Paint Fence - Problem
Imagine you're a decorator tasked with painting a beautiful wooden fence for a neighborhood. The fence consists of n posts arranged in a row, and you have k different paint colors to choose from.
However, there are some aesthetic rules you must follow:
- Every post must be painted exactly one color
- No three or more consecutive posts can have the same color (this would look monotonous!)
Your challenge is to determine: In how many different ways can you paint this fence?
Given two integers n (number of posts) and k (number of colors), return the total number of valid painting arrangements.
Example: With 3 posts and 2 colors (Red, Blue), you could paint: RRB, RBR, RBB, BRR, BRB, BBR - that's 6 different ways!
Input & Output
example_1.py — Basic Case
$
Input:
n = 3, k = 2
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Output:
6
💡 Note:
With 3 posts and 2 colors (Red=R, Blue=B), the valid arrangements are: RRB, RBR, RBB, BRR, BRB, BBR. Note that RRR and BBB are invalid because they have 3 consecutive same colors.
example_2.py — Single Post
$
Input:
n = 1, k = 1
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Output:
1
💡 Note:
With only 1 post and 1 color, there's exactly 1 way to paint it. No consecutive constraint applies.
example_3.py — Two Posts
$
Input:
n = 2, k = 4
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Output:
16
💡 Note:
With 2 posts and 4 colors, we can paint them in any combination since we need at least 3 consecutive posts to violate the rule. Total ways = 4 × 4 = 16.
Visualization
Tap to expand
Understanding the Visualization
1
Day 1 Setup
You have k outfit choices for the first day
2
Day 2 Planning
You can wear any of k outfits, giving k² total combinations
3
Pattern Recognition
From day 3 onward, track 'same as yesterday' vs 'different from yesterday'
4
State Transitions
Same = previous different count; Different = previous total × (k-1) choices
Key Takeaway
🎯 Key Insight: Instead of checking all k^n combinations, we only need to track two states at each position: endings with same vs different colors. This reduces complexity from exponential to linear!
Time & Space Complexity
Time Complexity
O(n)
Single pass through n posts, constant work per post
✓ Linear Growth
Space Complexity
O(1)
Only need to track previous two state values, not entire array
✓ Linear Space
Constraints
- 1 ≤ n ≤ 50
- 1 ≤ k ≤ 105
- No three consecutive posts can have the same color
- Every post must be painted exactly one color
💡
Explanation
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