Maximum Height of a Triangle - Practice Coding Problems

Maximum Height of a Triangle - Problem

Array Enumeration Easy

Imagine you're building a colorful ball pyramid! You have red and blue colored balls, and you want to stack them into a triangle where:

The 1st row has 1 ball
The 2nd row has 2 balls
The 3rd row has 3 balls
And so on...

Rules:

All balls in the same row must be the same color
Adjacent rows must have different colors

Given the count of red and blue balls, return the maximum height of the triangle you can build!

Example: With 2 red and 4 blue balls, you could build:
Row 1: 1 red ball
Row 2: 2 blue balls
Row 3: 3 red balls (but you only have 1 red left!)
So maximum height is 2.

Input & Output

example_1.py — Basic Case

$ Input: red = 2, blue = 4

› Output: 3

💡 Note: Starting with blue: Row 1 (1 blue), Row 2 (2 red), Row 3 (3 blue). Total height = 3. Starting with red only gives height 2.

example_2.py — Equal Balls

$ Input: red = 4, blue = 9

› Output: 3

💡 Note: Best pattern: Row 1 (1 red), Row 2 (2 blue), Row 3 (3 red). Uses 4 red and 2 blue balls. Height = 3.

example_3.py — Minimum Case

$ Input: red = 1, blue = 1

› Output: 1

💡 Note: Can only build 1 row with either 1 red ball or 1 blue ball. Height = 1.

Constraints

1 ≤ red, blue ≤ 100
Adjacent rows must have different colors
All balls in the same row must be the same color

Visualization

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Understanding the Visualization

🔴 Try Red First

Start with red balls in row 1, then blue in row 2, red in row 3...

🔵 Try Blue First

Start with blue balls in row 1, then red in row 2, blue in row 3...

📏 Compare Heights

Both approaches will give different heights - pick the maximum!

🏆 Optimal Solution

The winning pattern gives us the tallest possible pyramid

Key Takeaway

🎯 Key Insight: The alternating color constraint means we only need to check 2 scenarios - starting with red or starting with blue!

Asked in

G Google 15 a Amazon 12 f Meta 8 Apple 6

The key insight is recognizing there are only two possible patterns: starting with red or starting with blue. Since adjacent rows must alternate colors, we simulate both scenarios and return the maximum height achievable. The solution runs in O(√n) time where n is the total number of balls.

Common Approaches

Approach	Time	Space	Notes
✓ Simulation Approach	O(√n)	O(1)	Try both starting colors and simulate building the triangle

Simulation Approach — Algorithm Steps

Try starting with red balls in the first row
Alternate colors for each subsequent row (red→blue→red→...)
Keep track of remaining balls and stop when we can't fill a complete row
Try starting with blue balls and repeat the process
Return the maximum height from both attempts

Visualization

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Step-by-Step Walkthrough

Start with Red

Try building triangle starting with red balls in row 1

Alternate Colors

Place blue balls in row 2, red in row 3, etc.

Start with Blue

Try the opposite pattern: blue first, then red

Compare Heights

Return the maximum height achieved from both attempts

Code -

solution.c — C

#include <stdio.h>

int calculateHeight(int firstColor, int secondColor) {
    int height = 0;
    int row = 1;
    
    while (1) {
        if (row % 2 == 1) { // Odd rows use firstColor
            if (firstColor >= row) {
                firstColor -= row;
                height++;
            } else {
                break;
            }
        } else { // Even rows use secondColor
            if (secondColor >= row) {
                secondColor -= row;
                height++;
            } else {
                break;
            }
        }
        row++;
    }
    
    return height;
}

int maxHeightOfTriangle(int red, int blue) {
    int height1 = calculateHeight(red, blue);
    int height2 = calculateHeight(blue, red);
    return height1 > height2 ? height1 : height2;
}

int main() {
    int red, blue;
    scanf("%d %d", &red, &blue);
    printf("%d\n", maxHeightOfTriangle(red, blue));
    return 0;
}

Time & Space Complexity

Time Complexity

⏱️

O(√n)

We iterate through at most √(red+blue) rows since row i needs i balls, and the sum grows quadratically

✓ Linear Growth

Space Complexity

O(1)

Only using a few variables to track remaining balls and current height

✓ Linear Space

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Smart Actions

💡 Explanation

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Input & Output

Constraints

Visualization

Related Problems

Common Approaches

Simulation Approach — Algorithm Steps

Visualization

Code -

Time & Space Complexity

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