<p>Given a rectangle of size <code>n</code> x <code>m</code>, return <em>the minimum number of integer-sided squares that tile the rectangle</em>.</p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2019/10/17/sample_11_1592.png" style="width: 154px; height: 106px;" /></p>

<pre>
<strong>Input:</strong> n = 2, m = 3
<strong>Output:</strong> 3
<strong>Explanation:</strong> <code>3</code> squares are necessary to cover the rectangle.
<code>2</code> (squares of <code>1x1</code>)
<code>1</code> (square of <code>2x2</code>)</pre>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2019/10/17/sample_22_1592.png" style="width: 224px; height: 126px;" /></p>

<pre>
<strong>Input:</strong> n = 5, m = 8
<strong>Output:</strong> 5
</pre>

<p><strong class="example">Example 3:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2019/10/17/sample_33_1592.png" style="width: 224px; height: 189px;" /></p>

<pre>
<strong>Input:</strong> n = 11, m = 13
<strong>Output:</strong> 6
</pre>

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

<ul>
	<li><code>1 &lt;= n, m &lt;= 13</code></li>
</ul>

class Solution:
    def tilingRectangle(self, n: int, m: int) -> int:
        height = [0] * m
        q = []
        for i in range(min(m, n), 0, -1):
            heapq.heappush(q, (1, 1, -i, height))

        while q:
            guess, moves, neg_size, height = heapq.heappop(q)
            size = -neg_size
            idx = height.index(min(height))
            height = height[:]
            for i in range(size):
                height[idx + i] += size
            if all(h == n for h in height):
                return moves
            min_height = min(height)
            idx = height.index(min_height)
            ridx = idx + 1
            while ridx < m and height[ridx] == min_height:
                ridx += 1
            for i in range(min(ridx - idx, n - min_height), 0, -1):
                heapq.heappush(q, (moves + 1 + len(set(h for h in height if h < n)), 
                                moves + 1, -i, height))
17500