Construct the Rectangle

To solve this problem, we need to find a pair of factors (length and width) for the given area such that the difference between them is as small as possible. The best way to do this is to start from the square root of the area and then go down to 1 until we find a number that can divide the area without any remainder. This number will be our width, and the length will be the result of the area divided by the width.

Here is the Python code:

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from math import sqrt

class Solution:
    def constructRectangle(self, area: int) -> List[int]:
        width = int(sqrt(area))
        while area % width != 0:
            width -= 1
        length = area // width
        return [length, width]

In the code above, we use the sqrt function from the math module to get the square root of the area, which will be the initial value of the width. Then we enter a while loop that continues until we find a width that divides the area with no remainder. Inside the loop, we decrement the width by 1 on each iteration. Once we find the correct width, we calculate the length by integer division of the area by the width, and finally return a list with the length and width in this order.

Identifying Problem Isomorphism

“Construct the Rectangle” can be mapped to “Perfect Squares”.

Both problems involve considering factors of a given number. “Construct the Rectangle” requires you to find the length and width of a rectangle given its area. This involves understanding the factors of the area to find a pair that would result in minimum perimeter. In “Perfect Squares”, the objective is to find the least number of perfect square numbers that sum to a given number n, which also involves examining factors, but in a different way.

“Perfect Squares” is more complex due to its requirement for dynamic programming in order to find an optimal solution, while “Construct the Rectangle” requires more straightforward mathematical operations to achieve its result.

“Construct the Rectangle” requires knowledge of basic math, logic, and optimization. Below are simpler problems to build the skills necessary to solve it:

  1. 69. Sqrt(x): This problem will help you understand how to handle square roots, which are useful in the target problem for finding the sides of a square.

  2. 367. Valid Perfect Square: This problem further reinforces the understanding of squares and square roots.

  3. 258. Add Digits: This problem deals with digit manipulation, and while it doesn’t directly relate to the target problem, it helps improve problem-solving skills with numerical problems.

  4. 441. Arranging Coins: This problem involves the concept of arrangement which can provide insight for the target problem.

  5. 628. Maximum Product of Three Numbers: This problem requires finding optimal solutions from given numbers, similar to finding the optimal sides in the target problem.

  6. 204. Count Primes: Understanding prime numbers and how to find them can aid in understanding factors and divisors, which can be helpful in the target problem.

  7. 231. Power of Two: Understanding the properties of numbers and their factors can be helpful in solving the target problem.

  8. 292. Nim Game: This problem involves game theory and making optimal moves, similar to finding the optimal sides in the target problem.

  9. 507. Perfect Number: This problem helps in understanding the properties of perfect numbers and divisors.

  10. 415. Add Strings: This problem, though not directly related, deals with the manipulation of numbers as strings and can improve your numerical problem-solving skills.

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class Solution:
    def constructRectangle(self, area: int) -> List[int]:
        for l in range(int(area**0.5), 0, -1):            
            if area % l == 0: 
                return [area // l, l]

Problem Classification

This problem falls under the domain of optimization and number theory. It also relates to Web Design in a real-world context, but the primary problems to solve are mathematical.

The ‘What’ Components:

  • Input: The given input is a target area for a rectangular web page.
  • Output: The output must be an array that contains two values - the length (L) and width (W) of the web page.
  • Constraints:
    • The area of the web page must equal the target area.
    • The width (W) should not be larger than the length (L).
    • The difference between L and W should be as small as possible to minimize the difference between the dimensions.

We can classify this problem as an optimization problem in the field of Mathematics. The goal is to find the optimal dimensions of a rectangle that satisfy several conditions (a given area, a constraint on the relative sizes of length and width, and a minimization of the difference between length and width). It is also a problem of factoring in number theory. The length and width that satisfy the conditions are factors of the target area. Despite the context of Web Design, the problem does not involve the practicalities of web page design (e.g., usability, aesthetics), focusing solely on mathematical properties of the web page’s dimensions. Therefore, the problem is purely mathematical, not practical or design-oriented.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

Language Agnostic Coding Drills

  1. Dissecting the Code & Identifying Concepts:

    • Variables & Basic Arithmetic: Understanding of variable assignment and basic arithmetic operations is required to understand how the ’l’ and ‘area’ variables are used and manipulated.

    • List Creation & Assignment: The return statement creates a list of two integers and returns it, so understanding of list data structures is necessary.

    • Square Root Calculation: The square root operation (area**0.5) is used to calculate the starting point for ’l’. This requires knowledge of both exponentiation and its application in finding square roots.

    • Looping Constructs: The for loop is used to iterate over a range of values in reverse order, requiring understanding of how loops and the range() function work.

    • Conditional Statements: An if statement is used to check whether the area is divisible by ’l’, requiring understanding of conditional statements and the modulus operation (%).

  2. Listing Concepts by Difficulty:

    • Basic Arithmetic & Variable Assignment: This is a fundamental concept in programming and forms the base of understanding more complex concepts.

    • List Creation & Assignment: This concept builds on variable assignment and introduces the idea of data structures.

    • Looping Constructs: This is a slightly more complex concept that involves control flow and the generation of a sequence of numbers.

    • Square Root Calculation: This concept involves applying mathematical operations to achieve specific results.

    • Conditional Statements: This is a critical concept that allows for decision-making in code based on certain conditions.

  3. Problem-Solving Approach:

    The problem is solved by iterating over possible lengths for the rectangle, starting from the square root of the area and decreasing. The square root is chosen as the starting point because it’s the largest possible value that the width (and thus also the length) could have while still satisfying the constraints.

    In the loop, the program checks if the current length evenly divides the area (which means it’s a possible dimension for the rectangle). If it is, it calculates the corresponding width and returns the two values as a list. The first valid pair found will have the smallest difference between length and width due to the order of iteration (from larger to smaller values).

    Each of the identified concepts plays a part in this solution. Variables are used to hold and manipulate the input area, arithmetic is used to calculate potential lengths and widths, a list is used to store and return the final result, a loop is used to iterate through potential lengths, and a conditional statement is used to check if a length is valid. The specific application of these general programming concepts to this problem leads to the final solution.

Targeted Drills in Python

  1. Python Drills for each Concept:

    • Basic Arithmetic & Variable Assignment: Assign values to different variables and perform basic arithmetic operations such as addition, subtraction, multiplication, division, and modulus.

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      x = 10
y = 20
sum = x + y
difference = x - y
product = x * y
quotient = x / y
remainder = x % y

    • List Creation & Assignment: Create lists with different data types, access elements from lists, and modify elements in lists.

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      my_list = [1, 2, 3, "Hello", 3.14]
first_element = my_list[0]
last_element = my_list[-1]
my_list[3] = "World"

    • Looping Constructs: Use loops to iterate over a range of numbers or elements in a list.

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      for i in range(10):
    print(i)

    • Square Root Calculation: Use the math library’s sqrt function or the exponentiation operator to calculate the square root of a number.

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      import math
num = 25
sqrt_num = math.sqrt(num)
# or using exponentiation
sqrt_num = num**0.5

    • Conditional Statements: Use if, elif, and else statements to control the flow of the program based on different conditions.

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      num = 10
if num > 0:
    print("Positive number")
elif num == 0:
    print("Zero")
else:
    print("Negative number")

  2. Drills for Specific Needs of the Problem:

    • Finding Factors of a Number: The problem requires finding pairs of factors of the given area, so you should understand how to find factors of a number in Python.

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      num = 24
for i in range(1, num + 1):
    if num % i == 0:
        print(i)

    • Descending Loop: The problem involves looping from a high number down to a lower number, which is a less common use of range() but important for this problem.

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      for i in range(10, 0, -1):
    print(i)

  3. Merging the Drills for the Final Solution:

    Starting with the basic arithmetic and variable assignments, we set up our initial variables including the area and the starting point for our loop. We then move to our descending loop construct, where we iterate from the square root of the area down to 1. Within this loop, we apply our conditional statements to check if the current number is a factor of the area. If it is, we use list creation to put our length and width into a list and return it.

    With each of these drills, we build up to the solution step-by-step, combining general Python concepts with problem-specific requirements. The drills provide the fundamental tools we need to create the final solution, while the specific application of these tools to this problem’s requirements is what ultimately solves it.