<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:
        def is_valid(x, y, k):
            for i in range(x, x + k):
                for j in range(y, y + k):
                    if filled[i][j]:
                        return False
            return True

        def place_square(x, y, k, val):
            for i in range(x, x + k):
                for j in range(y, y + k):
                    filled[i][j] = val

        def backtrack(num_squares):
            if num_squares >= min_squares[0]:
                return
            x, y = -1, -1
            for i in range(n):
                for j in range(m):
                    if not filled[i][j]:
                        x, y = i, j
                        break
                if x != -1:
                    break
            if x == -1:
                min_squares[0] = min(min_squares[0], num_squares)
                return

            k = min(n - x, m - y)
            while k > 0:
                if is_valid(x, y, k):
                    place_square(x, y, k, True)
                    backtrack(num_squares + 1)
                    place_square(x, y, k, False)
                k -= 1

        filled = [[False] * m for _ in range(n)]
        min_squares = [n * m]
        backtrack(0)
        return min_squares[0]

# Example usage:
solution = Solution()
print(solution.tilingRectangle(2, 3))  # Output: 3
print(solution.tilingRectangle(5, 8))  # Output: 5
print(solution.tilingRectangle(11, 13))  # Output: 6
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