Minimum Garden Perimeter to Collect Enough Apples - Problem
The Apple Garden Challenge
Imagine an infinite 2D grid garden where apple trees are planted at every integer coordinate point. Each tree at position
You want to buy a square plot of land centered at the origin
Key Points:
• The square is centered at
• We count apples inside and on the perimeter of the square
• Return the perimeter (not the side length) of the minimum square
Example: A square with side length 2 has vertices at
Imagine an infinite 2D grid garden where apple trees are planted at every integer coordinate point. Each tree at position
(i, j) contains exactly |i| + |j| apples, where |x| represents the absolute value of x.You want to buy a square plot of land centered at the origin
(0, 0) that is axis-aligned (edges parallel to x and y axes). Your goal is to find the minimum perimeter of such a square that contains at least neededApples apples.Key Points:
• The square is centered at
(0, 0)• We count apples inside and on the perimeter of the square
• Return the perimeter (not the side length) of the minimum square
Example: A square with side length 2 has vertices at
(-1,-1), (-1,1), (1,-1), (1,1) and perimeter = 8. Input & Output
example_1.py — Small Garden
$
Input:
neededApples = 1
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Output:
8
💡 Note:
The smallest square centered at (0,0) has side length 1, containing coordinates from (-1,-1) to (1,1). This gives us 12 apples total, which is >= 1. Perimeter = 8 × 1 = 8.
example_2.py — Medium Garden
$
Input:
neededApples = 13
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Output:
16
💡 Note:
A square with side length 1 gives 12 apples (not enough). Side length 2 gives 60 apples (sufficient). Perimeter = 8 × 2 = 16.
example_3.py — Large Garden
$
Input:
neededApples = 1000000000
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Output:
5040
💡 Note:
For very large requirements, we need a much bigger square. Binary search efficiently finds the minimum side length needed, then returns 8 times that value.
Constraints
- 1 ≤ neededApples ≤ 2 × 1015
- All inputs are positive integers
- The answer will fit in a 64-bit signed integer
Visualization
Tap to expand
Understanding the Visualization
1
Grid Layout
Each point (i,j) has |i| + |j| apples - Manhattan distance from origin
2
Square Expansion
Squares grow outward from center, capturing more apple trees
3
Mathematical Pattern
Total apples follow a cubic polynomial based on square size
Key Takeaway
🎯 Key Insight: The apple distribution follows a predictable mathematical pattern, allowing us to use binary search instead of brute force calculation for optimal O(log n) performance.
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Explanation
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