12_nphard_basilisk_fountain.py

# The Clockwork Basilisks and the Octarine Fountain

'''In the mystical land of Discworld, there exists a legendary octarine fountain that emits a radiant glow of magical energy. However, this fountain is guarded by a group of clockwork basilisks, formidable creatures that have the power to turn anyone to stone upon making direct eye contact.

The clockwork basilisks are positioned in a circular formation around the octarine fountain. Each basilisk has a unique energy level, represented by a positive integer. The goal is to find a subset of basilisks such that their combined energy level is maximized while ensuring that no two basilisks in the subset are adjacent in the circular formation.

Your task is to determine whether it is possible to select a subset of basilisks that maximizes the total energy level while adhering to the constraint of not selecting adjacent basilisks.

This problem is known to be NP-Hard, meaning that it belongs to the class of problems that are at least as hard as the hardest problems in NP. While there is no known polynomial-time algorithm to solve all instances of this problem, it is possible to verify a solution in polynomial time.

To showcase this concept, we can use the "Subset Sum" problem, which is NP-Complete, to reduce it to the Clockwork Basilisks problem. The Subset Sum problem involves finding a subset of numbers from a given set that adds up to a specific target sum.

Here's an example code that demonstrates the reduction from Subset Sum to the Clockwork Basilisks problem:'''

def subset_sum_to_basilisks(numbers, target):
    n = len(numbers)

    # Reduce Subset Sum to Clockwork Basilisks problem
    basilisks = [num for num in numbers if num <= target]
    num_basilisks = len(basilisks)

    # Solve Clockwork Basilisks problem
    total_energy = sum(basilisks)

    # Check if subset sum equals the target
    if total_energy == target:
        return "It is possible to select a subset of basilisks with a total energy of {}.".format(target)
    else:
        return "It is not possible to select a subset of basilisks with a total energy of {}.".format(target)

# Example usage
numbers = [3, 7, 12, 5, 9]
target_sum = 22

result = subset_sum_to_basilisks(numbers, target_sum)
print(result)

'''In this code, the numbers list represents the set of numbers from which we want to find a subset that adds up to the target_sum. We reduce the Subset Sum problem to the Clockwork Basilisks problem by converting the numbers into basilisk energy levels and the target sum into the desired total energy level.

The subset_sum_to_basilisks() function takes the numbers and target as input and performs the reduction by selecting the basilisks with energy levels less than or equal to the target. It then solves the Clockwork Basilisks problem by summing up the energy levels of the selected basilisks.

Finally, the function checks if the total energy level equals the target sum and returns the corresponding result.

By reducing the NP-Complete Subset Sum problem to the Clockwork Basilisks problem, we demonstrate the concept of NP-Hardness. While it is challenging to find an optimal solution for the Clockwork Basilisks problem in polynomial time, we can verify a solution by performing a polynomial-time reduction and solving an NP-Complete problem.'''