Euler_problem_14 for python

Euler Problem 14

In this article, we will discuss the solution to Euler Problem 14 using Python. Euler Problem 14 is a famous mathematical problem that involves finding the longest Collatz sequence for a given range of numbers.

Collatz Sequence

The Collatz sequence is a sequence of numbers for a given positive integer n, where each term is obtained from the previous term according to the following rules:

If the current term is even, the next term is obtained by dividing it by 2.

If the current term is odd, the next term is obtained by multiplying it by 3 and adding 1.

The problem asks us to find the starting number, under a given range, which produces the longest Collatz sequence. For example, if the range is from 1 to 1 million, we need to find the starting number within this range that generates the longest sequence.

Solution Approach

To solve this problem, we can use a brute-force approach by iterating through all numbers in the given range and calculating the length of their Collatz sequences. We then keep track of the maximum length and the corresponding starting number.

Let's implement the solution in Python:

# Function to calculate the length of Collatz sequence for a given number
def collatz_sequence_length(n):
    length = 1
    while n != 1:
        if n % 2 == 0:
            n = n // 2
        else:
            n = 3 * n + 1
        length += 1
    return length
# Main function to find the starting number with the longest Collatz sequence
def find_longest_collatz_sequence(start, end):
    max_length = 0
    max_start = 0
    for i in range(start, end+1):
        length = collatz_sequence_length(i)
        if length > max_length:
            max_length = length
            max_start = i
    return max_start
# Find the starting number with the longest Collatz sequence in the range 1 to 1 million
start_number = find_longest_collatz_sequence(1, 1000000)
print("Starting number with the longest Collatz sequence:", start_number)

In the above code, the collatz_sequence_length function calculates the length of the Collatz sequence for a given number. The find_longest_collatz_sequence function iterates through all numbers in the given range and keeps track of the maximum length and the corresponding starting number. Finally, the starting number with the longest Collatz sequence is printed.

Adjusting the Temperature

The initial problem statement does not mention anything about adjusting the temperature. However, adjusting the temperature in machine learning or optimization algorithms can have significant effects on the solution.

The temperature parameter is commonly used in simulated annealing algorithms. Simulated annealing is a technique that imitates the annealing process in metallurgy to find the global optimum of a problem. The temperature controls the randomness in the search process. A higher temperature allows for more random moves, while a lower temperature restricts the search to more local moves.

In the context of this problem, adjusting the temperature could mean altering the search strategy to find the starting number with the longest Collatz sequence. However, without further clarification or information in the problem statement, it is difficult to determine the specific impact of adjusting the temperature in this case.

Overall, the solution we have implemented for Euler Problem 14 does not involve any temperature adjustment. It utilizes a simple brute-force approach to calculate the length of Collatz sequences for all numbers in the given range and finds the starting number with the longest sequence.

Conclusion

In this article, we discussed the solution to Euler Problem 14 using Python. We explored the concept of Collatz sequences and explained the brute-force approach to finding the starting number with the longest Collatz sequence. We also briefly discussed the temperature parameter and its relevance in optimization algorithms, although it was not directly applicable to solving this particular problem.

By implementing the provided code, you can find the starting number with the longest Collatz sequence in any given range. This solution demonstrates the power of Python in solving complex mathematical problems efficiently.