Arranging Coins

Arranging Coins - Problem

You have n coins and you want to build a staircase with these coins. The staircase consists of k rows where the i-th row has exactly i coins.

The last row of the staircase may be incomplete.

Given the integer n, return the number of complete rows of the staircase you will build.

Input & Output

Example 1 — Basic Case

$ Input: n = 5

› Output: 2

💡 Note: Row 1: 1 coin (4 left), Row 2: 2 coins (2 left), Row 3 needs 3 coins but only 2 left. So 2 complete rows.

Example 2 — Perfect Triangle

$ Input: n = 3

› Output: 2

💡 Note: Row 1: 1 coin (2 left), Row 2: 2 coins (0 left). Exactly 2 complete rows with no coins remaining.

Example 3 — Large Number

$ Input: n = 8

› Output: 3

💡 Note: Row 1: 1 coin, Row 2: 2 coins, Row 3: 3 coins (total 6), Row 4 needs 4 but only 2 left. So 3 complete rows.

Constraints

1 ≤ n ≤ 2³¹ - 1

Visualization

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Asked in

G Google 15 a Amazon 12 M Microsoft 8

The key insight is recognizing this as finding the largest k where k*(k+1)/2 ≤ n. Best approach uses the mathematical formula k = ⌊(-1 + √(1 + 8n))/2⌋ for O(1) time. Binary search provides an intuitive O(log n) alternative.

Common Approaches

✓ Optimized

⏱️ Time: N/A Space: N/A

Binary Search on Answer

⏱️ Time: O(log n) Space: O(1)

Since the answer is between 1 and n, we can binary search on this range. For each candidate k, check if k*(k+1)/2 ≤ n.

Mathematical Formula

⏱️ Time: O(1) Space: O(1)

The sum of first k natural numbers is k*(k+1)/2. We need to find the largest k such that k*(k+1)/2 ≤ n, which can be solved using the quadratic formula.

Row by Row Counting

⏱️ Time: O(√n) Space: O(1)

Start with row 1 (needs 1 coin), then row 2 (needs 2 coins), and so on. Keep subtracting coins until we can't complete a full row.

Algorithm Steps — Algorithm Steps

Code -

solution.c — C

Time & Space Complexity

Time Complexity

⏱️

✓ Linear Growth

Space Complexity

✓ Linear Space

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Smart Actions

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