Project Euler Problems 1-10 in Python

I’m working to bone up on my python skills so I decided to spend my Sunday doing problems 1-10 from Project Euler.  I’ve done them before with C or Java but this was my first time with Python.  Here are the problems and my commented code for each one in case it interests anybody.

Problem 1 – Multiples of 3 and 5

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.  ANSWER = 233168.

# Project Euler  -  Question 1  -  Multiples of 3 & 5
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=1
# 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.  ANSWER = 2318.

sum = 0 #variable to hold sum

# Iterate i from 0 to 99
# If i is divisible by 3 or 5, add to sum
for i in range(100):
    if (i%3 == 0 or i%5==0):
    sum += i

# Print answer
print('The sum is: ' + str(sum))

Problem 2 – Even Fibonacci Numbers

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.  ANSWER = 4613732


# Project Euler  -  Question 2  -  Even Fibonacci Numbers
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=2
# 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.
# Answer = 4613732

sum = 0 	# Variable to hold sum
num1 = 0;	# First number
num2 = 1;	# Second number

# While the second number is less than 4000000
# This ensures the first number is less after moving
while num2 < 4e6:
	# Method 1 of incrementing numbers
	temp = num1
	num1 = num2
	num2 = num1 + temp

	# Method 2:
	# num1, num2 = num2, num1+num2

	# If the number is eve, add to sum
	if(num1%2 == 0):
		sum += num1

# Print results
print('The sum is: ' + str(sum))

Problem 3 – Largest Prime Factor

The prime factors of 13195 are 5, 7, 13 and 29.  What is the largest prime factor of the number 600851475143?  ANSWER = 6857

# Project Euler  -  Question 3  -  Largest Prime Factor
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=3
# The prime factors of 13195 are 5, 7, 13 and 29.
# What is the largest prime factor of the number 600851475143?
# Answer = 6857

# Import math library to get sqrt
import math

# isPrime function - returns True or False
def isPrime(num):
	# Iterates from 2 to sqrt(num)+1 as discussed in #7
	# Make sure to convert sqrt to int for range
	# Using xrange will save considerable time for large numbers
	for i in xrange(2,int(math.sqrt(num))+1):
		if (num % i == 0):
			return False

	return True

# List with factors, starts with only one
factors = [600851475143]

# Infinite loop, will break out when necessary
while True:
	# If the largest factor (0 spot) is prime, break
	if(isPrime(factors[0])):
		break
	# Try to divide the largest factor by numbers starting with 2
	# If a number evenly divide, reduct the largest factor and
	#  append the divisor to the end of the list, break out of for
	#  loop and start again at 2 with the new largest factor
	for i in xrange(2,factors[0]):
		if (factors[0] % i == 0):
			factors[0] = factors[0] / i
			factors.append(i)
			break

# Sort factors and print them
factors.sort()
print(factors)

Problem 4 – Largest Palindrome Product

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.  ANSWER = 906609

# Project Euler  -  Question 4  -  Largest Palindromic Product
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=4
# A palindromic number reads the same both ways.
# The largest palindrome made from the product
#  of two 2-digit numbers is 9009 = 91 x 99.
# Find the largest palindrome made from the product of two 3-digit numbers.
# ANSWER: 913 * 993 = 906609.   Took 1.0s

# isPal function returns True or False
# Converts number to string and iterates through half of it
# If characters don't match return false
def isPal(num):
    numString = str(num)
    for i in range(0,len(numString)/2+1):
        if (numString[i] != numString[-i-1]):
            return False
    return True

# Keep track of max product
maxProduct = 0

# Keep track of values used for max prodcut
max1, max2 = 0,0

# Iterate from 999 to 99 going down for both values
# Generate a product and determine if it's a palindrome
# If it is a palindrom, determine if it is max
for i in range(999, 99, -1):
	for j in range(999, 99, -1):
		product = i * j
		if isPal(product):
			if(product > maxProduct):
				maxProduct = product
				max1, max2 = i, j

# Print snswers
print('The largest palindromic product is: ' + str(maxProduct))
print(str(maxProduct) + ' = ' + str(max1) + ' * ' + str(max2))

Problem 5 – Smallest Multiple

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?  ANSWER = 232,792,560

# Project Euler  -  Question 5  -  Smallest Multiple
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=5
# 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?
# Answer = 232,792,560

# First time I wrote this I started from 1 going up and
#  divided by all the numbers from 1 up to 20 and counted
#  number of multiples.  If it got to 20 it stopped and printed
#  Took about (290) to run.

# Second time I wrote it, I started from 1 going up and
#  divided by all the numbers from 20 down to 1 and counted
#  number of multiples and if I got to 20 it stopped and printed
#  Took about 224 seconds to run.  This speeds it up because a number
#  is less likely to be divisible by 20 or 19 than by 1 or 2 so by starting
#  at the top you rule out that number quicker and move on.

# Third time I wrote it, I started from 1 going up and
#  divided by all the numbers from 20 down to 11 and counted
#  number of multiples and if I got to 10 it stopped and printed
#  Took about 210 seconds to run.  This speeds it up because if a number
#  is divisible by 20, it's also divisible by 10, by 18 it's also by 9,
#  by 16, it's also by 8, etc.  So we don't need to test for 1-10

# Forth time I wrote it, I started from 20 going up by 20 and
#  divided by all the numbers from 20 down to 11 and counted
#  number of multiples and if I got to 10 it stopped and printed
#  Took about 14 seconds to run.  We can do this because all of these
#  numbers must be divisible by 20 so checking anything between the multiples
#  of twenty is a waste of time.

# Fifth time I wrote it, I started from 2520 going up by 2520 and
#  divided by all the numbers from 20 down to 11 and counted
#  number of multiples and if I got to 10 it stopped and printed
#  Took about 0.2 seconds to run.  We can start at 2520 since the
#  problem tells us that is the lowest number for 1-10 multiples
#  and since 1-20 includes 1-10, it can't be less. We can increase
#  by 2520 for the same reason.

# The code below is from the forth time I wrote it with two
#  solutions using different techniques

# ------------------------------------------------------
# First solution --------------------------------------
# ------------------------------------------------------

num = 0

# Infinite Loop because we don't know the max
# We will use a break when we find the first number
# Could also do a for loop up to 20! but this is easier
while True:
	#increment num by 20 each time
	num += 20

	# NumMultiples keeps track of the number of multiples for num
	numMultiples = 0

	# Iterate from 20 to 11 going down by 1
	for i in range(20,10, -1):
		# If the number is divisible by i then increase
		#  the number of multiples num has.
		if (num % i == 0):
			numMultiples+=1
		else:
			# If not evenly divisible, break to save time
			break;

	# If there were 10 multiples that means it's evenly
	#  divisibly by all the numbers so we break out of while
	if (numMultiples == 10):
		break;

	# Print out current num to keep track (in millions)
	if num % 1000000 == 0:
		print(num/1000000)

print('The lowest common multiple is: ' + str(num))

# ------------------------------------------------------
# Second solution --------------------------------------
# ------------------------------------------------------

num = 0
keepGoing = True

# In this version, we go until the boolean keepGoing is changed to False
while keepGoing:
	#increment num by 20 each time
	num += 20

	# NumMultiples keeps track of the number of multiples for num
	numMultiples = 0

	# Iterate from 20 to 11 going down by 1
	for i in range(20,10, -1):
		if (num % i != 0):
			# If not evenly divisible, break out of for loop
			#  because we know it isn't the LCM
			break
		if (i == 11):
			# If we reached here without breaking, then we've
			#  found the number we're looking for.
			# Change keepGoing to False to break out of while loop
			keepGoing = False
			break

	# Print out current num to keep track (in millions)
	if num % 1000000 == 0:
		print(num/1000000)

print('The lowest common multiple is: ' + str(num))

Problem 6 – Sum Square Difference

The sum of the squares of the first ten natural numbers is, 12 + 22 + … + 102 = 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.  ANSWER = 25164150

# Project Euler  -  Question 6  -  Sum Square Difference
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=6
# The sum of the squares of the first ten natural numbers is,
#  1^2 + 2^2 + ... + 10^2 = 385
# The square of the sum of the first ten natural numbers is,
#  (1 + 2 + ... + 10)^2 = 55^2 = 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.
# ANSWER = 25,164,150

# Keep track of the sum of the squares
sumSquare = 0

# Keep track of the the sums
sums = 0

# Iterate through numbers [1:100] 1-100 inclusevely
for i in range(1,101):
	sumSquare += i*i
	sums += i

# Square the individuals sums to find square of sums
squareSum = sums * sums

# Print answers
print('The sum of the squares is: ' + str(sumSquare))
print('The square of the sums is: ' + str(sums))
print('The difference is: ' + str(squareSum - sumSquare))

Problem 7 – 10001st prime

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?  ANSWER = 104,743

# Project Euler  -  Question 7  -  10,001st prime
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=7
# 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?
# ANSWER = 104,743
# Took 102s in first version.
# In the first version, I checked for primes from
#  two to num.
# In the second version, I checked for primes from
#  two to num/2+1.
# In the third version, I checked for primes from
#  two to sqrt(num)+1.
# Version 1: 102s
# Version 2: 44s
# Version 3: 0.5s

# Import math library to get sqrt
import math

# isPrime function - returns True or False
def isPrime(num):
	# Iterates from 2 to sqrt(num)+1 as discussed above
	# Make sure to convert sqrt to int for range
	for i in range(2,int(math.sqrt(num))+1):
		if (num % i == 0):
			return False

	return True

count = 1 	# Number of primes
num = 2 	# Prime number (count)

# While loop to continue until we reach 100001th prime
while (count<10001):
	num+=1
	if isPrime(num):
		#print('Found a prime: ' + str(num) + ', prime number: ' + str(count))
		# If a prime is found, increase count and continue
		count+=1

# Print answer
print('The 10001th prime is: ' + str(num))

Problem 8 – Largest Product in a Series

The four adjacent digits in the 1000-digit number (below) that have the greatest product are 9 × 9 × 8 × 9 = 5832.  Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?  ANSWER = 23,514,624,000

7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450

# Project Euler  -  Question 8  -  Largest Product in a Series
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=8
# The four adjacent digits in the 1000-digit number that
#  have the greatest product are 9 x 9 x 8 x 9 = 5832.

# 73167176531330624919225119674426574742355349194934
# 96983520312774506326239578318016984801869478851843
# 85861560789112949495459501737958331952853208805511
# 12540698747158523863050715693290963295227443043557
# 66896648950445244523161731856403098711121722383113
# 62229893423380308135336276614282806444486645238749
# 30358907296290491560440772390713810515859307960866
# 70172427121883998797908792274921901699720888093776
# 65727333001053367881220235421809751254540594752243
# 52584907711670556013604839586446706324415722155397
# 53697817977846174064955149290862569321978468622482
# 83972241375657056057490261407972968652414535100474
# 82166370484403199890008895243450658541227588666881
# 16427171479924442928230863465674813919123162824586
# 17866458359124566529476545682848912883142607690042
# 24219022671055626321111109370544217506941658960408
# 07198403850962455444362981230987879927244284909188
# 84580156166097919133875499200524063689912560717606
# 05886116467109405077541002256983155200055935729725
# 71636269561882670428252483600823257530420752963450

# Find the thirteen adjacent digits in the 1000-digit
#  number that have the greatest product.
# What is the value of this product?
# ANSWER = 23,514,624,000.  Took 0.1s

# findProduct function takes a string and returns
#  the product of every digit
def findProduct(numString):
	product = 1
	for i in range(len(numString)):
		product = product * int(numString[i])
	return product

# The number we need in int and string form
bigNum = 7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450
bigString = str(bigNum)

# Variables to track things
maxProduct = 0
product = 0
numChars = 13

# Iterate over the 1000 digits except for the end digits
# For each iteration find the string of numChar length and
#  pass it to the function to find product
for i in range(len(bigString)-numChars):
	product = findProduct(bigString[i:i+numChars])
	# If a product is larger than max, store it
	if (product > maxProduct):
		maxProduct = product

# Print answers
print('Max Product of ' + str(numChars) + ' digits is: ' + str(maxProduct))

Problem 9 – Special Pythagorean Triplet

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,

a2 + b2 = c2

For example, 32 + 42 = 9 + 16 = 25 = 52.

There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.  ANSWER = 31,875,000

# Project Euler  -  Question 9  -  Special Pythagorean Triplet
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=9
# A Pythagorean triplet is a set of three natural numbers,
#   a < b < c, for which, a2 + b2 = c2
# For example, 32 + 42 = 9 + 16 = 25 = 52.
# There exists exactly one Pythagorean triplet for which
#   a + b + c = 1000.  Find the product abc.
# Answer = 31,875,000  Took 21 seconds. (10s with exit())

# Iterate a from 1 to 1000
# Then iterate b from a to 1000
# Then iterate c from b to 1000
# This is because a < b < c. Otherwise it would take
#  much much longer and return two answers
for a in range(1,1000):
	for b in range(a,1000):
		for c in range (b, 1000):
			# Once we have an iteration of a, b, and c
			# Determine if it fits the criteria of a+b+c==1000 and
			#  a^2 + b^2 == c^2
			# Test a+b+c first because it is a faster test.  Takes ~half the time
			if (a+b+c == 1000):
				if (a*a + b*b == c*c):
					# Print answers
					print('A: ' + str(a) + ' B: ' + str(b) + ' C: ' + str(c))
					print('Product is: ' + str(a*b*c))
					# If we found it, exit to save time
					exit()

Problem 10 – Summation of Primes

The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.  Find the sum of all the primes below two million.  ANSWER = 142,913,828,922

# Project Euler  -  Question 10  -  Summation of Primes
# Written by Matthew Walker, 20 August 2017

# https://projecteuler.net/problem=10
# The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
# Find the sum of all the primes below two million.
# ANSWER = 142,913,828,922    Took 14 seconds.
# Using xrange in isPrime function saves considerable time.
# 29s using just range, 20s using range but omitting odds
# 14s using xrange, 13.5s using xrange and omitting odds

# Import math library to get sqrt
import math

# isPrime function - returns True or False
def isPrime(num):
	# Iterates from 2 to sqrt(num)+1 as discussed in #7
	# Make sure to convert sqrt to int for range
	# Using xrange will save considerable time for large numbers
	for i in xrange(2,int(math.sqrt(num))+1):
		if (num % i == 0):
			return False

	return True

sum = 0

# Iterate from 2 to two million
# You can increase speed slightly by starting at 3 and
#  iterating by 2 to skip evens but only saves 0.5s
for i in range(2,2000000):
	# If number is prime, add to sum
	if isPrime(i):
		sum += i

# Print out results
print('The sum of all primes below 2 million is: ' + str(sum))
Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s