<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:
    n = len(cookies)
    # Each child gets one cookie.
    if n == k:
      return max(cookies)

    dp = {}
    done_mask = (1 << n) - 1
    def helper(mask, cur_sum, cur_child, cur_cookie):
      # Early return: some children don't have any cookies (must not be an optimal solution).
      if mask == done_mask and cur_child < k - 1:
        return math.inf
      # Only one child left and we assign the remaining cookies to this child.
      if cur_child == k-1:
        last_sum = cur_sum
        for i in range(n):
          if mask & (1 << i) == 0:
            last_sum += cookies[i]
        # print("last_sum: ", last_sum)
        return last_sum

      state = (mask, cur_sum, cur_child)
      if state in dp:
        return dp[state]

      ans = math.inf
      for i in range(cur_cookie, n):
        if mask & (1 << i):
          continue
        next_mask = mask | (1 << i)
        # Assign to the current child.
        ans1 = max(helper(next_mask, cur_sum + cookies[i], cur_child, cur_cookie+1), cur_sum)
        ans2 = max(helper(next_mask, cookies[i], cur_child+1, 1), cur_sum)  # Start to assign to the next child.
        ans = min(ans, ans1, ans2)
        # print("ans1: ", ans1, ", ans2: ", ans2, ", ans: ", ans)
      dp[state] = ans
      return ans

    return helper(0, 0, 0, 0)
150