Tiling a Rectangle with the Fewest Squares - Solution & Explanation

Use dynamic programming to solve the problem by maintaining a DP table where each entry dp[i][j] represents the minimum number of squares needed to tile a rectangle of size i x j. The base case is straightforward with small squares, and we build up from there by iterating through each rectangle size up to n x m.

This Python function defines a DP table where dp[i][j] gives the minimum number of tiles to tile a rectangle of size i x j. We iterate through rectangle sizes and fill this table based on smaller subproblems, considering potential square tiles within it.

This approach leverages recursion coupled with memorization to try placing the largest possible square within a rectangle and then recursively solve the remainder. The memorization helps optimize by storing previously computed results for specific rectangle dimensions.

This Python function recursively tries all possible square positions and sizes, and stores intermediate results in a memo dictionary to avoid recalculation. This significantly reduces the redundant computation.