# 679. 24 Game

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

public boolean judgePoint24(int[] cards) {
List<Double> nums = new ArrayList<>();
for (int num : cards) {
}
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) {
}
}
for (char op : ops) {
switch (op) {
case '/' -> {
if (nums.get(j) == 0) {
continue;
}
}
case '*' -> {
}
case '+' -> {
}
case '-' -> {
}
}
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;
}
};

• 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)


• 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);
}