Tag Archives: math

project euler problem #6

Aside

The sum of the squares of the first ten natural numbers is,

12 + 22 + … + 10^2 = 385
The square of the sum of the first ten natural numbers is,

(1 + 2 + … + 10)^2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

http://projecteuler.net/problem=6

highlight below for my solution:


# A = (sum[1..n])^2 = [n(n+1)/2] ^ 2
# B = sum[1^2..n^2] = n(n+1)(2n+1)/6
# A-B = n(n+1)(3n^2-n-2)/12


def delta_sum_products_product_sum(n):
    return n * (n + 1) * (3 * (n ** 2) - n - 2) / 12


print delta_sum_products_product_sum(100)

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project euler problem #7

Aside

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

http://projecteuler.net/problem=7

highlight below for my solution:


#using a prime number set datastructure - https://gist.github.com/aausch/6709819
p =  PrimeSet()

i = 10001

while (len(p)<10001):
    i in p
    i += i

print sorted(list(p))[10000]

(see also my solution to problem #5)

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project euler problem #5

Aside

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

http://projecteuler.net/problem=5

highlight below for my solution:


#using a prime number set datastructure - https://gist.github.com/aausch/6709819
p = PrimeSet()

def min_product(n):
    n in p #initialize the PrimeSet with all primes less than n
    product = 1
    for prime in p:
        product = product * (prime ** (int(n ** (1.0/prime))))
    return product

print min_product(20)

(see also my solution to problem #3)

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project euler problem #3

Aside

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600851475143 ?

http://projecteuler.net/problem=3

highlight below for my solution:


#using a prime number set datastructure - https://gist.github.com/aausch/6709819
p_set = PrimeSet()
n = 600851475143
sqrt = int(n ** 0.5) 
p_set[sqrt]
max_factor = 1
for x in p_set:
    if n % x == 0 and x > max_factor:
        max_factor = x       
print max_factor


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project euler problem #2

Aside

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

http://projecteuler.net/problem=2

  • try solving it as a python one-liner – i couldn’t figure out a clean solution
  • try optimizing your solution for speed

highlight below for my solution:


#using a fibonacci dictionary - https://gist.github.com/aausch/6707846
fib_dict = FibDict()
j = 3
while (fib_dict[j] < 4000000):
    j = j + 3
print sum([fib_dict[i] for i in range(3,j,3)])  # j = 36, probably


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project euler problem #4

Aside

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.

Find the largest palindrome made from the product of two 3-digit numbers.

http://projecteuler.net/problem=4

highlight below for my solution:


def is_palindrome(num):
    return str(num) == str(num)[::-1]

def fn(n):
    max_palindrome = 1
    for x in range(n,1,-1):
        if x*n < max_palindrome:
            break
        for y in range(n,x-1,-1):
            if is_palindrome(x*y) and x*y > max_palindrome:
                max_palindrome = x*y
            elif x * y < max_palindrome:
                break
    return max_palindrome

print fn(999)


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