Category Archives: programming

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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sorting for humans

Aside

how would you go about implementing natural string sort in python?
— seeknuance

natural sort order – it’s like encryption. everyone tries to roll their own, and it never works out well (if you didn’t follow the link above, jeff atwood said it best).

don’t reinvent the wheel unless you really have to. for the code in post linked above, i’d say, “don’t forget about capital letters”, and also, “length is not as important as you’d think”. imagine, for example, this [low-end,value,high-end] setup for the range:

['a','m15','z']

is m15 in that range or not?

(John, from seeknuance, observes in the comments that this and other cases are not relevant for his specific implementation; he can afford to stay much simpler, and more efficient, than my code below)

anyways, this is what i’d do:


import re

def in_natural_range(low, value, high):
    result = []
    for s in [low,value,high]:
        result.append( [int(c) if c.isdigit() else c.lower() for c in re.split('([0-9]+)', s)])
    return ''.join([str(i) for i in sorted(result)[1]]) == value.lower() #fixed based on comment below

print in_natural_range('a','b','c')
print in_natural_range('1','2','3')
print in_natural_range('a1','b','c1')
print in_natural_range('a1','a1','a1')
print in_natural_range('a1','a2','a3')
print in_natural_range('a1','a22','a3')
print in_natural_range('11','12','13')
print in_natural_range('11','123','13')

well, if i had to i’d do that. in practice, i’d just stick to python’s natsort library.

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project euler problem #1

Aside

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000.

http://projecteuler.net/problem=1

  • try solving it as a python one-liner
  • try optimizing your solution for speed

highlight below for my solution:


sum(range(0,1000,3)) + sum(range(0,1000,5)) - sum(range(0,1000,15))

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