679. 24点问题
今天这个题我们会经常遇到，即经典的24点问题。
题目
你有 4 张写有 1 到 9 数字的牌。你需要判断是否能通过 *，/，+，-，(，) 的运算得到 24。
示例 1:
输入: [4, 1, 8, 7]
输出: True
解释: (8-4) * (7-1) = 24
示例 2:
输入: [1, 2, 1, 2]
输出: False
注意:
除法运算符 / 表示实数除法，而不是整数除法。例如 4 / (1 - 2/3) = 12 。
每个运算符对两个数进行运算。特别是我们不能用 - 作为一元运算符。例如，[1, 1, 1, 1] 作为输入时，表达式 -1 - 1 - 1 - 1 是不允许的。
你不能将数字连接在一起。例如，输入为 [1, 2, 1, 2] 时，不能写成 12 + 12 。
思路
这个题目的解法有很多种，笔者水平有限，仅能列出两条：
- DFS递归。四个数取出任意两个数之后，做加减乘除计算后将结果放回原数组，并将新数组作为参数传给下一层去处理。最终判断返回的结果是否为24。
- 超暴力思路。对给定数组进行排序，并且将排序后的结果与把所有成立的数组一一做对比，从而返回判断值。若排序后的结果在所有成立的数组，则返回True；否则返回False.
代码
DFS递归
class Solution:
def judgePoint24(self, nums: List[int]) -> bool:
if not nums: return False
def helper(nums):
if len(nums) == 1: return abs(nums[0]-24) < 1e-6
for i in range(len(nums)):
for j in range(len(nums)):
if i != j:
newnums = [nums[k] for k in range(len(nums)) if i != k != j]
if helper(newnums + [nums[i] + nums[j]]): return True
if helper(newnums + [nums[i] - nums[j]]): return True
if helper(newnums + [nums[i] * nums[j]]): return True
if nums[j] != 0 and helper(newnums + [nums[i] / nums[j]]): return True
return False
return helper(nums)
超暴力思路
class Solution:
def judgePoint24(self, nums: List[int]) -> bool:
nums.sort()
return nums in [[1, 1, 1, 8], [1, 1, 2, 6], [1, 1, 2, 7], [1, 1, 2, 8], [1, 1, 2, 9], [1, 1, 3, 4], [1, 1, 3, 5], [1, 1, 3, 6], [1, 1, 3, 7], [1, 1, 3, 8], [1, 1, 3, 9], [1, 1, 4, 4], [1, 1, 4, 5], [1, 1, 4, 6], [1, 1, 4, 7], [1, 1, 4, 8], [1, 1, 4, 9], [1, 1, 5, 5], [1, 1, 5, 6], [1, 1, 5, 7], [1, 1, 5, 8], [1, 1, 6, 6], [1, 1, 6, 8], [1, 1, 6, 9], [1, 1, 8, 8], [1, 2, 2, 4], [1, 2, 2, 5], [1, 2, 2, 6], [1, 2, 2, 7], [1, 2, 2, 8], [1, 2, 2, 9], [1, 2, 3, 3], [1, 2, 3, 4], [1, 2, 3, 5], [1, 2, 3, 6], [1, 2, 3, 7], [1, 2, 3, 8], [1, 2, 3, 9], [1, 2, 4, 4], [1, 2, 4, 5], [1, 2, 4, 6], [1, 2, 4, 7], [1, 2, 4, 8], [1, 2, 4, 9], [1, 2, 5, 5], [1, 2, 5, 6], [1, 2, 5, 7], [1, 2, 5, 8], [1, 2, 5, 9], [1, 2, 6, 6], [1, 2, 6, 7], [1, 2, 6, 8], [1, 2, 6, 9], [1, 2, 7, 7], [1, 2, 7, 8], [1, 2, 7, 9], [1, 2, 8, 8], [1, 2, 8, 9], [1, 3, 3, 3], [1, 3, 3, 4], [1, 3, 3, 5], [1, 3, 3, 6], [1, 3, 3, 7], [1, 3, 3, 8], [1, 3, 3, 9], [1, 3, 4, 4], [1, 3, 4, 5], [1, 3, 4, 6], [1, 3, 4, 7], [1, 3, 4, 8], [1, 3, 4, 9], [1, 3, 5, 6], [1, 3, 5, 7], [1, 3, 5, 8], [1, 3, 5, 9], [1, 3, 6, 6], [1, 3, 6, 7], [1, 3, 6, 8], [1, 3, 6, 9], [1, 3, 7, 7], [1, 3, 7, 8], [1, 3, 7, 9], [1, 3, 8, 8], [1, 3, 8, 9], [1, 3, 9, 9], [1, 4, 4, 4], [1, 4, 4, 5], [1, 4, 4, 6], [1, 4, 4, 7], [1, 4, 4, 8], [1, 4, 4, 9], [1, 4, 5, 5], [1, 4, 5, 6], [1, 4, 5, 7], [1, 4, 5, 8], [1, 4, 5, 9], [1, 4, 6, 6], [1, 4, 6, 7], [1, 4, 6, 8], [1, 4, 6, 9], [1, 4, 7, 7], [1, 4, 7, 8], [1, 4, 7, 9], [1, 4, 8, 8], [1, 4, 8, 9], [1, 5, 5, 5], [1, 5, 5, 6], [1, 5, 5, 9], [1, 5, 6, 6], [1, 5, 6, 7], [1, 5, 6, 8], [1, 5, 6, 9], [1, 5, 7, 8], [1, 5, 7, 9], [1, 5, 8, 8], [1, 5, 8, 9], [1, 5, 9, 9], [1, 6, 6, 6], [1, 6, 6, 8], [1, 6, 6, 9], [1, 6, 7, 9], [1, 6, 8, 8], [1, 6, 8, 9], [1, 6, 9, 9], [1, 7, 7, 9], [1, 7, 8, 8], [1, 7, 8, 9], [1, 7, 9, 9], [1, 8, 8, 8], [1, 8, 8, 9], [2, 2, 2, 3], [2, 2, 2, 4], [2, 2, 2, 5], [2, 2, 2, 7], [2, 2, 2, 8], [2, 2, 2, 9], [2, 2, 3, 3], [2, 2, 3, 4], [2, 2, 3, 5], [2, 2, 3, 6], [2, 2, 3, 7], [2, 2, 3, 8], [2, 2, 3, 9], [2, 2, 4, 4], [2, 2, 4, 5], [2, 2, 4, 6], [2, 2, 4, 7], [2, 2, 4, 8], [2, 2, 4, 9], [2, 2, 5, 5], [2, 2, 5, 6], [2, 2, 5, 7], [2, 2, 5, 8], [2, 2, 5, 9], [2, 2, 6, 6], [2, 2, 6, 7], [2, 2, 6, 8], [2, 2, 6, 9], [2, 2, 7, 7], [2, 2, 7, 8], [2, 2, 8, 8], [2, 2, 8, 9], [2, 3, 3, 3], [2, 3, 3, 5], [2, 3, 3, 6], [2, 3, 3, 7], [2, 3, 3, 8], [2, 3, 3, 9], [2, 3, 4, 4], [2, 3, 4, 5], [2, 3, 4, 6], [2, 3, 4, 7], [2, 3, 4, 8], [2, 3, 4, 9], [2, 3, 5, 5], [2, 3, 5, 6], [2, 3, 5, 7], [2, 3, 5, 8], [2, 3, 5, 9], [2, 3, 6, 6], [2, 3, 6, 7], [2, 3, 6, 8], [2, 3, 6, 9], [2, 3, 7, 7], [2, 3, 7, 8], [2, 3, 7, 9], [2, 3, 8, 8], [2, 3, 8, 9], [2, 3, 9, 9], [2, 4, 4, 4], [2, 4, 4, 5], [2, 4, 4, 6], [2, 4, 4, 7], [2, 4, 4, 8], [2, 4, 4, 9], [2, 4, 5, 5], [2, 4, 5, 6], [2, 4, 5, 7], [2, 4, 5, 8], [2, 4, 5, 9], [2, 4, 6, 6], [2, 4, 6, 7], [2, 4, 6, 8], [2, 4, 6, 9], [2, 4, 7, 7], [2, 4, 7, 8], [2, 4, 7, 9], [2, 4, 8, 8], [2, 4, 8, 9], [2, 4, 9, 9], [2, 5, 5, 7], [2, 5, 5, 8], [2, 5, 5, 9], [2, 5, 6, 6], [2, 5, 6, 7], [2, 5, 6, 8], [2, 5, 6, 9], [2, 5, 7, 7], [2, 5, 7, 8], [2, 5, 7, 9], [2, 5, 8, 8], [2, 5, 8, 9], [2, 6, 6, 6], [2, 6, 6, 7], [2, 6, 6, 8], [2, 6, 6, 9], [2, 6, 7, 8], [2, 6, 7, 9], [2, 6, 8, 8], [2, 6, 8, 9], [2, 6, 9, 9], [2, 7, 7, 8], [2, 7, 8, 8], [2, 7, 8, 9], [2, 8, 8, 8], [2, 8, 8, 9], [2, 8, 9, 9], [3, 3, 3, 3], [3, 3, 3, 4], [3, 3, 3, 5], [3, 3, 3, 6], [3, 3, 3, 7], [3, 3, 3, 8], [3, 3, 3, 9], [3, 3, 4, 4], [3, 3, 4, 5], [3, 3, 4, 6], [3, 3, 4, 7], [3, 3, 4, 8], [3, 3, 4, 9], [3, 3, 5, 5], [3, 3, 5, 6], [3, 3, 5, 7], [3, 3, 5, 9], [3, 3, 6, 6], [3, 3, 6, 7], [3, 3, 6, 8], [3, 3, 6, 9], [3, 3, 7, 7], [3, 3, 7, 8], [3, 3, 7, 9], [3, 3, 8, 8], [3, 3, 8, 9], [3, 3, 9, 9], [3, 4, 4, 4], [3, 4, 4, 5], [3, 4, 4, 6], [3, 4, 4, 7], [3, 4, 4, 8], [3, 4, 4, 9], [3, 4, 5, 5], [3, 4, 5, 6], [3, 4, 5, 7], [3, 4, 5, 8], [3, 4, 5, 9], [3, 4, 6, 6], [3, 4, 6, 8], [3, 4, 6, 9], [3, 4, 7, 7], [3, 4, 7, 8], [3, 4, 7, 9], [3, 4, 8, 9], [3, 4, 9, 9], [3, 5, 5, 6], [3, 5, 5, 7], [3, 5, 5, 8], [3, 5, 5, 9], [3, 5, 6, 6], [3, 5, 6, 7], [3, 5, 6, 8], [3, 5, 6, 9], [3, 5, 7, 8], [3, 5, 7, 9], [3, 5, 8, 8], [3, 5, 8, 9], [3, 5, 9, 9], [3, 6, 6, 6], [3, 6, 6, 7], [3, 6, 6, 8], [3, 6, 6, 9], [3, 6, 7, 7], [3, 6, 7, 8], [3, 6, 7, 9], [3, 6, 8, 8], [3, 6, 8, 9], [3, 6, 9, 9], [3, 7, 7, 7], [3, 7, 7, 8], [3, 7, 7, 9], [3, 7, 8, 8], [3, 7, 8, 9], [3, 7, 9, 9], [3, 8, 8, 8], [3, 8, 8, 9], [3, 8, 9, 9], [3, 9, 9, 9], [4, 4, 4, 4], [4, 4, 4, 5], [4, 4, 4, 6], [4, 4, 4, 7], [4, 4, 4, 8], [4, 4, 4, 9], [4, 4, 5, 5], [4, 4, 5, 6], [4, 4, 5, 7], [4, 4, 5, 8], [4, 4, 6, 8], [4, 4, 6, 9], [4, 4, 7, 7], [4, 4, 7, 8], [4, 4, 7, 9], [4, 4, 8, 8], [4, 4, 8, 9], [4, 5, 5, 5], [4, 5, 5, 6], [4, 5, 5, 7], [4, 5, 5, 8], [4, 5, 5, 9], [4, 5, 6, 6], [4, 5, 6, 7], [4, 5, 6, 8], [4, 5, 6, 9], [4, 5, 7, 7], [4, 5, 7, 8], [4, 5, 7, 9], [4, 5, 8, 8], [4, 5, 8, 9], [4, 5, 9, 9], [4, 6, 6, 6], [4, 6, 6, 7], [4, 6, 6, 8], [4, 6, 6, 9], [4, 6, 7, 7], [4, 6, 7, 8], [4, 6, 7, 9], [4, 6, 8, 8], [4, 6, 8, 9], [4, 6, 9, 9], [4, 7, 7, 7], [4, 7, 7, 8], [4, 7, 8, 8], [4, 7, 8, 9], [4, 7, 9, 9], [4, 8, 8, 8], [4, 8, 8, 9], [4, 8, 9, 9], [5, 5, 5, 5], [5, 5, 5, 6], [5, 5, 5, 9], [5, 5, 6, 6], [5, 5, 6, 7], [5, 5, 6, 8], [5, 5, 7, 7], [5, 5, 7, 8], [5, 5, 8, 8], [5, 5, 8, 9], [5, 5, 9, 9], [5, 6, 6, 6], [5, 6, 6, 7], [5, 6, 6, 8], [5, 6, 6, 9], [5, 6, 7, 7], [5, 6, 7, 8], [5, 6, 7, 9], [5, 6, 8, 8], [5, 6, 8, 9], [5, 6, 9, 9], [5, 7, 7, 9], [5, 7, 8, 8], [5, 7, 8, 9], [5, 8, 8, 8], [5, 8, 8, 9], [6, 6, 6, 6], [6, 6, 6, 8], [6, 6, 6, 9], [6, 6, 7, 9], [6, 6, 8, 8], [6, 6, 8, 9], [6, 7, 8, 9], [6, 7, 9, 9], [6, 8, 8, 8], [6, 8, 8, 9], [6, 8, 9, 9], [7, 8, 8, 9]]
复杂度比较
DFS递归：
时间复杂度：O(n^3)
空间复杂度：O(n^4)
超暴力思路
时间复杂度：O(nlogn)，其实就是sort()函数的时间复杂度。
空间复杂度：O(n^4)