Tiling a Rectangle with the Fewest Squares

class Solution:
    def tilingRectangle(self, n: int, m: int) -> int:
        def fill_missing(a: List[List[int]], n: int, ra: int) -> int:
            c = 0
            if n > 0:
                while ra > 0:
                    b = min(ra, n)
                    t = ra // b
                    # Multiple adjacent squares of same size
                    # can be treated as a rectangle of width b * t and height b
                    a.append([b * t, b])
                    ra -= b * t
                    c += t
            return c

        def find_rects_count(n, m, initial) -> int:
            a = [ [initial, initial] ]

            cnt = 1 + fill_missing(a, n, m - initial)

            while n > 0:
                min_n = min( x[1] for x in a )
                ra = m
                for i, item in zip(itertools.count(len(a)-1, -1), reversed(a)):
                    item[1] -= min_n
                    if item[1] == 0:
                        ra -= item[0]
                n -= min_n
                cnt += fill_missing(a, n, ra)

            return cnt

        boundary = min(n, m)
        r = n * m
        for b in range(boundary//2, boundary+1):
            r = min(r, find_rects_count(n, m, b))

        return r

Identifying Problem Isomorphism

“Perfect Squares” is an approximate isomorphism to “Tiling a Rectangle with the Fewest Squares”. Both involve finding an optimal way to represent a number or area with the fewest possible square numbers.

In “Perfect Squares”, the task is to determine the least number of perfect square numbers that sum to a given integer ’n’. A perfect square is an integer that is the square of an integer; for example 1, 4, 9, 16 are all perfect squares. For instance, given ’n’ as 12, the least number of perfect square numbers is 3, since 12 can be represented as 4 + 4 + 4.

In “Tiling a Rectangle with the Fewest Squares”, you have to tile a rectangle of width ’n’ and height ’m’ with the smallest number of perfect square tiles. This involves finding an optimal way to arrange square tiles such that the entire rectangle is covered with the fewest number of tiles.

While both problems deal with representing a given value using the least number of perfect squares, “Perfect Squares” is simpler due to its lesser complexity in the dimensions involved (1-D vs 2-D) and the more straightforward nature of the problem.

The main concept shared between these two problems is the optimal utilization of square numbers, which may involve concepts from number theory and dynamic programming.

10 Prerequisite LeetCode Problems

The “Tiling a Rectangle with the Fewest Squares” problem involves dynamic programming. Here are 10 problems to prepare:

  1. “Perfect Squares” (LeetCode Problem #279): This problem gives you a sense of how to break down a number into squares.

  2. “Unique Paths” (LeetCode Problem #62): This problem helps to understand dynamic programming in a grid, which can be helpful in “Tiling a Rectangle with the Fewest Squares”.

  3. “Minimum Path Sum” (LeetCode Problem #64): This is another dynamic programming problem that works with a grid and will give you practice on DP concepts.

  4. “Unique Paths II” (LeetCode Problem #63): This problem introduces obstacles, adding a layer of complexity to grid-based dynamic programming.

  5. “Coin Change” (LeetCode Problem #322): This problem will help you think about dynamic programming in terms of combinations and permutations, which can be helpful for the “Tiling a Rectangle with the Fewest Squares” problem.

  6. “Maximal Square” (LeetCode Problem #221): This problem gives you a sense of how to work with squares in a grid and find the maximum square.

  7. “Partition Equal Subset Sum” (LeetCode Problem #416): This problem introduces the concept of partitioning, which is a key component in the “Tiling a Rectangle with the Fewest Squares” problem.

  8. “Count Square Submatrices with All Ones” (LeetCode Problem #1277): This problem further practices the concept of finding squares in a grid.

  9. “Target Sum” (LeetCode Problem #494): This is a more complex dynamic programming problem that requires a bit more thinking in terms of combinations.

  10. “Longest Increasing Subsequence” (LeetCode Problem #300): This problem helps understand the concept of dynamic programming in terms of sequences, which will help in devising the DP solution for “Tiling a Rectangle with the Fewest Squares”.

Problem Classification

Problem Statement: Given a rectangle of size n x m, return the minimum number of integer-sided squares that tile the rectangle.

Example 1:

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

Example 2:

Input: n = 5, m = 8 Output: 5 Example 3:

Input: n = 11, m = 13 Output: 6


1 <= n, m <= 13

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Alternatively, if you’re working on a specific problem, you might ask something like:

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?


Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?


Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.


Similar Problems

Given the problem [provide the problem], identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.