<p>You are given an integer array <code>cards</code> of length <code>4</code>. You have four cards, each containing a number in the range <code>[1, 9]</code>. You should arrange the numbers on these cards in a mathematical expression using the operators <code>[&#39;+&#39;, &#39;-&#39;, &#39;*&#39;, &#39;/&#39;]</code> and the parentheses <code>&#39;(&#39;</code> and <code>&#39;)&#39;</code> to get the value 24.</p>

<p>You are restricted with the following rules:</p>

<ul>
	<li>The division operator <code>&#39;/&#39;</code> represents real division, not integer division.

	<ul>
		<li>For example, <code>4 / (1 - 2 / 3) = 4 / (1 / 3) = 12</code>.</li>
	</ul>
	</li>
	<li>Every operation done is between two numbers. In particular, we cannot use <code>&#39;-&#39;</code> as a unary operator.
	<ul>
		<li>For example, if <code>cards = [1, 1, 1, 1]</code>, the expression <code>&quot;-1 - 1 - 1 - 1&quot;</code> is <strong>not allowed</strong>.</li>
	</ul>
	</li>
	<li>You cannot concatenate numbers together
	<ul>
		<li>For example, if <code>cards = [1, 2, 1, 2]</code>, the expression <code>&quot;12 + 12&quot;</code> is not valid.</li>
	</ul>
	</li>
</ul>

<p>Return <code>true</code> if you can get such expression that evaluates to <code>24</code>, and <code>false</code> otherwise.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> cards = [4,1,8,7]
<strong>Output:</strong> true
<strong>Explanation:</strong> (8-4) * (7-1) = 24
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> cards = [1,2,1,2]
<strong>Output:</strong> false
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>cards.length == 4</code></li>
	<li><code>1 &lt;= cards[i] &lt;= 9</code></li>
</ul>

class Solution {

    bool valid(vector<int>& cards) {
        double a = cards[0], b = cards[1], c = cards[2],d = cards[3];
        if(valid(a+b,c,d) || valid(a-b,c,d) || valid(a*b,c,d) || valid(a/b,c,d)) return true;
        if(valid(a,b+c,d) || valid(a,b-c,d) || valid(a,b*c,d) || valid(a,b/c,d)) return true;
        if(valid(a,b,c+d) || valid(a,b,c-d) || valid(a,b,c*d) || valid(a,b,c/d)) return true;
        return false;
    }

    bool valid(double a, double b, double c) {
        if(valid(a+b,c) || valid(a-b,c) || valid(a*b,c) || valid(a/b,c)) return true;
        if(valid(a,b+c) || valid(a,b-c) || valid(a,b*c) || valid(a,b/c)) return true;
        return false;
    }

    bool valid(double a, double b) {
        if(abs(a+b-24.0) < 0.0001 || abs(a-b-24.0) < 0.0001 || abs(a*b-24.0) < 0.0001 || b&&abs(a/b-24.0) < 0.0001) return true;
        return false;
    }

public:
    bool judgePoint24(vector<int>& cards) {
        
        sort(cards.begin(), cards.end());
        do {
            if(valid(cards)) return true;
        } while(next_permutation(cards.begin(), cards.end()));
        return false;
    }
};
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