Tiling a Rectangle with the Fewest Squares

Tiling a Rectangle with the Fewest Squares

What is this problem about?

Imagine you have a rectangular floor of dimensions n by m. You want to cover the entire floor with the minimum number of square tiles. The tiles can be of any integer size, but they cannot overlap or extend beyond the boundaries of the rectangle. This is a classic optimization problem that is surprisingly difficult to solve for all cases.

Why is this asked in interviews?

This tiling a rectangle with the fewest squares interview question is a hard-level backtracking problem asked by Amazon, Google, and Bloomberg. It tests your ability to explore a complex state space and use pruning to find an optimal solution. It evaluates your recursion skills and whether you can identify that a simple greedy approach (always placing the largest possible square) does not always yield the minimum number of tiles.

Algorithmic pattern used

The problem utilizes the Backtracking interview pattern.

1. Use a recursive function to fill the rectangle. The state can be represented as the current heights of each column of the floor.
2. Find the column with the minimum height. This is where you should attempt to place the next square.
3. Try placing squares of all possible sizes at this location (from the largest possible down to 1).
4. For each size, update the state and recurse.
5. Pruning: If the current number of squares used is already greater than or equal to the best result found so far, terminate that branch of the search.
6. The base case is when all columns reach the target height n.

Example explanation

Rectangle 11x13.

• A greedy approach might use a 11x11 square and then try to fill the remaining 11x2 area with 1x1 squares. Total = 1 + 22 = 23.
• However, there's a more optimal way to tile an 11x13 rectangle that uses only 6 squares (this is a famous case in tiling theory). Backtracking explores these non-greedy arrangements by trying different square sizes at each "lowest" point.

Common mistakes candidates make

In "Tiling a Rectangle with the Fewest Squares coding problem," the most common mistake is assuming that a greedy approach works. Another error is not efficiently representing the "filled" state of the rectangle, which can make the backtracking extremely slow. Finally, failing to implement effective pruning will often result in a "Time Limit Exceeded" error on larger test cases.

Interview preparation tip

Backtracking is often used for "exact cover" or "tiling" problems. Practice representing the state of a 2D grid effectively. Learning about bitmasking for small grids or height-arrays for larger ones can be very beneficial. Always remember that optimization problems often require exploring many possibilities before confirming the best one.