Question

679. 24 Game

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Description

You are given an integer array cards of length 4. You have four cards, each containing a number in the range [1, 9]. You should arrange the numbers on these cards in a mathematical expression using the operators ['+', '-', '*', '/'] and the parentheses '(' and ')' to get the value 24.

You are restricted with the following rules:

• The division operator '/' represents real division, not integer division.
  • For example, 4 / (1 - 2 / 3) = 4 / (1 / 3) = 12.
• Every operation done is between two numbers. In particular, we cannot use '-' as a unary operator.
  • For example, if cards = [1, 1, 1, 1], the expression "-1 - 1 - 1 - 1" is not allowed.
• You cannot concatenate numbers together
  • For example, if cards = [1, 2, 1, 2], the expression "12 + 12" is not valid.

Return true if you can get such expression that evaluates to 24, and false otherwise.

 

Example 1:

Input: cards = [4,1,8,7]
Output: true
Explanation: (8-4) * (7-1) = 24

Example 2:

Input: cards = [1,2,1,2]
Output: false

 

Constraints:

• cards.length == 4
• 1 <= cards[i] <= 9

Solutions

Solution 1: DFS

We design a function $dfs(nums)$, where $nums$ represents the current number sequence. The function returns a boolean value indicating whether there exists a permutation that makes this number sequence equal to $24$.

If the length of $nums$ is $1$, we return $true$ only when this number is $24$, otherwise we return $false$.

Otherwise, we can enumerate any two numbers $a$ and $b$ in $nums$ as the left and right operands, and enumerate the operator $op$ between $a$ and $b$. The result of $a\ op\ b$ can be used as an element of the new number sequence. We add it to the new number sequence and remove $a$ and $b$ from $nums$, then recursively call the $dfs$ function. If it returns $true$, it means we have found a permutation that makes this number sequence equal to $24$, and we return $true$.

If none of the enumerated cases return $true$, we return $false$.

class Solution:
  def judgePoint24(self, cards: List[int]) -> bool:
    def dfs(nums: List[float]):
      n = len(nums)
      if n == 1:
        if abs(nums[0] - 24) < 1e-6:
          return True
        return False
      ok = False
      for i in range(n):
        for j in range(n):
          if i != j:
            nxt = [nums[k] for k in range(n) if k != i and k != j]
            for op in ops:
              match op:
                case "/":
                  if nums[j] == 0:
                    continue
                  ok |= dfs(nxt + [nums[i] / nums[j]])
                case "*":
                  ok |= dfs(nxt + [nums[i] * nums[j]])
                case "+":
                  ok |= dfs(nxt + [nums[i] + nums[j]])
                case "-":
                  ok |= dfs(nxt + [nums[i] - nums[j]])
              if ok:
                return True
      return ok

    ops = ("+", "-", "*", "/")
    nums = [float(x) for x in cards]
    return dfs(nums)

class Solution {
  private final char[] ops = {'+', '-', '*', '/'};

  public boolean judgePoint24(int[] cards) {
    List<Double> nums = new ArrayList<>();
    for (int num : cards) {
      nums.add((double) num);
    }
    return dfs(nums);
  }

  private boolean dfs(List<Double> nums) {
    int n = nums.size();
    if (n == 1) {
      return Math.abs(nums.get(0) - 24) < 1e-6;
    }
    boolean ok = false;
    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < n; ++j) {
        if (i != j) {
          List<Double> nxt = new ArrayList<>();
          for (int k = 0; k < n; ++k) {
            if (k != i && k != j) {
              nxt.add(nums.get(k));
            }
          }
          for (char op : ops) {
            switch (op) {
              case '/' -> {
                if (nums.get(j) == 0) {
                  continue;
                }
                nxt.add(nums.get(i) / nums.get(j));
              }
              case '*' -> {
                nxt.add(nums.get(i) * nums.get(j));
              }
              case '+' -> {
                nxt.add(nums.get(i) + nums.get(j));
              }
              case '-' -> {
                nxt.add(nums.get(i) - nums.get(j));
              }
            }
            ok |= dfs(nxt);
            if (ok) {
              return true;
            }
            nxt.remove(nxt.size() - 1);
          }
        }
      }
    }
    return ok;
  }
}

class Solution {
public:
  bool judgePoint24(vector<int>& cards) {
    vector<double> nums;
    for (int num : cards) {
      nums.push_back(static_cast<double>(num));
    }
    return dfs(nums);
  }

private:
  const char ops[4] = {'+', '-', '*', '/'};

  bool dfs(vector<double>& nums) {
    int n = nums.size();
    if (n == 1) {
      return abs(nums[0] - 24) < 1e-6;
    }
    bool ok = false;
    for (int i = 0; i < n; ++i) {
      for (int j = 0; j < n; ++j) {
        if (i != j) {
          vector<double> nxt;
          for (int k = 0; k < n; ++k) {
            if (k != i && k != j) {
              nxt.push_back(nums[k]);
            }
          }
          for (char op : ops) {
            switch (op) {
            case '/':
              if (nums[j] == 0) {
                continue;
              }
              nxt.push_back(nums[i] / nums[j]);
              break;
            case '*':
              nxt.push_back(nums[i] * nums[j]);
              break;
            case '+':
              nxt.push_back(nums[i] + nums[j]);
              break;
            case '-':
              nxt.push_back(nums[i] - nums[j]);
              break;
            }
            ok |= dfs(nxt);
            if (ok) {
              return true;
            }
            nxt.pop_back();
          }
        }
      }
    }
    return ok;
  }
};

func judgePoint24(cards []int) bool {
  ops := [4]rune{'+', '-', '*', '/'}
  nums := make([]float64, len(cards))
  for i, num := range cards {
    nums[i] = float64(num)
  }
  var dfs func([]float64) bool
  dfs = func(nums []float64) bool {
    n := len(nums)
    if n == 1 {
      return math.Abs(nums[0]-24) < 1e-6
    }
    ok := false
    for i := 0; i < n; i++ {
      for j := 0; j < n; j++ {
        if i != j {
          var nxt []float64
          for k := 0; k < n; k++ {
            if k != i && k != j {
              nxt = append(nxt, nums[k])
            }
          }
          for _, op := range ops {
            switch op {
            case '/':
              if nums[j] == 0 {
                continue
              }
              nxt = append(nxt, nums[i]/nums[j])
            case '*':
              nxt = append(nxt, nums[i]*nums[j])
            case '+':
              nxt = append(nxt, nums[i]+nums[j])
            case '-':
              nxt = append(nxt, nums[i]-nums[j])
            }
            ok = ok || dfs(nxt)
            if ok {
              return true
            }
            nxt = nxt[:len(nxt)-1]
          }
        }
      }
    }
    return ok
  }

  return dfs(nums)
}

function judgePoint24(cards: number[]): boolean {
  const ops: string[] = ['+', '-', '*', '/'];
  const dfs = (nums: number[]): boolean => {
    const n: number = nums.length;
    if (n === 1) {
      return Math.abs(nums[0] - 24) < 1e-6;
    }
    let ok: boolean = false;
    for (let i = 0; i < n; i++) {
      for (let j = 0; j < n; j++) {
        if (i !== j) {
          const nxt: number[] = [];
          for (let k = 0; k < n; k++) {
            if (k !== i && k !== j) {
              nxt.push(nums[k]);
            }
          }
          for (const op of ops) {
            switch (op) {
              case '/':
                if (nums[j] === 0) {
                  continue;
                }
                nxt.push(nums[i] / nums[j]);
                break;
              case '*':
                nxt.push(nums[i] * nums[j]);
                break;
              case '+':
                nxt.push(nums[i] + nums[j]);
                break;
              case '-':
                nxt.push(nums[i] - nums[j]);
                break;
            }
            ok = ok || dfs(nxt);
            if (ok) {
              return true;
            }
            nxt.pop();
          }
        }
      }
    }
    return ok;
  };

  return dfs(cards);
}