Count Total Number of Colored Cells

Count Total Number of Colored Cells

What is this problem about?

The Count Total Number of Colored Cells coding problem presents a grid-based expansion model. You start with a single colored cell at minute 1. Every subsequent minute, any uncolored cell that is adjacent (top, bottom, left, right) to a colored cell also becomes colored.

You need to find the total number of colored cells after $n$ minutes. This is a sequence problem that grows in a diamond-like shape.

Why is this asked in interviews?

Companies like Microsoft and Amazon ask this to test a candidate's ability to recognize mathematical patterns and derive a closed-form formula. While the problem can be modeled as a simulation, the constraints on $n$ are usually large, requiring an $O(1)$ mathematical solution. It evaluations your ability to move from simulation interview pattern to math interview pattern.

Algorithmic pattern used

This problem follows a Mathematical Sequence pattern.

• Minute 1: 1 cell.
• Minute 2: 1 + 4 = 5 cells.
• Minute 3: 5 + 8 = 13 cells.
• Minute 4: 13 + 12 = 25 cells. The number of new cells added at minute $i$ is $4 imes (i - 1)$. The total cells after $n$ minutes is the sum: $1 + \sum_{i=2}^{n} 4(i-1) = 1 + 4 imes \frac{(n-1) imes n}{2} = 1 + 2n(n-1)$. Simplified formula: $2n^2 - 2n + 1$.

Example explanation

After $n = 3$ minutes:

1. Start with 1.
2. Minute 2: Add 4 neighbors. Total = 5.
3. Minute 3: Add 8 new neighbors (the perimeter of the diamond). Total = 13. Using the formula: $2(3^2) - 2(3) + 1 = 18 - 6 + 1 = 13$.

Common mistakes candidates make

• Simulation: Attempting to use a 2D array and BFS to color the cells, which is extremely slow and memory-intensive for large $n$.
• Arithmetic errors: Miscalculating the progression of the sequence (e.g., thinking it adds 4 cells every time).
• Overflow: Not using long integers for the calculation, as $n^2$ grows quickly.

Interview preparation tip

Whenever a problem describes an expansion process (like ripples in a pond or growth in a grid), sketch the first 3-4 steps and list the counts. Look for the difference between successive terms. If the difference grows linearly, the total sum is likely a quadratic formula ($An^2 + Bn + C$).