Title: Solving the "Fishing" Game with Dynamic Programming
An Indian Strategy Game and Its Optimal Solution via DP
Introduction
The "Fishing" game is a classic Indian board game where players collect fish from a pond to maximize their score. The pond contains fish of different colors (e.g., red, blue, green) with varying values. Players take turns to catch fish, but they cannot catch the same color consecutively. The goal is to calculate the optimal strategy for a player to achieve the highest possible score. This problem can be efficiently solved using Dynamic Programming (DP).
Problem Statement
Given:
A pond with n fish, each of a specific color and value.
A rule: A player cannot catch the same color consecutively.
Players take turns until no fish remain.
Objective:
Determine the maximum score a player can achieve using an optimal strategy.
Dynamic Programming Approach
1. State Representation
Define the state as dp[c], representing the maximum score achievable starting from the i-th fish, with the last caught fish color being c (where c is 0 for red, 1 for blue, 2 for green).
2. Base Case
If there are no fish left (i == n), the score is 0.
3. State Transition
For each fish i and color c, the player has two choices:
Catch the current fish:
If the current fish’s color is not c, add its value to dp[i-1][current_color].
Skip the current fish:
Move to the next fish with the same last color c.
The recurrence relation is:
dp[c] = max(
(fish.value + dp[i-1][fish.color]) if fish.color != c else 0,
dp[i-1][c]
)
4. Initialization
Initialize a DP table with dimensions (n+1) x 3 (3 colors).
Code Implementation
def max_fishing_score(fish):
n = len(fish)
dp = [[0]*3 for _ in range(n+1)] # dp[c] for fish 0..i-1
for i in range(1, n+1):
current_color = fish[i-1][0]
current_value = fish[i-1][1]
for c in range(3):
# Option 1: Catch the fish if color differs
if current_color != c:
option1 = current_value + dp[i-1][c]
else:
option1 = 0 # Cannot catch same color consecutively
# Option 2: Skip the fish
option2 = dp[i-1][c]
dp[c] = max(option1, option2)
return max(dp[n])
# Example Input: List of (color, value) tuples
fish = [(0, 3), (1, 5), (0, 2), (2, 7), (1, 4)]
print("Maximum Score:", max_fishing_score(fish)) # Output: 15
Explanation
State Transition: The DP table tracks the best possible score for each position and color constraint.
Optimal Substructure: The solution for fish i depends on the optimal solution for fish i-1.
Complexity: Time complexity is O(n), and space complexity is O(n) (can be optimized to O(1) using rolling arrays).
Key Takeaways
Dynamic Programming is ideal for sequential decision-making problems with overlapping subproblems.
The "Fishing" game highlights how real-world puzzles can be modeled using state transitions.
This approach can be extended to handle more colors, penalties, or additional game rules.
Let me know if you need further clarification or variations! 🎣
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