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Leetcode 3200. Maximum Height of a Triangle

Problem Statement

You are given an input integer N, which represents the number of units of material you can use to build a triangle. You need to determine the maximum height of a triangle that you can construct using these units, where every additional height level i (starting from 1) requires exactly i units to construct.

Clarifying Questions

  1. Input Constraints:
    • What is the range of the input integer N?
    • Is there a minimum value for N?
  2. Output:
    • Are we supposed to return the height as an integer?

Assuming the constraints like most problems of this nature:

Strategy

  1. Understanding the Problem:
    • We are dealing with a problem where adding levels to the triangle requires an increasing number of units.
    • Assuming we want to build a triangle with height h, the total number of units required will be the sum of the first h natural numbers.
      • This can be calculated using the formula for the sum of the first h natural numbers: (\text{Sum} = \frac{h(h + 1)}{2})
  2. Finding the Maximum Height:
    • We want to maximize the height h such that (\frac{h(h + 1)}{2} \leq N).
    • Rearranging gives a quadratic inequality in terms of h: [ h^2 + h - 2N \leq 0 ]
    • We can solve this quadratic equation to find the maximum height.
  3. Implementation Plan:
    • Use the quadratic formula ( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), for this specific case.
      • Here ( a = 1 ), ( b = 1 ), and ( c = -2N ).
    • The feasible solution for h will be within the range of: [ h = \frac{-1 + \sqrt{1 + 8N}}{2} ]

Time Complexity

Code

public class MaximumHeightOfTriangle {
    public static int maximumHeight(int N) {
        if (N == 0) return 0;

        // Calculate the discriminant of the quadratic equation
        double discriminant = 1 + 8 * (double) N;

        // Calculate the positive root using the quadratic formula
        double h = (-1 + Math.sqrt(discriminant)) / 2;

        // The height must be an integer, so we take the floor value of h
        return (int) Math.floor(h);
    }

    public static void main(String[] args) {
        int N = 10;
        System.out.println("Maximum height of a triangle with " + N + " units is: " + maximumHeight(N));
    }
}

Explanation of Code

  1. Special Case: If ( N ) is 0, return 0 because no triangle can be formed.
  2. Discriminant: Compute the discriminant of the quadratic equation, which in this case is (1 + 8N).
  3. Quadratic Formula: Compute the height using the rearranged quadratic formula and take the floor value to ensure we work with whole units.
  4. Result: Return the floor value of computed h to comply with the integer height requirement.

This approach ensures that we can efficiently find the maximum height within the constraints.

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