Burst Balloons - Problem
๐ŸŽˆ Balloon Popping Challenge

You're at a carnival game with n balloons arranged in a line, each balloon has a number painted on it. Your goal is to burst all balloons to collect the maximum possible coins!

How the scoring works:
โ€ข When you burst balloon i, you earn nums[i-1] ร— nums[i] ร— nums[i+1] coins
โ€ข If a neighbor doesn't exist (out of bounds), treat it as having value 1
โ€ข Once a balloon is burst, its neighbors become adjacent

The Challenge: Find the optimal order to burst all balloons to maximize your coin collection!

Example: With balloons [3,1,5,8], you could earn up to 167 coins by choosing the right sequence.

Input & Output

example_1.py โ€” Basic Case
$ Input: [3,1,5,8]
โ€บ Output: 167
๐Ÿ’ก Note: Optimal sequence: First burst 1 (3ร—1ร—5=15), then 5 (3ร—5ร—8=120), then 3 (1ร—3ร—8=24), finally 8 (1ร—8ร—1=8). Total: 15+120+24+8=167 coins.
example_2.py โ€” Single Balloon
$ Input: [9]
โ€บ Output: 9
๐Ÿ’ก Note: Only one balloon to burst. Since boundaries are treated as 1, we get 1ร—9ร—1 = 9 coins.
example_3.py โ€” Two Balloons
$ Input: [7,2]
โ€บ Output: 23
๐Ÿ’ก Note: Two possible orders: (1) Burst 7 first: 1ร—7ร—2=14, then 2: 1ร—2ร—1=2, total=16. (2) Burst 2 first: 1ร—2ร—1=2, then 7: 1ร—7ร—1=7, total=9. Wait! That's wrong. Correct: (1) 1ร—7ร—2=14, then 1ร—2ร—1=2, total=16. (2) 1ร—2ร—7=14, then 1ร—7ร—1=7, total=21. Actually optimal is burst 2 first (1ร—2ร—7=14), then 7 (1ร—7ร—1=7), giving 21. But let me recalculate: if we burst 2 first, remaining is [7], boundaries become 1ร—2ร—7=14. Then burst 7: 1ร—7ร—1=7. Total=21. If we burst 7 first: 1ร—7ร—2=14, then remaining [2]: 1ร—2ร—1=2. Total=16. So answer should be 21, but let me verify with DP: with boundaries [1,7,2,1], we get max of trying 7 or 2 as last in range (0,3). If 7 is last: dp[0][1] + 1ร—7ร—2 + dp[1][3] = 0 + 14 + 2 = 16. If 2 is last: dp[0][2] + 1ร—2ร—1 + dp[2][3] = 7 + 2 + 0 = 9. So 16 is correct, but let me double-check the calculation...

Constraints

  • n == nums.length
  • 1 โ‰ค n โ‰ค 300
  • 0 โ‰ค nums[i] โ‰ค 100

Visualization

Tap to expand
๐ŸŽˆ Carnival Balloon Strategy3158Dynamic Programming StrategyStep 1: Add Boundaries [1, 3, 1, 5, 8, 1]The '1' boundaries make calculation easier - no special cases!Step 2: Build DP Tabledp[i][j] = maximum coins from bursting all balloons between positions i and jStep 3: Choose Last Balloon WiselyFor each range, try every balloon as the 'last to burst'This gives us: coins = left_boundary ร— last_balloon ร— right_boundaryMaximum Score: 167 coins! ๐Ÿ†Sequence: 1โ†’5โ†’3โ†’8 (reading positions, not values)
Understanding the Visualization
1
Set Up the Game
Add invisible '1-point' balloons at both ends as boundaries
2
Plan Small Ranges
Start with ranges of 3 balloons, decide optimal last balloon to burst
3
Build Up Larger Ranges
Use solutions from smaller ranges to solve larger ranges
4
Find Global Optimum
The final answer covers the entire range of original balloons
Key Takeaway
๐ŸŽฏ Key Insight: By thinking backwards (what to burst LAST), we avoid the complexity of tracking changing neighbors and transform this into a clean dynamic programming problem with optimal substructure.
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