Container With Most Water — Beginner Two Pointer Breakdown (That “Ohhh” Moment)

When I first saw Container With Most Water, I froze.

The problem looked simple.
The brute force was easy.
But the optimized solution?

After seeing it, I had only one reaction:

“Ohhh… so THAT’S why the pointer moves.”

If you’re struggling with this problem — you’re in the right place.

🧩 Problem: Container With Most Water

Description

You are given an array where each element represents the height of a vertical line.

Choose two lines such that they form a container holding the maximum amount of water.

Example

Input:  [1,8,6,2,5,4,8,3,7]
Output: 49

🧠 First Thought (Every Beginner’s Brain)

👉 “Check every pair and calculate area.”

That works… but it’s O(n²).

Interviewers expect O(n).

So we need a smarter approach.

🔍 Key Observation (THIS is the breakthrough)

The area depends on two things:

Area = min(height[left], height[right]) × (right - left)

So:

Width → distance between pointers

Height → shorter line

👉 The shorter line limits the water.

This one sentence changes everything.

🚨 Why Brute Force Fails

Brute force checks all pairs:

(0,1), (0,2), (0,3)...

But most of these are useless, because:

Width keeps decreasing

Height doesn’t improve

We need to discard bad pairs early.

✍️ Simple Two Pointer Approach (Beginner English)

Step 1

Start with two pointers:

• left at start
• right at end

Why?
👉 Maximum width at the beginning.

Step 2

Calculate current area

area = min(height[left], height[right]) × (right - left)

Store the maximum.

Step 3 (THE MOST IMPORTANT STEP 🔥)

Move only the shorter height pointer.

Why?

Because:

The taller line is already good
The shorter line is the bottleneck
Moving the taller one can NEVER increase area

This is the “ohhh” moment 😮

Step 4

Repeat until left < right

🔄 Pointer Movement Example (Visual Thinking)

Heights: [1,8,6,2,5,4,8,3,7]
           L               R

min(1,7) × width → small
Move L (shorter)

           8               7
           L               R

min(8,7) × width → better
Move R (shorter)

Repeat...

Every move is logical, not guesswork.

💻 Convert Simple Approach to Code (C++)

#include <iostream>
#include <vector>
using namespace std;

int maxArea(vector<int>& height) {
    int left = 0;
    int right = height.size() - 1;
    int maxWater = 0;

    while (left < right) {
        int width = right - left;
        int h = min(height[left], height[right]);
        int area = width * h;

        maxWater = max(maxWater, area);

        // Move the shorter pointer
        if (height[left] < height[right])
            left++;
        else
            right--;
    }

    return maxWater;
}

int main() {
    vector<int> height = {1,8,6,2,5,4,8,3,7};
    cout << maxArea(height);
}

⏱️ Complexity

Time: O(n)
Space: O(1)

❌ Common Beginner Mistakes

❌ Moving both pointers
❌ Moving the taller pointer
❌ Thinking height increase > width decrease
❌ Trying sliding window here (wrong pattern)

🎯 Final Mental Rule (Memorize This)

Water is always limited by the shorter line.
So move the pointer with the smaller height.

This rule alone solves the entire problem.