如何运用算法实现“24点”数学游戏？

发布时间：2020-05-27 16:23:07 作者：鸽子
来源：亿速云阅读：302

“24点”是一种数学游戏，正如象棋、围棋一样是一种人们喜闻乐见的娱乐活动。它始于何年何月已无从考究，但它以自己独具的数学魅力和丰富的内涵正逐渐被越来越多的人们所接受。今天就为大家分享一道关于“24点”的算法题目。

话不多说，直接看题。

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 。

题目分析

拿到题目，第一反应就可以想到暴li求解。如果我们要判断给出的4张牌是否可以通过组合得到24，那我们只需找出所有的可组合的方式进行遍历。

4个数字，3个操作符，外加括号，基本目测就能想到组合数不会大到超出边界。所以，我们只要把他们统统列出来，不就可以进行求解了吗？说干就干！

我们首先定义个方法，用来判断两个数的的所有操作符组合是否可以得到24。

func judgePoint24_2(a, b float64) bool {
    return a+b == 24 || a*b == 24 || a-b == 24 || b-a == 24 || a/b == 24 || b/a == 24 
}

但是这个方法写的正确吗？其实不对！因为在计算机中，实数在计算和存储过程中会有一些微小的误差，对于一些与零作比较的语句来说，有时会因误差而导致原本是等于零但结果却小于或大于零之类的情况发生，所以常用一个很小的数 1e-6 代替 0，进行判读！

(1e-6：表示1乘以10的负6次方。Math.abs(x)<1e-6 其实相当于x==0。1e-6(也就是0.000001)叫做epslon，用来抵消浮点运算中因为误差造成的相等无法判断的情况。这个知识点需要掌握！)

举个例子：

func main() {
    var a float64
    var b float64
    b = 2.0
    //math.Sqrt：开平方根
    c := math.Sqrt(2)
    a = b - c*c
    fmt.Println(a == 0)                  //false
    fmt.Println(a < 1e-6 && a > -(1e-6)) //true
}

这里直接用 a==0 就会得到false，但是通过 a < 1e-6 && a > -(1e-6) 却可以进行准确的判断。

所以我们将上面的方法改写：

 //go语言
 //judgePoint24_2：判断两个数的所有操作符组合是否可以得到24
 func judgePoint24_2(a, b float64) bool {
     return (a+b < 24+1e-6 && a+b > 24-1e-6) ||
         (a*b < 24+1e-6 && a*b > 24-1e-6) ||
         (a-b < 24+1e-6 && a-b > 24-1e-6) ||
         (b-a < 24+1e-6 && b-a > 24-1e-6) ||
         (a/b < 24+1e-6 && a/b > 24-1e-6) ||
         (b/a < 24+1e-6 && b/a > 24-1e-6) 
}

完善了通过两个数来判断是否可以得到24的方法，现在我们加一个判断三个数是否可以得到24的方法。

//硬核代码，不服来辩！
func judgePoint24_3(a, b, c float64) bool {
    return judgePoint24_2(a+b, c) ||
        judgePoint24_2(a-b, c) ||
        judgePoint24_2(a*b, c) ||
        judgePoint24_2(a/b, c) ||
        judgePoint24_2(b-a, c) ||
        judgePoint24_2(b/a, c) ||

        judgePoint24_2(a+c, b) ||
        judgePoint24_2(a-c, b) ||
        judgePoint24_2(a*c, b) ||
        judgePoint24_2(a/c, b) ||
        judgePoint24_2(c-a, b) ||
        judgePoint24_2(c/a, b) ||

        judgePoint24_2(c+b, a) ||
        judgePoint24_2(c-b, a) ||
        judgePoint24_2(c*b, a) ||
        judgePoint24_2(c/b, a) ||
        judgePoint24_2(b-c, a) ||
        judgePoint24_2(b/c, a)
}

好了。三个数的也出来了，我们再加一个判断4个数为24点的方法：（排列组合，我想大家都会....）

前方高能！！！

//硬核代码，不服来辩！
func judgePoint24(nums []int) bool {
    return judgePoint24_3(float64(nums[0])+float64(nums[1]), float64(nums[2]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[0])-float64(nums[1]), float64(nums[2]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[0])*float64(nums[1]), float64(nums[2]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[0])/float64(nums[1]), float64(nums[2]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[1])-float64(nums[0]), float64(nums[2]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[1])/float64(nums[0]), float64(nums[2]), float64(nums[3])) ||

        judgePoint24_3(float64(nums[0])+float64(nums[2]), float64(nums[1]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[0])-float64(nums[2]), float64(nums[1]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[0])*float64(nums[2]), float64(nums[1]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[0])/float64(nums[2]), float64(nums[1]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[2])-float64(nums[0]), float64(nums[1]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[2])/float64(nums[0]), float64(nums[1]), float64(nums[3])) ||

        judgePoint24_3(float64(nums[0])+float64(nums[3]), float64(nums[2]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[0])-float64(nums[3]), float64(nums[2]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[0])*float64(nums[3]), float64(nums[2]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[0])/float64(nums[3]), float64(nums[2]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[3])-float64(nums[0]), float64(nums[2]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[3])/float64(nums[0]), float64(nums[2]), float64(nums[1])) ||

        judgePoint24_3(float64(nums[2])+float64(nums[3]), float64(nums[0]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[2])-float64(nums[3]), float64(nums[0]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[2])*float64(nums[3]), float64(nums[0]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[2])/float64(nums[3]), float64(nums[0]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[3])-float64(nums[2]), float64(nums[0]), float64(nums[1])) ||
        judgePoint24_3(float64(nums[3])/float64(nums[2]), float64(nums[0]), float64(nums[1])) ||

        judgePoint24_3(float64(nums[1])+float64(nums[2]), float64(nums[0]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[1])-float64(nums[2]), float64(nums[0]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[1])*float64(nums[2]), float64(nums[0]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[1])/float64(nums[2]), float64(nums[0]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[2])-float64(nums[1]), float64(nums[0]), float64(nums[3])) ||
        judgePoint24_3(float64(nums[2])/float64(nums[1]), float64(nums[0]), float64(nums[3])) ||

        judgePoint24_3(float64(nums[1])+float64(nums[3]), float64(nums[2]), float64(nums[0])) ||
        judgePoint24_3(float64(nums[1])-float64(nums[3]), float64(nums[2]), float64(nums[0])) ||
        judgePoint24_3(float64(nums[1])*float64(nums[3]), float64(nums[2]), float64(nums[0])) ||
        judgePoint24_3(float64(nums[1])/float64(nums[3]), float64(nums[2]), float64(nums[0])) ||
        judgePoint24_3(float64(nums[3])-float64(nums[1]), float64(nums[2]), float64(nums[0])) ||
        judgePoint24_3(float64(nums[3])/float64(nums[1]), float64(nums[2]), float64(nums[0]))
}

Go语言示例

搞定收工，我们整合全部代码如下：