Statistics

Problem Statement for "Reroll"

Problem Statement

This problem is about dice. All dice in this problem are the standard six-sided dice with numbers 1 through 6 on their faces.

Consider the following game that is played with n dice:

  1. Your opponent chooses a positive integer target (between n and 6n, inclusive).
  2. You roll all n dice.
  3. You select any subset of the dice you rolled and you reroll them.
  4. You add the numbers that are now shown on the n dice. If the sum is exactly equal to target, you win, otherwise you lose.

This game is often played as a drinking game with one extra rule: for each die you rerolled in step 3 you have to drink a shot.

Monika is playing this game. Her priorities are as follows:

  1. She wants to have a chance to win the game.
  2. She wants to drink as few shots as possible.

You are given the int target. You are also given the int[] dice. The elements of dice are the values rolled by Monika in step 2 of the game. Monika is now in step 3. Compute and return the smallest number of shots she will have to drink in order to have a positive probability to win the game.

Definition

Class:
Reroll
Method:
minimumDice
Parameters:
int, int[]
Returns:
int
Method signature:
int minimumDice(int target, int[] dice)
(be sure your method is public)

Constraints

  • target will be between n and 6n, inclusive, where n is the number of elements in dice.
  • dice will contain between 1 and 20 elements, inclusive.
  • Each element of dice will be between 1 and 6, inclusive.

Examples

  1. 7

    {2,6,4}

    Returns: 1

    The current sum is 12. In order to have a shot at having sum of 7, Monika has to reroll the 6 and hope for a 1.

  2. 10

    {4,2,4,5}

    Returns: 2

    The current sum is 15. Monika can choose, for instance, to reroll the two fours. She will win if she gets a 1 and a 2.

  3. 42

    {6,6,6,6,6,6,6}

    Returns: 0

    Monika has already won - no need to reroll anything.

  4. 42

    {1,3,5,3,6,4,2}

    Returns: 6

    Out of all the dice, just the single 6 is useful. Monika has to reroll all other dice and hope they all land a six.

  5. 120

    {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

    Returns: 20

  6. 20

    {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}

    Returns: 20

  7. 3

    {1}

    Returns: 1

  8. 2

    {3}

    Returns: 1

  9. 6

    {5}

    Returns: 1

  10. 2

    {2}

    Returns: 0

  11. 13

    {6,5,4,5,3,3,1,6,5,4}

    Returns: 8

  12. 21

    {5,4,1,3,3,2,6,1,3,1,2,2,6,6,4,1,2,6,3,6}

    Returns: 15

  13. 95

    {5,3,2,3,4,1,4,6,6,5,5,4,4,6,5,4,5,1,1,4}

    Returns: 4

  14. 49

    {6,2,6,2,1,1,5,2,4,5,1,2,2,4,3,4,1,1,4,4}

    Returns: 3

  15. 24

    {1,3,2,4,6,3,6,3,1,3,6,4,3,2,4,4,2,6,2,3}

    Returns: 14

  16. 34

    {3,6,4,5,6,3,3,4,2,4,3,2,5,2,1,4,4,1,2,1}

    Returns: 9

  17. 83

    {6,1,3,4,2,1,4,3,5,2,1,4,2,3,3,2,4,5,1,3}

    Returns: 5

  18. 51

    {4,6,4,6,6,4,5,3,4,3,2,5,2,3,6,3,6,3,6,3}

    Returns: 7

  19. 47

    {1,4,3,5,2,3,1,2,5,3,6,3,2,5,5,3,3,6,2,6}

    Returns: 5

  20. 58

    {3,1,3,2,4,3,1,5,2,3,6,5,2,3,4,2,3,5,3,4}

    Returns: 2

  21. 106

    {3,4,2,4,3,2,3,5,5,3,4,1,3,6,2,2,6,4,3,1}

    Returns: 11

  22. 34

    {2,3,2,6,6,2,4,2,2,2,5,2,1,2,5,3,6,5,6,2}

    Returns: 8

  23. 24

    {3,2,6,6,5,2,2,1,4,4}

    Returns: 3

  24. 44

    {4,6,6,4,5,5,6,2,2,5}

    Returns: 1

  25. 42

    {4,3,4,3,6,6,2,4,4,1}

    Returns: 1

  26. 28

    {4,1,5,6,3,5,3,2,2,5}

    Returns: 2

  27. 25

    {6,3,6,5,4,5,5,3,5,3}

    Returns: 5

  28. 43

    {3,3,3,6,4,6,5,5,6,5}

    Returns: 1

  29. 60

    {2,1,2,6,5,5,2,6,4,3}

    Returns: 8

  30. 43

    {1,2,6,6,2,4,2,6,2,3}

    Returns: 2

  31. 57

    {5,2,1,5,4,1,3,1,2,1}

    Returns: 8

  32. 25

    {1,1,4,5,4}

    Returns: 2

  33. 8

    {3,5,6,2,3}

    Returns: 3

  34. 14

    {6,4,2,5,4}

    Returns: 2

  35. 12

    {6,3,3,4,2}

    Returns: 2

  36. 30

    {4,2,6,4,2}

    Returns: 4

  37. 3

    {3,2}

    Returns: 1

  38. 5

    {2,5}

    Returns: 1

  39. 10

    {4,3}

    Returns: 1

  40. 12

    {2,4}

    Returns: 2

  41. 2

    {1,2}

    Returns: 1

  42. 2

    {2,4}

    Returns: 2

  43. 2

    {5,3}

    Returns: 2

  44. 9

    {2,3}

    Returns: 1

  45. 7

    {1,4}

    Returns: 1

  46. 2

    {2,6}

    Returns: 2

  47. 2

    {1,5}

    Returns: 1

  48. 12

    {1,1}

    Returns: 2

  49. 3

    {4,5}

    Returns: 2

  50. 78

    {5,4,5,5,6,2,6,5,5,4,2,5,2,4,3,5,6,2,1}

    Returns: 1

  51. 95

    {4,6,2,2,6,4,3,6,5,2,6,1,1,3,5,5,5,5,4}

    Returns: 5

  52. 67

    {1,6,1,3,6,3,4,6,5,3,4,4,6,3,6,3,1,2}

    Returns: 0

  53. 79

    {2,3,1,3,6,5,4,5,1,2,6,5,5,5,1,5,3,6}

    Returns: 3

  54. 68

    {6,3,1,3,4,1,5,1,2,4,2,1,5,5,5,3,4}

    Returns: 3

  55. 63

    {4,2,3,1,4,5,2,4,5,6,4,4,5,2,2,3,5}

    Returns: 1

  56. 82

    {1,6,2,4,5,2,1,5,3,4,2,1,3,2,4,6}

    Returns: 7

  57. 40

    {2,1,6,3,4,5,5,4,4,4,5,3,6,3,4,5}

    Returns: 6

  58. 24

    {4,5,3,4,3,4,2,5,6,3,2,6,6,5,2}

    Returns: 9

  59. 17

    {1,6,1,4,4,1,4,2,3,3,3,2,3,3,2}

    Returns: 10

  60. 42

    {1, 3, 5, 3, 6, 4, 2 }

    Returns: 6

  61. 10

    {4, 2, 4, 5 }

    Returns: 2

  62. 12

    {1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6 }

    Returns: 11

  63. 1

    {6 }

    Returns: 1

  64. 12

    {1, 2, 3, 4 }

    Returns: 1

  65. 7

    {2, 6, 4 }

    Returns: 1

  66. 10

    {6, 6, 6, 6, 6, 6, 6 }

    Returns: 7

  67. 32

    {6, 6, 6, 6, 6, 6, 6, 1 }

    Returns: 3

  68. 8

    {2, 1, 6 }

    Returns: 1

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