {"-?", "?-" }

    Returns: 0.25

  93. {"-WWL", "L-WW", "LL-?", "WL?-" }

    Returns: 0.08333333333333331

  94. {"-??", "?-?", "??-" }

    Returns: 0.16666666666666669

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