<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:
        
        res = m * n
        tile = [[False] * m for _ in range(n)]

        def setTile(r, c, k, v):
            for i in range(k):
                for j in range(k):
                    tile[r + i][c + j] = v

        def dfs(r, c, cnt):
            nonlocal res
            if cnt >= res:
                return
            if r >= n:
                res = cnt
                return
            if c >= m:
                dfs(r + 1, 0, cnt)
                return
            if tile[r][c]:
                dfs(r, c + 1, cnt)
                return
            for k in range(min(n - r, m - c), 0, -1):
                if all(tile[r + i][c + j] == False for i in range(k) for j in range(k)):
                    setTile(r, c, k, True)
                    dfs(r, c + k, cnt + 1)
                    setTile(r, c, k, False)

        dfs(0, 0, 0)
        return res

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