Minimum Garden Perimeter to Collect Enough Apples

Minimum Garden Perimeter to Collect Enough Apples

1. What is this problem about?

The Minimum Garden Perimeter to Collect Enough Apples is a mathematical word problem. You have a square garden centered at $(0, 0)$. In each plot $(i, j)$ in the garden, there are $|i| + |j|$ apples. Given a target number of apples, you need to find the minimum perimeter of a square garden that contains at least that many apples. The garden expands in units of 1 from the center, so a garden with "radius" $k$ spans from $-k$ to $k$ in both $x$ and $y$.

2. Why is this asked in interviews?

Amazon asks the Minimum Garden Perimeter to Collect Enough Apples interview question to test a candidate's ability to derive mathematical formulas and use Binary Search on a monotonic function. It evaluates whether you can recognize that as the garden size increases, the total number of apples increases strictly, making it a perfect candidate for binary search rather than a brute-force simulation.

3. Algorithmic pattern used

The algorithmic pattern is Math (Summation) and Binary Search.

1. Derive a formula for the total apples in a square of radius $k$. The total apples in a square of side length $2k$ is $2k(k+1)(2k+1)$.
2. Since this function is monotonically increasing with $k$, we can binary search for the smallest $k$ such that $f(k) \geq target\_apples$.
3. The perimeter of such a square is $8k$. This "Math, Binary Search interview pattern" allows solving the problem in $O(\log N)$ time, where $N$ is the target number of apples.

4. Example explanation

Target apples = 60.

• Radius $k=1$: Square is from -1 to 1.
  • Apples at $(0,0)=0$, $(-1,0)=1$, $(1,0)=1$, $(0,-1)=1$, $(0,1)=1$, $(-1,-1)=2$, $(1,-1)=2$, $(-1,1)=2$, $(1,1)=2$.
  • Total = 12.
• Radius $k=2$: Total apples = $2(2)(3)(5) = 60$.
• Perimeter for $k=2$ is $8 \times 2 = 16$. The result is 16.

5. Common mistakes candidates make

In the Minimum Garden Perimeter to Collect Enough Apples coding problem, the biggest challenge is deriving the summation formula correctly. Many candidates try to use nested loops to count apples, which is $O(k^2)$ and will time out for large targets (like $10^{15}$). Another mistake is not setting the binary search bounds correctly or failing to account for the fact that the perimeter is $8k$ (the side length is $2k$, and there are 4 sides, but the question defines perimeter in a specific grid-aligned way).

6. Interview preparation tip

When you see a problem with a large target and a clear growth pattern, try to find a closed-form formula. If you can't find one, see if you can at least calculate the value for a given $k$ in $O(1)$ or $O(\log k)$. This almost always points to a "Binary Search on Answer" pattern. Practice sum of squares and sum of integers formulas, as they are often the basis for these types of "Math interview questions."