<p>You are given an integer array <code>cookies</code>, where <code>cookies[i]</code> denotes the number of cookies in the <code>i<sup>th</sup></code> bag. You are also given an integer <code>k</code> that denotes the number of children to distribute <strong>all</strong> the bags of cookies to. All the cookies in the same bag must go to the same child and cannot be split up.</p>

<p>The <strong>unfairness</strong> of a distribution is defined as the <strong>maximum</strong> <strong>total</strong> cookies obtained by a single child in the distribution.</p>

<p>Return <em>the <strong>minimum</strong> unfairness of all distributions</em>.</p>

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

<pre>
<strong>Input:</strong> cookies = [8,15,10,20,8], k = 2
<strong>Output:</strong> 31
<strong>Explanation:</strong> One optimal distribution is [8,15,8] and [10,20]
- The 1<sup>st</sup> child receives [8,15,8] which has a total of 8 + 15 + 8 = 31 cookies.
- The 2<sup>nd</sup> child receives [10,20] which has a total of 10 + 20 = 30 cookies.
The unfairness of the distribution is max(31,30) = 31.
It can be shown that there is no distribution with an unfairness less than 31.
</pre>

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

<pre>
<strong>Input:</strong> cookies = [6,1,3,2,2,4,1,2], k = 3
<strong>Output:</strong> 7
<strong>Explanation:</strong> One optimal distribution is [6,1], [3,2,2], and [4,1,2]
- The 1<sup>st</sup> child receives [6,1] which has a total of 6 + 1 = 7 cookies.
- The 2<sup>nd</sup> child receives [3,2,2] which has a total of 3 + 2 + 2 = 7 cookies.
- The 3<sup>rd</sup> child receives [4,1,2] which has a total of 4 + 1 + 2 = 7 cookies.
The unfairness of the distribution is max(7,7,7) = 7.
It can be shown that there is no distribution with an unfairness less than 7.
</pre>

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

<ul>
	<li><code>2 &lt;= cookies.length &lt;= 8</code></li>
	<li><code>1 &lt;= cookies[i] &lt;= 10<sup>5</sup></code></li>
	<li><code>2 &lt;= k &lt;= cookies.length</code></li>
</ul>

class Solution:
    def distributeCookies(self, cookies: List[int], k: int) -> int:
        # total_cookies = [0]*k
        # n = len(cookies)
        # min_unfairness = math.inf
        # # @lru_cache(None)
        # def backtracking(cookie_index, total_cookies):
        #     nonlocal min_unfairness
        #     if cookie_index == n:
        #         unfairness = max(total_cookies)
        #         min_unfairness = unfairness
        #         return
        #     for i in range(k):
        #         if total_cookies[i]+cookies[cookie_index] <= min_unfairness:
        #             total_cookies[i] += cookies[cookie_index]
        #             backtracking(cookie_index+1, total_cookies)
        #             total_cookies[i] -= cookies[cookie_index]
        # backtracking(0, total_cookies)
        # return min_unfairness


        n = len(cookies)
        total_sum = sum(cookies)
        total_cookies = [0]*k
        min_unfairness = math.inf
        @lru_cache(None)
        def backtracking(mask, curr_unfairness):
            nonlocal min_unfairness
            if mask == (1 << n) - 1:
                min_unfairness = min(min_unfairness, curr_unfairness)
                return curr_unfairness
            if curr_unfairness >= min_unfairness:
                return curr_unfairness
            for i in range(n):
                if not (mask & (1 << i)):
                    for child in range(k):
                        new_mask = mask | (1 << i)
                        total_cookies[child] += cookies[i]
                        if total_cookies[child] <= min_unfairness:
                            backtracking(new_mask, max(curr_unfairness, total_cookies[child]))
                        total_cookies[child] -= cookies[i]
                    if min_unfairness == total_sum // k:
                        return min_unfairness
            return min_unfairness
        backtracking(0, 0)
        return min_unfairness

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