<p>Alice and Bob are opponents in an archery competition. The competition has set the following rules:</p>

<ol>
	<li>Alice first shoots <code>numArrows</code> arrows and then Bob shoots <code>numArrows</code> arrows.</li>
	<li>The points are then calculated as follows:
	<ol>
		<li>The target has integer scoring sections ranging from <code>0</code> to <code>11</code> <strong>inclusive</strong>.</li>
		<li>For <strong>each</strong> section of the target with score <code>k</code> (in between <code>0</code> to <code>11</code>), say Alice and Bob have shot <code>a<sub>k</sub></code> and <code>b<sub>k</sub></code> arrows on that section respectively. If <code>a<sub>k</sub> &gt;= b<sub>k</sub></code>, then Alice takes <code>k</code> points. If <code>a<sub>k</sub> &lt; b<sub>k</sub></code>, then Bob takes <code>k</code> points.</li>
		<li>However, if <code>a<sub>k</sub> == b<sub>k</sub> == 0</code>, then <strong>nobody</strong> takes <code>k</code> points.</li>
	</ol>
	</li>
</ol>

<ul>
	<li>
	<p>For example, if Alice and Bob both shot <code>2</code> arrows on the section with score <code>11</code>, then Alice takes <code>11</code> points. On the other hand, if Alice shot <code>0</code> arrows on the section with score <code>11</code> and Bob shot <code>2</code> arrows on that same section, then Bob takes <code>11</code> points.</p>
	</li>
</ul>

<p>You are given the integer <code>numArrows</code> and an integer array <code>aliceArrows</code> of size <code>12</code>, which represents the number of arrows Alice shot on each scoring section from <code>0</code> to <code>11</code>. Now, Bob wants to <strong>maximize</strong> the total number of points he can obtain.</p>

<p>Return <em>the array </em><code>bobArrows</code><em> which represents the number of arrows Bob shot on <strong>each</strong> scoring section from </em><code>0</code><em> to </em><code>11</code>. The sum of the values in <code>bobArrows</code> should equal <code>numArrows</code>.</p>

<p>If there are multiple ways for Bob to earn the maximum total points, return <strong>any</strong> one of them.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/24/ex1.jpg" style="width: 600px; height: 120px;" />
<pre>
<strong>Input:</strong> numArrows = 9, aliceArrows = [1,1,0,1,0,0,2,1,0,1,2,0]
<strong>Output:</strong> [0,0,0,0,1,1,0,0,1,2,3,1]
<strong>Explanation:</strong> The table above shows how the competition is scored. 
Bob earns a total point of 4 + 5 + 8 + 9 + 10 + 11 = 47.
It can be shown that Bob cannot obtain a score higher than 47 points.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/02/24/ex2new.jpg" style="width: 600px; height: 117px;" />
<pre>
<strong>Input:</strong> numArrows = 3, aliceArrows = [0,0,1,0,0,0,0,0,0,0,0,2]
<strong>Output:</strong> [0,0,0,0,0,0,0,0,1,1,1,0]
<strong>Explanation:</strong> The table above shows how the competition is scored.
Bob earns a total point of 8 + 9 + 10 = 27.
It can be shown that Bob cannot obtain a score higher than 27 points.
</pre>

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

<ul>
	<li><code>1 &lt;= numArrows &lt;= 10<sup>5</sup></code></li>
	<li><code>aliceArrows.length == bobArrows.length == 12</code></li>
	<li><code>0 &lt;= aliceArrows[i], bobArrows[i] &lt;= numArrows</code></li>
	<li><code>sum(aliceArrows[i]) == numArrows</code></li>
</ul>

class Solution:
    def maximumBobPoints(self, numArrows: int, aliceArrows: List[int]) -> List[int]:
        # Bob does not shoot at 0 point
        # Bob either shoots exactly one more arrow than alice to win a point or
        # does not shoot at all to sacrifice the point
        # bob's decision can given as bit vector with 11 bits or up to 2**11. 
        # Note we can ignore 0 points for Bob does not shoot there at all

        def points_obtained(bit_vector):
            # points obtained where bob's result is encoded with bit_vector
            # if bob does not have enough arraows for this it returns -1
            pt = 0
            arrows = 0
            for b in range(12):
                if bit_vector & (1 << b):
                    pt += (b+1)
                    arrows += (aliceArrows[b+1] + 1)
                    if arrows > numArrows:
                        return -1
            return pt
        
        max_pt = 0
        max_bit_vector = 0
        for bit_vector in range(1<<11):
            pts_ob = points_obtained(bit_vector)
            # print(f"vector {bit_vector}, points {pts_ob}")
            if pts_ob > max_pt:
                max_pt, max_bit_vector = pts_ob, bit_vector
        # now encode bit vector as number of arrows bob needs to shoot
        bobArrows = [0]*12
        remArrows = numArrows
        for b in range(max_bit_vector.bit_length()):
            if max_bit_vector & (1 << b):
                bobArrows[b+1] = aliceArrows[b+1] + 1
                remArrows -= bobArrows[b+1]
        # add remaining arrow if any to maximum 1 bit
        bobArrows[max_bit_vector.bit_length()] += remArrows
        return bobArrows






        
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