Statistics

Problem Statement for "ProbabilityTree"

Problem Statement

Probability trees are a useful tool in mathematics to help determine conditional probabilities of complex events. Each node in a probability tree represents an event. We write P(X) to mean "the probability that event X occurs", and P(~X) to mean "the probability that event X does not occur". If a node is at the root of the tree, it has a single number associated with it (between 0.0 and 1.0, inclusive) which represents the likelihood of this event taking place. All other nodes on the tree have two numbers attached to them: the probability of this event happening if the parent node's event occurs, and the probability of this event happening if the parent node's event does not occur. If the current node is A and the parent node is B, these probabilities are referred to as P(A|B) (probability of A given B) and P(A|~B) (probability of A given not-B), respectively.

For example, here is a tree with only two nodes:

To calculate the probability of event 1 occurring (whether or not event 0 occurs), one can use this rule:
P(A) = P(A|B) * P(B) + P(A|~B) * P(~B)
The following diagram shows a more complete list of probabilities which can be derived:

Note that the sum of two opposite events is always equal to 1. P(1) was calculated as follows:
P(1) = P(1|0) * P(0) + P(1|~0) * P(~0) = (0.4 * 0.7) + (0.6 * 0.1) = 0.34

You will be given a String[] tree, representing a probability tree. The first element of tree represents the root node, and will be a single integer between 1 and 99, inclusive, representing the percent likelihood of the event taking place. Each successive element of tree will be of the form "<parent> <prob1> <prob2>", where <parent> is the zero-based index of the parent node in tree, and <prob1> and <prob2> are P(A|B) and P(A|~B) for this node, respectively. You will also be given two percentages, lowerBound and upperBound. Your method should return a int[] containing every node X, in ascending order, such that lowerBound < P(X) < upperBound.

Definition

Class:
ProbabilityTree
Method:
getOdds
Parameters:
String[], int, int
Returns:
int[]
Method signature:
int[] getOdds(String[] tree, int lowerBound, int upperBound)
(be sure your method is public)

Constraints

  • tree will contain between 1 and 50 elements, inclusive.
  • Each element of tree will contain between 1 and 8 characters, inclusive.
  • The first element of tree will contain a single integer between 1 and 99, inclusive, with no leading zeroes.
  • All other elements of tree will contain three space-separated integers of the form " ", with no unnecessary leading zeroes.
  • will be an integer between 0 and N-1, inclusive, where N is the number of elements in tree.
  • and will be between 1 and 99, inclusive.
  • All nodes in tree will be connected, directly or indirectly, to the root node.
  • lowerBound will be between 0 and 99, inclusive.
  • upperBound will be between 1 and 100, inclusive.
  • lowerBound will be strictly less than upperBound.
  • No event's likelihood will be within 1e-9 of lowerBound or upperBound.

Examples

  1. {"40","0 70 10"}

    30

    50

    Returns: { 0, 1 }

    This is the example from the problem statement. The probability of event 0 is 0.4, and the probability of event 1 is 0.34, both of which are within the specified range.

  2. {"20","2 50 50","0 50 50"}

    49

    51

    Returns: { 1, 2 }

    Both event 1 and event 2 have 50% probabilities of happening, regardless of what happens at their parents.

  3. {"10","0 99 41","1 40 3","2 91 43"}

    81

    88

    Returns: { }

    event | probability (rounded) ------+------------ 0 | 0.10 1 | 0.47 2 | 0.20 3 | 0.53

  4. {"81","0 54 56","0 15 26","1 93 63"}

    99

    100

    Returns: { }

  5. {"60","0 48 1","1 43 58"}

    48

    50

    Returns: { }

  6. {"25","0 79 99","1 12 96","2 85 67"}

    65

    98

    Returns: { 1, 3 }

  7. {"40","0 24 96","0 45 12","1 40 8","2 34 83"}

    78

    97

    Returns: { }

  8. {"79","0 64 52","1 70 87","0 38 99","1 24 8"}

    47

    81

    Returns: { 0, 1, 2, 3 }

  9. {"94","0 15 16","0 16 8","0 40 7"}

    6

    72

    Returns: { 1, 2, 3 }

  10. {"27","0 31 72"}

    13

    69

    Returns: { 0, 1 }

  11. {"56","0 14 59","0 93 95","2 43 35","2 52 54"}

    12

    79

    Returns: { 0, 1, 3, 4 }

  12. {"74","0 99 33"}

    55

    100

    Returns: { 0, 1 }

  13. {"98","0 27 45","1 87 65","0 50 89","2 20 98","2 54 15","5 88 60","1 33 19","0 36 27","2 39 32","7 1 33","2 55 55","0 64 72","6 78 44","7 16 33","3 90 99","12 31 54","8 21 14","16 48 53","2 56 29","5 44 40","5 59 84","10 37 48","21 65 12","16 86 37","4 86 6","20 27 39","26 90 86","21 79 85","13 37 26","29 83 35","18 44 59","6 5 4","16 79 25","29 86 18","15 60 35","32 63 15","33 22 92","29 46 77"}

    48

    82

    Returns: { 2, 3, 6, 11, 12, 13, 18, 19, 21, 23, 24, 28, 30, 31, 35, 37, 38 }

  14. {"45","0 22 7","1 83 32","1 7 67","2 91 43","1 11 31","0 76 9","6 7 94","2 17 37","5 5 49","2 3 98","10 70 13","10 15 4","6 22 50","5 29 72","9 34 77","6 16 63","10 31 48","9 86 87","0 19 43","3 53 47","16 42 1","3 40 67","1 70 18","8 37 7","5 95 93","21 91 65","2 36 78","20 4 83","22 47 21","7 62 84","23 99 68","8 55 33","19 38 54","0 75 60","5 80 45","31 31 75","1 33 78","23 98 87","8 55 14","38 35 55","33 78 68","41 21 97","13 81 93","18 53 41"}

    52

    61

    Returns: { 3, 7, 10, 14, 35 }

  15. {"27","0 5 3","0 6 13","1 82 32","0 62 24","1 78 69","1 55 47","2 23 5","0 73 45","2 89 87","3 30 55","0 48 61","1 84 44","8 9 2"}

    25

    46

    Returns: { 0, 3, 4, 12 }

  16. {"2","0 27 23","1 91 49","2 45 22","3 14 54","2 21 47","5 49 12","5 34 7","5 26 41","2 75 37","3 60 80","2 68 36","1 13 24","3 66 76","7 58 61","9 40 67","9 98 96","10 12 30","2 56 91","17 50 75","15 91 37","1 77 9","19 52 14","1 8 90","5 16 15","2 17 73","8 4 59","14 82 65","10 7 11","23 38 38","6 89 29","6 47 82","11 28 8"}

    37

    50

    Returns: { 4, 22, 25, 26, 29, 30 }

  17. {"9","0 43 11","0 66 91","2 83 14","0 71 97","1 41 24","2 42 46","3 69 27","6 65 2","8 29 77","3 37 96","1 5 6","7 70 9","3 51 84","8 96 24","13 81 13","12 91 63","8 22 53","2 68 63","9 72 42","7 34 97","3 65 22","19 70 87","6 78 45","21 19 35","0 2 99","24 75 8","21 13 59","14 3 66","13 37 88","8 32 89","9 79 71","22 17 13","17 24 95","10 83 12","29 66 27","15 69 44","2 4 64","20 47 96","24 23 21","33 1 15","32 41 1","23 94 98","14 77 52","39 41 53","30 47 48","45 47 63","24 94 40","20 13 74"}

    43

    59

    Returns: { 7, 10, 12, 14, 15, 17, 21, 29, 34, 35, 36, 44, 45, 46, 47 }

  18. {"41","0 53 19","1 88 43","2 44 75","0 75 57","4 66 57","3 48 62","3 29 12","2 98 77","7 15 96","9 22 73","0 96 30","5 11 62","12 58 73","3 55 12","10 7 82","5 2 66","5 11 42","2 46 89","14 19 27","1 50 44","16 34 98","21 9 21","11 46 91","21 79 51","16 98 32","18 99 46","16 74 95","16 37 33","14 21 33","15 44 3","20 1 1","8 45 7","25 86 72","11 82 10","30 47 49","28 65 12","25 5 35","8 57 51","28 19 37","8 80 28","34 72 84"}

    50

    57

    Returns: { 6, 34, 38 }

  19. {"7","0 64 62","0 55 3","1 49 69","1 9 46","3 79 69","1 72 74","0 42 42","2 91 96","2 87 72","9 86 62","0 64 89","6 17 1","7 13 80","11 7 38","5 14 69","1 79 82","1 82 37","4 77 46","0 83 90","2 52 70","18 72 19","19 78 57","15 46 62","14 64 98","20 90 65","4 89 28","19 72 43","5 73 47","27 63 46","3 43 87","18 80 93","30 37 7","0 9 41","7 49 67","28 24 23","19 71 25","28 33 60","27 2 39","13 66 17","1 46 45","38 75 32","16 55 22","6 75 10","37 99 76","4 71 30","33 76 76","40 71 39","42 94 42"}

    68

    87

    Returns: { 5, 6, 9, 10, 16, 20, 22, 25, 27, 31, 44, 46 }

  20. {"35","0 10 57","0 12 68","0 97 4","1 79 59","4 18 21","2 47 74","0 84 44","2 37 64","2 17 13","3 63 52","3 18 91","3 46 25","3 31 34","11 86 29","1 8 47","7 53 32","16 25 98","14 45 64","4 95 76","19 75 69","13 7 29"}

    33

    63

    Returns: { 0, 1, 2, 3, 6, 7, 8, 10, 16, 18 }

  21. {"24","0 40 91","1 22 43","1 11 14","1 81 54","1 69 76"}

    19

    71

    Returns: { 0, 2, 5 }

  22. {"62","0 40 26","0 60 88","2 94 60","0 33 12","4 65 71","2 47 52","2 73 61","6 30 87","6 43 68","4 33 22","8 78 75","4 25 65","5 79 14","7 38 42","11 68 67","11 2 98","10 47 87","14 76 88","6 41 33","5 6 86","10 26 64","18 63 94","9 58 97","7 37 12","14 99 23","3 2 79","24 73 91","7 62 55","8 98 47","28 54 30","6 43 33","14 54 99","16 75 11","19 4 75","10 78 38","1 9 62","6 65 56","32 83 76","22 16 91","34 70 34","28 72 22","41 38 5","23 79 92","13 22 72","30 98 90","32 15 20"}

    25

    72

    Returns: { 0, 1, 2, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 19, 20, 21, 22, 24, 25, 28, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 44 }

  23. {"56","0 5 11","0 87 16","1 6 20","3 91 60","0 70 6","5 56 64","2 92 64","2 98 21","0 15 24"}

    59

    72

    Returns: { 4, 6, 8 }

  24. {"51","0 70 76","1 76 53","0 92 57","1 31 2","4 56 16","1 90 68","2 27 19","6 69 24","6 63 46","5 72 51","9 72 88","8 53 23","9 59 30","7 88 12","1 48 78","4 42 69","12 13 12","3 7 71","4 98 70","1 88 16","18 62 24","15 78 5","21 20 61","22 63 83","1 36 7"}

    20

    53

    Returns: { 0, 4, 5, 7, 12, 13, 14, 18, 21, 22, 23, 25 }

  25. {"45","0 24 10","1 6 62","1 70 41","2 25 23","0 28 81","4 39 81","4 66 15","2 62 10","8 46 61","7 41 46","9 8 42","11 48 54","4 72 45","7 51 47","5 34 97","2 11 42","3 9 43","1 97 2","13 84 40","5 82 20","7 89 95","17 95 70","6 19 48","11 29 80","14 68 93","6 10 84","9 23 50","9 78 27","22 83 44","24 22 39","1 24 73","26 47 72","10 63 82","18 19 71","27 93 54"}

    62

    75

    Returns: { 6, 19, 24, 29, 31, 32, 33, 35 }

  26. {"86","0 34 20","1 74 47","2 95 29","1 13 74","4 80 91","0 60 52","3 35 60","1 6 60","5 27 34","8 90 39","4 92 17","4 56 85","6 75 53","8 34 57","1 12 43","13 7 78","11 86 23","4 4 90","6 15 83","19 74 59","11 22 77","7 4 33","18 56 23","11 42 18","16 45 80","25 93 65","16 87 93","25 56 43","26 36 70","22 22 89","1 6 43","5 90 30","3 62 1","29 85 19","20 3 19","8 71 41","28 79 92","25 91 74","10 56 49"}

    11

    93

    Returns: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39 }

  27. {"64","0 20 18","0 24 55","0 40 45","3 16 67","0 16 64","2 75 17","0 20 37","5 23 55","8 68 59","0 77 26","4 56 9","11 83 1"}

    10

    56

    Returns: { 1, 2, 3, 4, 5, 6, 7, 8, 11, 12 }

  28. {"89","0 86 89","0 38 13"}

    76

    91

    Returns: { 0, 1 }

  29. {"12","0 18 42","1 95 27","0 6 60","1 42 83","2 74 77","3 63 51","4 7 40","7 99 85","8 77 92","5 4 48","3 41 73","3 10 85","11 70 34","2 95 73","9 30 3","9 48 59","4 35 9","16 96 3","9 31 55","19 31 65","16 30 17","10 65 71","11 46 49","11 51 52","13 39 67","24 67 40","18 74 38","16 97 52","13 65 82","1 65 28","12 93 57","0 93 35","31 46 73","30 28 8","10 93 64","2 54 4","0 9 66","23 87 82","26 62 43","32 2 66","38 10 90","11 82 7","19 7 84","42 46 68"}

    28

    96

    Returns: { 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44 }

  30. {"54","0 33 98","0 12 56","1 28 73","3 26 24","2 86 74","3 72 21","0 70 47","4 66 60","8 38 2","2 19 95","9 30 42","1 3 67","6 48 69","5 3 81","12 93 67","0 7 16","14 27 31"}

    25

    70

    Returns: { 0, 1, 2, 3, 6, 7, 8, 11, 12, 13, 17 }

  31. {"37","0 61 49","0 61 33","1 56 67","1 68 21","1 50 49","0 94 80","5 37 84","1 36 74","6 15 17","1 10 51","0 95 50","7 37 76","1 84 41","1 94 93","8 1 67","8 21 27","4 58 99","13 30 22"}

    78

    98

    Returns: { 6, 14, 17 }

  32. {"41","0 42 7","0 3 11","2 50 47","0 96 67","2 12 40","0 36 62","5 60 59","5 66 25","2 64 96","3 51 56","0 16 98","4 94 29","3 97 12","0 37 24","0 65 98","4 88 97","0 37 26","0 91 61","2 62 13"}

    11

    32

    Returns: { 1, 14, 17, 19 }

  33. {"85","0 47 91","1 54 50","0 79 19","1 27 21","0 33 87","3 39 11","1 64 83","3 81 71","6 30 39","5 71 2","8 62 51","11 28 2","6 10 84","5 55 33","10 53 35","0 32 38","8 44 32","12 70 96","16 42 3","6 14 24","13 94 86","10 90 25","13 68 67","11 6 41","6 81 17","3 59 63","9 44 26","25 62 64","9 15 85","0 12 50","8 9 79","1 43 69","16 46 49","24 88 39","4 81 8","31 7 50","30 79 56","25 86 7","8 63 14","0 86 91","24 59 98","12 3 46","27 68 14","36 5 38","26 34 85","39 42 46","29 22 86","17 59 74","29 35 53"}

    58

    96

    Returns: { 0, 3, 7, 8, 11, 13, 18, 21, 23, 26, 28, 29, 37, 40, 41, 48 }

  34. {"14","0 97 36","1 94 35","0 34 36","3 57 72","3 50 63","4 99 88","3 61 3","1 85 75","7 89 25","6 48 12","8 76 46","6 66 69","12 82 23","6 93 97","1 26 86","12 84 3","2 2 2","12 46 97"}

    21

    78

    Returns: { 1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 15, 16, 18 }

  35. {"46","0 51 50","0 91 28","2 57 59","0 31 63","2 98 90","2 79 79","0 7 35","6 1 37","3 2 73","0 29 16","2 23 16","3 2 90","0 15 76","7 46 90","9 85 97","12 91 84","15 77 72","16 80 15","6 24 59","4 81 48","4 57 71","14 56 3","9 36 11","2 28 91","11 58 28","23 75 43","18 69 70","10 27 83","28 5 52","23 86 29","3 43 21","31 24 23","13 58 88","12 60 14","21 27 67","15 60 9","11 8 48","33 76 63","29 11 51","4 37 65","22 41 70","25 79 15"}

    87

    97

    Returns: { 5, 15 }

  36. {"55","0 99 32","0 3 14","0 10 13","0 19 2","4 70 51","4 79 88","0 46 11","1 73 63","1 71 49","7 36 20","3 32 21","4 34 60","8 86 59","1 59 33","8 85 50","10 82 32","11 65 41"}

    58

    97

    Returns: { 1, 6, 8, 9, 13, 15 }

  37. {"79","0 67 90","1 38 73","0 73 67","3 11 66","4 48 59","2 63 71","2 83 81","7 16 4","8 44 17","4 61 31","1 89 9","7 96 19","0 10 1","12 1 9","9 92 89","2 3 35","14 24 33","11 87 35","18 49 31","3 47 70","9 87 20","0 57 29","9 86 99","10 43 42","9 72 55"}

    11

    84

    Returns: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 24, 25 }

  38. {"30","0 50 3","0 75 45","2 79 98","2 38 26","3 28 21","1 98 78","4 55 67","1 21 98","0 34 53","9 39 87","3 91 89","11 80 42","4 25 94","8 81 79"}

    37

    88

    Returns: { 2, 3, 6, 7, 8, 9, 10, 12, 13, 14 }

  39. {"78","0 65 9","0 94 84","1 45 11","0 77 31","1 34 18","0 65 68","3 83 67","5 63 53","8 57 3","2 97 74","7 97 47","5 72 39","5 54 48","8 69 59","1 20 21","1 10 61","8 8 28","2 32 87","0 43 20"}

    38

    70

    Returns: { 1, 4, 6, 8, 12, 13, 14 }

  40. {"33","0 56 39","0 51 36","2 60 74","1 95 64","4 80 10","5 15 24","1 1 13","0 22 96","1 95 44","5 98 22","8 58 62","7 94 38","11 87 68","10 81 83","3 63 49","14 63 72","14 70 81","14 95 19","12 29 51","19 97 40","8 38 23","14 73 27","14 83 31","12 23 53","16 25 41","0 9 95","18 96 33","10 89 67","3 88 78","19 45 17","22 10 48","16 34 68","29 45 78","3 13 65","34 6 31"}

    66

    92

    Returns: { 3, 4, 8, 9, 10, 13, 14, 17, 18, 23, 26, 27, 28, 29 }

  41. {"47","0 68 58","1 42 81","2 9 57","0 51 55","0 71 80","4 79 42","1 86 53","1 31 41","8 71 80","5 65 53","3 56 64","6 88 7","6 6 46","13 36 85","10 44 40","11 44 93","10 95 3","15 18 67","4 6 57","9 4 65","0 86 68"}

    66

    95

    Returns: { 5, 7, 9, 14, 21 }

  42. {"49","0 35 98","1 80 2","0 95 83","3 90 75","3 59 79","0 12 12","5 14 40","3 94 18","5 76 6","7 15 32","4 12 8","3 58 28","11 41 62","7 39 11","9 87 45","0 81 99","8 9 80","15 11 18","6 38 34"}

    33

    80

    Returns: { 0, 1, 2, 5, 9, 12, 13, 15, 19 }

  43. {"51","0 80 21","1 80 86","0 51 77","1 7 85","3 69 10","1 70 28"}

    16

    92

    Returns: { 0, 1, 2, 3, 4, 5, 6 }

  44. {"8","0 22 27","1 11 86","0 47 8","1 86 31","1 24 87","3 81 47","4 19 74","5 47 2","1 79 27","5 77 26","1 23 69","5 4 22","1 80 37","11 82 55","4 70 36","5 7 53","5 77 79","7 11 77","17 51 43","5 56 45","5 54 55","16 29 75","11 99 50","8 21 48","2 52 17"}

    56

    96

    Returns: { 2, 5, 10, 11, 14, 17, 22, 23 }

  45. {"38","0 2 16","1 50 27","2 89 9","3 68 84","0 45 36","0 71 12","0 39 69","5 90 33","8 12 70","6 84 18","3 60 35","0 26 2","2 88 59","11 9 69","10 61 68","13 11 74"}

    4

    42

    Returns: { 0, 1, 2, 3, 5, 6, 9, 10, 12, 16 }

  46. {"82","0 1 55","1 42 77","1 51 46","1 72 76","3 17 74","4 82 27","5 17 53","0 62 91","2 5 95","5 23 88","7 59 51","7 6 10","10 61 54","10 59 55","9 91 6","9 47 50","12 2 86","12 8 15","18 43 33","16 37 71","19 47 51","18 19 43","20 74 81","21 12 64","5 94 57","16 30 29","4 98 63","23 17 59","3 33 82","4 57 50","21 48 86","26 8 8","14 89 58","24 67 6","11 57 67","24 14 41","20 55 25","33 65 14","36 95 21","10 9 53","16 50 72","31 22 39","21 36 48","33 26 12","16 37 82","6 11 33","28 71 49","39 46 84"}

    30

    93

    Returns: { 0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 45, 47, 48 }

  47. {"28","0 86 71","0 16 24","2 82 68","1 22 54","2 93 80","1 89 52","1 15 55","4 73 54","8 9 42","6 97 61","3 4 62","0 90 82","10 60 65","6 75 55","2 10 72","12 30 73","16 91 45","17 42 78","2 68 74","13 80 26","17 2 4","4 47 95","11 24 75","10 52 43","22 76 47","20 28 32","11 67 49","4 19 55","22 25 12","20 44 59","29 68 16","5 59 71","28 73 1","10 57 6","2 58 68","17 76 1","17 38 80","3 65 19","5 76 48","36 7 77","15 84 28","32 66 68","26 33 37","20 54 50","2 19 60","25 18 5","32 19 61","35 94 74","15 21 34"}

    20

    79

    Returns: { 0, 1, 2, 3, 4, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49 }

  48. {"22","0 34 47","0 20 85","2 36 5","2 21 67","4 99 30","5 37 54","1 13 50","0 13 92"}

    58

    82

    Returns: { 2, 8 }

  49. {"10","0 76 23","0 52 64","0 79 47","0 59 59","4 57 30","1 71 13","0 70 54","6 98 65","6 19 92","7 81 8","0 12 88","5 56 88","12 22 43","3 21 35","14 15 40","8 70 6","6 71 71","8 19 92","12 16 13","1 30 87","1 72 46","18 69 61","21 53 40","9 73 48","11 88 29","2 6 81","21 84 45","0 96 84","17 45 81","25 44 47","28 31 4","5 84 23","4 16 29","9 80 20","3 15 10","31 57 84","6 16 37","20 63 99","0 91 34","20 74 74","16 15 3"}

    69

    82

    Returns: { 8, 9, 11, 12, 17, 20, 25, 36, 38, 40 }

  50. {"41","0 31 30","1 26 55","1 52 7","2 74 83","0 43 93","0 71 29","3 33 91","7 74 63","6 27 43","2 57 11","6 90 74","11 49 84","0 1 46","1 46 67","2 14 47","1 98 66","15 76 38","6 24 65","14 14 29","13 29 41","3 40 81","3 93 30","22 39 63","3 84 99","2 9 37","18 21 10"}

    12

    77

    Returns: { 0, 1, 2, 3, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26 }

  51. {"38","0 32 5","0 73 87","1 57 10","2 4 28","3 89 23","1 14 73","4 58 78","4 24 27","4 34 13","8 77 75","1 97 30","1 82 68","2 55 55","2 15 98","5 4 14","12 35 13","14 91 69","5 56 80","9 35 43","11 65 13","0 70 62","16 65 63","17 1 22","21 12 53","4 36 4","3 4 95","10 93 67","10 75 59","28 59 29","13 16 72","6 95 90","27 91 32","29 42 76","2 44 34","19 81 52"}

    56

    80

    Returns: { 6, 7, 10, 12, 17, 18, 21, 22, 26, 28, 33, 35 }

  52. {"57","0 76 85","1 50 12","0 67 85","2 8 98","1 58 32","5 8 20","2 68 98","6 78 98","8 92 79","0 97 20","7 46 25","6 68 42","4 71 9","4 39 25","11 7 57","12 18 2"}

    25

    95

    Returns: { 0, 1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 14, 15 }

  53. {"90","0 6 78","0 1 12","0 26 57","3 84 64","1 74 4","4 40 6","4 62 50","7 2 86","3 14 93","6 73 73","9 88 5","3 93 28","9 18 87","13 62 42","2 52 44","2 9 2","15 27 21","5 60 24","5 28 41","16 15 63","2 58 88","20 21 69","17 75 60","8 59 90","10 55 63","24 51 84","1 80 32","1 28 63","2 34 94","2 21 63","20 51 57","24 42 28","28 2 54","17 77 5","26 36 86","25 96 12","29 68 70","19 14 47"}

    5

    43

    Returns: { 1, 3, 5, 6, 8, 13, 17, 18, 19, 22, 27, 32, 33, 34, 38 }

  54. {"83","0 39 11","0 19 4","0 51 42","2 55 11","3 41 4","5 94 60"}

    20

    61

    Returns: { 1, 3, 5 }

  55. {"34","0 93 29","0 69 15","2 28 1","3 86 51","4 92 67","1 77 77","0 33 9","5 86 90","4 84 6","3 59 89","7 54 19","3 86 54","10 81 81","9 61 38","9 80 23","7 87 4","5 52 14","17 86 1","9 80 34","6 60 55","9 62 6","9 3 3","4 70 95","17 78 4","4 39 32","18 38 62","21 95 19","1 25 33","17 94 86","17 38 10","0 68 65","15 76 97"}

    43

    97

    Returns: { 1, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 19, 20, 23, 26, 27, 29, 31, 32 }

  56. {"93","0 84 42","1 81 86","2 89 74","2 20 23","3 96 45","5 81 59","1 81 24","2 16 2","0 4 45","3 44 37","0 52 71","8 22 48","9 88 34","1 34 91","8 17 13","2 99 30","4 32 5","3 51 38","14 87 11","2 52 76","13 63 37","1 30 41","17 25 84","0 59 5","17 98 24","7 20 50","10 98 46","10 51 69","23 88 23","14 69 50","25 75 2","4 41 55","24 87 33","6 94 18","30 12 58"}

    61

    69

    Returns: { 27, 28, 33 }

  57. {"35","0 12 91","1 82 22","1 96 27","2 15 83","4 58 39","0 26 73","6 11 84","6 16 47","6 93 22","8 61 51","3 75 93","10 33 96","6 74 27","7 5 99","11 58 11","4 62 19","11 7 81","5 72 97","5 58 49","0 90 44","19 93 59","9 8 77","21 29 91","15 34 48","7 88 87","20 6 89"}

    31

    57

    Returns: { 0, 4, 5, 6, 7, 10, 13, 15, 16, 19, 22, 23, 24, 26 }

  58. {"93","0 99 24","0 17 5","1 14 40","2 16 96","0 70 38","0 29 51","2 64 78","2 83 82","6 55 73","1 81 86","1 62 71","2 60 10","5 10 38","13 64 80","5 98 67","7 56 12","7 85 86","3 16 50","13 62 72","3 9 58","5 24 55","13 14 70","6 41 36","4 42 87","17 99 32","2 50 89","10 37 45","24 55 58","4 77 49","24 62 39","1 16 25","23 36 81","19 31 78","7 51 27","9 13 89","30 89 12","27 63 91","36 95 70","7 37 6","8 16 85","32 32 45","13 46 7"}

    69

    89

    Returns: { 4, 7, 8, 10, 14, 15, 17, 19, 26, 29, 37, 38 }

  59. {"80","0 63 85","0 44 4","1 15 48","3 29 15","1 75 61","4 31 54","6 71 18","3 90 4","8 78 77","7 75 70","3 37 11","7 3 66","11 23 50","4 5 27","0 15 11","1 43 24","2 48 74","4 66 48","10 63 69","16 80 9","0 12 78","2 67 15","7 39 65","6 48 54","0 73 63","3 90 22","12 49 49","11 9 52","27 4 99","7 51 8","19 56 93","18 36 62","26 48 36","20 66 35","26 18 61","0 4 5","13 46 12"}

    21

    77

    Returns: { 1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37 }

  60. {"14","0 61 78","0 52 80","2 16 3","0 79 93","2 94 34","3 31 65","0 13 11","5 66 97","8 28 85"}

    19

    64

    Returns: { 6, 9 }

  61. {"44","0 90 34","0 61 67","1 40 22","3 58 70","4 55 44","1 13 84","2 96 61","7 10 59","8 7 31","6 46 9","5 99 58","11 7 80","12 59 94","3 26 80","13 1 53","8 61 13","9 21 69","14 68 81","2 90 15","12 35 69","2 27 40","8 41 63","10 1 99","2 51 29","0 27 95","9 65 36","11 30 32","2 86 54","18 61 96","17 7 32","14 19 67","5 9 90","16 15 28","1 70 89","2 37 41","15 71 61","14 49 2","11 57 96","18 47 92","27 9 97","12 42 16","6 43 68"}

    41

    51

    Returns: { 0, 6, 24, 26, 32 }

  62. {"33","0 94 5","1 2 48","2 37 56","2 29 38","3 30 50","5 38 1","3 85 46","3 97 19","4 95 80","5 97 77","1 22 43","0 9 30","7 92 52","7 44 65","6 12 46","4 44 72","4 17 40","16 51 57","10 52 76","6 53 13","3 99 56","1 96 54","13 70 32","9 85 75","6 14 91","10 17 40","0 7 10","6 47 79","15 10 72","5 76 30","24 36 25","28 6 40","9 81 34","15 11 62","5 31 1","17 92 76","20 49 52","24 93 16","33 71 46","14 64 45","25 54 66","7 28 2","5 15 21","36 59 36","44 51 41","37 45 43","29 45 88"}

    5

    99

    Returns: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47 }

  63. {"90", "8 11 19", "0 58 54", "8 18 9", "0 96 35", "1 70 22", "0 40 1", "6 78 2", "2 74 29"}

    7

    42

    Returns: { 1, 3, 5, 6, 7 }

  64. {"70", "5 12 19", "9 45 37", "10 83 33", "3 59 58", "0 75 44", "10 25 24", "6 64 47", "13 34 79", "5 13 32", "0 9 34", "10 37 4", "1 54 80", "12 62 39", "9 67 49", "3 6 58"}

    30

    61

    Returns: { 2, 3, 4, 7, 8, 13, 14, 15 }

  65. {"51", "29 58 3", "6 56 86", "18 97 1", "44 99 25", "33 69 90", "27 67 49", "32 15 19", "33 1 21", "45 12 33", "29 24 40", "45 86 74", "40 30 65", "0 18 27", "1 90 65", "0 47 62", "40 81 72", "42 25 56", "45 16 81", "8 94 92", "29 41 92", "24 4 29", "32 56 91", "20 16 77", "1 35 79", "45 77 61", "6 50 19", "20 69 43", "4 6 16", "15 55 26", "42 73 90", "40 8 49", "33 16 33", "15 95 47", "9 66 40", "25 80 39", "35 72 70", "27 10 36", "40 36 10", "32 2 48", "33 44 23", "22 51 45", "25 8 43", "18 32 96", "45 41 74", "0 51 6", "18 48 15"}

    8

    82

    Returns: { 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46 }

  66. {"74", "13 21 67", "0 96 8", "7 96 2", "14 46 41", "21 88 59", "0 32 99", "2 98 26", "7 95 29", "21 67 53", "11 30 8", "0 17 70", "9 10 2", "9 29 36", "7 98 44", "11 73 52", "9 55 29", "11 79 56", "9 15 92", "0 78 82", "0 27 69", "0 14 75"}

    24

    99

    Returns: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21 }

  67. {"66", "2 6 34", "9 87 24", "9 33 55", "9 78 25", "16 70 73", "8 37 29", "34 9 50", "7 6 86", "8 32 28", "15 14 93", "7 16 24", "22 71 91", "11 43 36", "16 57 15", "9 95 93", "0 91 67", "22 68 41", "29 54 62", "14 44 16", "33 29 89", "11 6 5", "16 35 21", "10 80 42", "7 92 66", "6 69 48", "27 50 23", "33 56 21", "16 35 39", "7 85 71", "6 62 93", "11 67 68", "4 42 84", "15 53 49", "0 32 2", "7 77 50", "16 84 67", "34 65 76", "6 1 68", "12 54 33"}

    0

    70

    Returns: { 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33, 34, 35, 38, 39 }

  68. { "51", "29 58 3", "6 56 86", "18 97 1", "44 99 25", "33 69 90", "27 67 49", "32 15 19", "33 1 21", "45 12 33", "29 24 40", "45 86 74", "40 30 65", "0 18 27", "1 90 65", "0 47 62", "40 81 72", "42 25 56", "45 16 81", "8 94 92", "29 41 92", "24 4 29", "32 56 91", "20 16 77", "1 35 79", "45 77 61", "6 50 19", "20 69 43", "4 6 16", "15 55 26", "42 73 90", "40 8 49", "33 16 33", "15 95 47", "9 66 40", "25 80 39", "35 72 70", "27 10 36", "40 36 10", "32 2 48", "33 44 23", "22 51 45", "25 8 43", "18 32 96", "45 41 74", "0 51 6", "18 48 15" }

    8

    82

    Returns: { 0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46 }

  69. { "51", "29 58 3", "6 56 86", "18 97 1", "44 99 25", "33 69 90", "27 67 49", "32 15 19", "33 1 21", "45 12 33", "29 24 40", "45 86 74", "40 30 65", "0 18 27", "1 90 65", "0 47 62", "40 81 72", "42 25 56", "45 16 81", "8 94 92", "29 41 92", "24 4 29", "32 56 91", "20 16 77", "1 35 79", "45 77 61", "6 50 19", "20 69 43", "4 6 16", "15 55 26", "42 73 90", "40 8 49", "33 16 33", "15 95 47", "9 66 40", "25 80 39", "35 72 70", "27 10 36", "40 36 10", "32 2 48", "33 44 23", "22 51 45", "25 8 43", "18 32 96", "45 41 74", "0 51 6", "18 48 15" }

    33

    89

    Returns: { 0, 2, 3, 4, 5, 6, 10, 11, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 39, 40, 41, 43, 44, 46 }

  70. { "40", "0 70 10" }

    30

    50

    Returns: { 0, 1 }

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