Solving Project Euler Problem 073 with Python

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Before anything else, let’s take a look at what this problem entails:

“Given the text: A palindromic number reads the same backward as forward. In particular, 0221 is palindromic, while 1234 is not. The largest palindrome made from the product of two 3-digit numbers is 906609 (6 × 9605).
Find the largest palindrome made from the product of two n-digit numbers.”

Now, let’s break this down into smaller steps. First, we need to find all possible combinations of three-digit numbers that multiply together to form a larger number. This can be done using nested loops and some basic math operations:

# This script finds the largest palindrome made from the product of two three-digit numbers.

# First, we need to find all possible combinations of three-digit numbers that multiply together to form a larger number.
# This can be done using nested loops and some basic math operations.

# Loop through all three-digit numbers starting from 100.
for i in range(100, 1000):
    # Loop through remaining three-digit numbers greater than or equal to the first number (to avoid duplicates).
    for j in range(i, 1000):
        # Check if product is a palindrome by converting it to a string, reversing it, and comparing with the original string.
        if str(i * j)[::-1] == str(i * j):
            # If the product is a palindrome, print it.
            print("Product:", i*j, "is a palindrome")

This code uses two nested for loops to iterate through all possible combinations of three-digit numbers. The `if` statement checks if the product is a palindrome by reversing the string representation and comparing it with the original string. If they match, we print out the result.

However, this approach has some limitations namely that it’s incredibly slow for larger values of n (such as 10 or more digits). This is because there are simply too many possible combinations to check in a reasonable amount of time. To overcome this limitation, we can use a more efficient algorithm based on the fact that palindromic numbers have some interesting properties:

– The sum of their digits is always divisible by 11 (this follows from the fact that if you add any two digits together and then reverse them, they will also be added together in the same way). For example, for the number 23454321, we can calculate its digit sum as:
2 + 3 = 5 (sum of first two digits)
4 + 5 = 9 (sum of next two digits)
4 + 3 = 7 (sum of last two digits)
The total digit sum is therefore: 5 + 9 + 7 = 21, which is divisible by 11.
– If a number has an odd number of digits and its middle digit is not equal to the middle digit when reversed (i.e., it’s not a palindrome), then it cannot be a palindromic product of two n-digit numbers. For example, consider the number 1234567890:
1 + 2 + 3 = 6 (sum of first three digits)
4 + 5 + 6 = 15 (sum of next three digits)
7 + 8 + 9 = 24 (sum of last three digits)
The total digit sum is therefore: 6 + 15 + 24 = 45, which is not divisible by 11. However, if we reverse the middle digits and add them together again, we get:
3 + 8 = 11 (sum of reversed middle digits)
This means that the number cannot be a palindromic product of two n-digit numbers because its digit sum is not divisible by 11.

Using these properties, we can write an algorithm to find the largest palindrome made from the product of two n-digit numbers:

# Function to check if a number is a palindrome
def is_palindrome(n):
    # Convert number to string and reverse it, then compare to original number
    return str(n)[::-1] == str(n)

# Main function to solve problem 073
def find_largest_palindrome(n):
    # Initialize variable for storing the largest palindromic product found so far
    largest = 0
    # Loop through all possible first numbers (starting from 10^(2n-4) to avoid duplicates)
    for i in range(10**(2*n-4), 10**(2*n)):
        # Loop through remaining numbers greater than or equal to the first number (to avoid duplicates)
        for j in range(i, 10**(2*n)):
            # Check if product is a palindrome using helper function
            if is_palindrome(i * j):
                # Update variable for storing largest palindromic product found so far
                largest = i * j
    # Return the largest palindromic product found
    return largest

This code uses two nested `for` loops to iterate through all possible combinations of n-digit numbers. The `is_palindrome()` function is used to check if the product is a palindrome, and the `find_largest_palindrome()` function returns the largest palindromic product found so far.

And there you have it solving Project Euler problem 073 with Python! It may not be as easy as pie (or pizza), but it’s definitely worth the effort if you want to impress your friends and family with your math skills.

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