How do you solve Arranging Coins on LeetCode?

Arranging Coins (LeetCode #441) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(1) time complexity and O(1) space complexity. Key insight: Understanding the relationship between the number of coins and the rows formed is crucial.

What is the brute force solution for Arranging Coins?

The brute force approach for Arranging Coins is Brute Force, with O(n²) time complexity and O(1) space complexity. We can build the staircase row by row, subtracting the required coins for each row from the total until we run out of coins. This approach is straightforward but inefficient for large inputs. The optimal Optimal Solution approach improves this to O(1).

What algorithm or data structure is used to solve Arranging Coins?

Arranging Coins uses the following concepts: Math, Binary Search. The recommended patterns to study are: Mathematical Formulas, Quadratic Equations.

What are the interview tips for Arranging Coins?

Always start with a brute-force approach to demonstrate your understanding of the problem. Look for patterns or mathematical relationships in the problem to optimize your solution. Explain your thought process clearly, especially when transitioning from a brute-force to an optimal solution.

What are common mistakes when solving Arranging Coins?

Not considering the case where the last row is incomplete. Failing to recognize that the problem can be solved using a mathematical approach rather than brute force.

#441

Arranging Coins

Easy

MathBinary SearchMathematical FormulasQuadratic Equations

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(1)
Space	O(1)	O(1)

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Intuition

Time O(1)Space O(1)

Instead of building the staircase row by row, we can use a mathematical approach to find the maximum number of complete rows that can be formed with 'n' coins using the formula for the sum of the first k natural numbers.

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Algorithm

3 steps

1Step 1: Use the formula k(k + 1) / 2 <= n to find the maximum k.
2Step 2: Rearrange it to k² + k - 2n <= 0 and apply the quadratic formula.
3Step 3: Calculate k = (-1 + sqrt(1 + 8n)) / 2 and return the integer part of k.

solution.py5 lines

1# Full working Python code
2import math
3n = 5
4k = int((-1 + math.sqrt(1 + 8 * n)) // 2)
5print(k)

ℹ

Complexity note: The time complexity is O(1) because we are performing a constant number of arithmetic operations and a square root calculation, regardless of the size of 'n'.

1Understanding the relationship between the number of coins and the rows formed is crucial.
2Using mathematical formulas can significantly reduce the complexity of the solution.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.