March 6th, 2009
Using names.txt, a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.
For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938 × 53 = 49714.
What is the total of all the name scores in the file?
I love abusing map.
import re
def word_sum(word):
word_sum = sum(map(lambda letter: ord(letter)-ord('A')+1, word))
return word_sum
def name_scores_sum(path):
f = file(path)
names = f.read()
names = sorted(re.findall('([A-Z]+)',names))
sums = map(word_sum,names)
scores = map(lambda (index,sum): index*sum, enumerate(sums,1))
return sum(scores)
>>> name_scores_sum(path)
871198282 |
Tags: file, map, problem22, project euler, python, strings, sum
Posted in map | 3 Comments »
March 3rd, 2009
Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.
For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.
Evaluate the sum of all the amicable numbers under 10000.
def proper_divisors(n):
"""
Find all divisors of n by trial division
"""
divisors = set([1])
for i in range(2, math.ceil(n ** 0.5)+1):
if n % i == 0:
divisors.add(i)
divisors.add(n/i)
return divisors
def amicable_numbers(n):
candidates = range(1,n)
amicables = []
for a in candidates:
b = sum(proper_divisors(a))
if a != b \
and sum(proper_divisors(a)) == b \
and sum(proper_divisors(b)) == a:
amicables.append(a)
amicables.append(b)
candidates.remove(b)
return amicables
>>> sum(amicable_numbers(10000))
31626 |
Tags: amicable, divisors, euler, factors, problem21, python
Posted in amicable numbers | No Comments »
February 12th, 2009
I noticed I left some of the timing decorator @wrapper statements in some solutions, so here is the timing function I have been using to determine the approximate solution time.
def print_timing(func):
def wrapper(*arg):
t1 = time.time()
res = func(*arg)
t2 = time.time()
print '%s took %0.3f ms' % (func.func_name, (t2-t1)*1000.0)
return res
return wrapper |
In use:
@print_timing
def some_function()
return "Hi, I'm a function. And I'm gonna get timed!" |
Tags: decorator, python, timing, wrapper
Posted in python | No Comments »
February 11th, 2009
The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.
There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.
How many circular primes are there below one million?
Using python sets, the sieve of Eratosthenes from problem 10 and some pre-filtering for primes with even digits.
# sieve of Eratosthenes
def sieve(n):
primes = range(2,n)
x = 0
while primes[x] < n**0.5:
# remove all multiples of the current prime from primes
primes = [y for y in primes if y==primes[x] or y%primes[x]]
x = x+1
return primes
def get_rotations(x):
strx = str(x)
rotations = [strx[i:] + strx[:i] for i in range(len(strx))]
rotations = map(int,rotations)
return rotations
def all_digits_odd(n):
n = str(n)
for digit in n:
if int(digit) % 2 == 0:
return False
return True
@print_timing
def num_circular_primes(x):
primes = sieve(x)
# optimize: remove all primes with any even digits
primes = filter(all_digits_odd,primes)
primes = set(primes)
cprimes = 0
max_prime = max(primes)
for p in primes:
rotations = set(get_rotations(p))
intersection = primes & rotations
# if all rotations are primes, they are all circular
if len(intersection) == len(rotations):
cprimes += len(intersection)
#remove all the rotations from the working set
primes = primes - rotations
return cprimes
>>> num_circular_primes(1000000)
num_circular_primes took 6531.000 ms
54 |
Tags: circular primes, primes, problem35, project euler, python, sets, sieve of eratosthenes
Posted in primes | No Comments »
February 11th, 2009
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
3
7 5
2 4 6
8 5 9 3
That is, 3 + 7 + 4 + 9 = 23.
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)
Using the functional programming concept of an aggregator (with reduce and a custom aggregating function):
triangle = \
"""75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23"""
def get_row_maximal_sums(prow,row):
"""
Given the previous and current rows, find the sum of
the maximal path leading to each node of the current row.
"""
sums = [0]*len(row)
for index, item in enumerate(row):
try:
left_pred = prow[index-1]
except IndexError:
# node has no left predecessor
left_pred = 0
try:
right_pred = prow[index]
except IndexError:
# node has no right predecessor
right_pred = 0
sums[index] = item + max(left_pred,right_pred)
return sums
# process triangle into list of row lists
t = triangle.split("\n")
for i,row in enumerate(t):
t[i] = map(int,row.split(" "))
# problem 18
>>> max(reduce(get_row_maximal_sums,t))
1074
# problem 67
>>> max(reduce(get_row_maximal_sums,t))
7273 |
Tags: aggregator, maximal sum, problem 18, problem 67, project euler, python, triangle
Posted in triangle | No Comments »
February 10th, 2009
The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
Let us list the factors of the first seven triangle numbers:
1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28
We can see that 28 is the first triangle number to have over five divisors.
What is the value of the first triangle number to have over five hundred divisors?
Iterators again.
import math
def divisors(n):
"""
Find all divisors of n by trial division
"""
divisors = set([1])
for i in range(1, math.ceil(n ** 0.5)+1):
if n % i == 0:
divisors.add(i)
divisors.add(n/i)
return divisors
class Triangle:
def __init__(self):
self.count = 1
self.sum = 0
def __iter__(self):
return self
def next(self):
self.sum += self.count
self.count += 1
return self.sum
def problem12(n):
t = Triangle()
while True:
x = t.next()
num = len(divisors(x))
if num > n:
return x, num
>>> problem12(500)
problem12 took 6172.000 ms
(76576500, 576) |
Tags: divisors, factors, iterator, problem12, project euler, python, triangle number
Posted in factors | No Comments »
February 5th, 2009
A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:
012 021 102 120 201 210
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
This one was fun.
from math import factorial
@print_timing
def problem24(n):
"""
Determine the nth lexicographic
permutation of 0123456789
"""
answer = ""
digits = [0,1,2,3,4,5,6,7,8,9]
for i in range(9,-1,-1):
ifact = factorial(i)
digit = (n-1) / ifact
answer += str(digits.pop(digit))
n = n - (digit * ifact)
return answer
>>> problem24(1000000)
problem24 took 0.000 ms
'2783915460'
>>> |
Tags: permutations, problem24, project euler, python
Posted in permutations | No Comments »
February 5th, 2009
“The decimal number, 585 = 10010010012 (binary), is palindromic in both bases.
Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.
(Please note that the palindromic number, in either base, may not include leading zeros.)”
Python just makes these too easy.
def is_palindrome(x):
"""
Returns true if the digits of x are the same both backward
and forward in both base 2 and base 10.
"""
x10 = str(x)
x2 = bin(x)[2:]
return x10 == x10[::-1] and x2 == x2[::-1]
# check only odd numbers since even base 2 numbers end in 0
>>> sum(filter(is_palindrome, range(1,1000000,2)))
872187 |
Tags: binary, palindrome, project euler, python
Posted in palindrome | No Comments »
February 5th, 2009
Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:
21 22 23 24 25
20 07 08 09 10
19 06 01 02 11
18 05 04 03 12
17 16 15 14 13
It can be verified that the sum of both diagonals is 101.
What is the sum of both diagonals in a 1001 by 1001 spiral formed in the same way?
Observing that the four corners of the spiral can be expressed by:
k2,
k2-k+1,
k2-2k+2,
k2-3k+3,
We get:
def spiralx(n):
sum = 0
for k in range(3,n+1,2):
sum = sum + (4 * k**2) - 6*k + 6
print sum + 1
>>> spiralx(5)
101
>>> spiralx(3)
25
>>> spiralx(1001)
669171001 |
Tags: diagonals, problem28, project euler, python, spiral
Posted in spiral | No Comments »
February 5th, 2009
Starting in the top left corner of a 2×2 grid, there are 6 routes (without backtracking) to the bottom right corner.
How many routes are there through a 20×20 grid?
Solved using combinatorics. João Ferreira has a good description of the problem and general solution.
Tags: binomial coefficient, combinatorics, grid, paths, problem15, project euler
Posted in combinatorics | No Comments »
February 5th, 2009
“The Fibonacci sequence is defined by the recurrence relation:
Fn = Fn−1 + Fn−2, where F1 = 1 and F2 = 1.
Hence the first 12 terms will be:
F1 = 1
F2 = 1
F3 = 2
F4 = 3
F5 = 5
F6 = 8
F7 = 13
F8 = 21
F9 = 34
F10 = 55
F11 = 89
F12 = 144
The 12th term, F12, is the first term to contain three digits.
What is the first term in the Fibonacci sequence to contain 1000 digits?
“
It is possible to do this mathematically, but I brute forced this one for the chance to implement a Fibonacci iterator (used a Fibonacci generator back in Problem 2).
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| class Fibonacci:
"Iterator for fibonacci sequence"
def __init__(self):
self.first = 1
self.second = 1
self.counter = 0
def __iter__(self):
return self
def count(self):
return self.counter
def next(self):
self.current = self.first
self.first = self.second
self.second = self.current + self.first
self.counter += 1
return self.current
def problem25(n):
f = Fibonacci()
while True:
x = f.next()
xlength = len(str(x))
if xlength == n:
return f.count()
>>> problem25(1000)
4782 |
Tags: fibonacci, iterator, problem25, project euler, python
Posted in fibonacci | No Comments »
February 4th, 2009
“The series, 11 + 22 + 33 + ... + 1010 = 10405071317.
Find the last ten digits of the series, 11 + 22 + 33 + ... + 10001000.”
A (somewhat obfuscated) little one-liner in python. I don’t know whether or not taking the last ten digits before adding makes a difference in execution time.
1
2
| >>> str(sum([int(str(x**x)[-10:]) for x in range(1,1001)]))[-10:]
'9110846700' |
A prettier solution:
1
2
| >>> sum([n**n for n in range(1,1001)]) % 10**10
9110846700L |
Tags: problem3, project euler, python, series
Posted in series | No Comments »
February 4th, 2009
“The following iterative sequence is defined for the set of positive integers:
n → n/2 (n is even)
n → 3n + 1 (n is odd)
Using the rule above and starting with 13, we generate the following sequence:
13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
Which starting number, under one million, produces the longest chain?
NOTE: Once the chain starts the terms are allowed to go above one million.”
This poor solution takes quite a while. A better method is to cache the lengths of sequences for starting values, avoiding recalculating those sequences again and again. This solution does that to an extent, but only so far as to not recalculate parts of the current maxchain.
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| def iterative_seq(x):
chain = [x]
while x > 1:
# optimization: & is faster than %
if x & 1:
# optimization for x*3 + 1
x += (x << 1) + 1
else:
# optimization for x/2
x = x >> 1
chain.append(x)
return chain
def problem14(limit):
maxchain = []
for x in range(limit/2,limit):
if x not in maxchain:
newchain = iterative_seq(x)
if len(newchain) > len(maxchain):
maxchain = newchain
return maxchain |
Tags: o, problem14, project euler, python, sequence
Posted in sequence | No Comments »
February 4th, 2009
Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.
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5
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9
10
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13
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100
| 37107287533902102798797998220837590246510135740250
46376937677490009712648124896970078050417018260538
74324986199524741059474233309513058123726617309629
91942213363574161572522430563301811072406154908250
23067588207539346171171980310421047513778063246676
89261670696623633820136378418383684178734361726757
28112879812849979408065481931592621691275889832738
44274228917432520321923589422876796487670272189318
47451445736001306439091167216856844588711603153276
70386486105843025439939619828917593665686757934951
62176457141856560629502157223196586755079324193331
64906352462741904929101432445813822663347944758178
92575867718337217661963751590579239728245598838407
58203565325359399008402633568948830189458628227828
80181199384826282014278194139940567587151170094390
35398664372827112653829987240784473053190104293586
86515506006295864861532075273371959191420517255829
71693888707715466499115593487603532921714970056938
54370070576826684624621495650076471787294438377604
53282654108756828443191190634694037855217779295145
36123272525000296071075082563815656710885258350721
45876576172410976447339110607218265236877223636045
17423706905851860660448207621209813287860733969412
81142660418086830619328460811191061556940512689692
51934325451728388641918047049293215058642563049483
62467221648435076201727918039944693004732956340691
15732444386908125794514089057706229429197107928209
55037687525678773091862540744969844508330393682126
18336384825330154686196124348767681297534375946515
80386287592878490201521685554828717201219257766954
78182833757993103614740356856449095527097864797581
16726320100436897842553539920931837441497806860984
48403098129077791799088218795327364475675590848030
87086987551392711854517078544161852424320693150332
59959406895756536782107074926966537676326235447210
69793950679652694742597709739166693763042633987085
41052684708299085211399427365734116182760315001271
65378607361501080857009149939512557028198746004375
35829035317434717326932123578154982629742552737307
94953759765105305946966067683156574377167401875275
88902802571733229619176668713819931811048770190271
25267680276078003013678680992525463401061632866526
36270218540497705585629946580636237993140746255962
24074486908231174977792365466257246923322810917141
91430288197103288597806669760892938638285025333403
34413065578016127815921815005561868836468420090470
23053081172816430487623791969842487255036638784583
11487696932154902810424020138335124462181441773470
63783299490636259666498587618221225225512486764533
67720186971698544312419572409913959008952310058822
95548255300263520781532296796249481641953868218774
76085327132285723110424803456124867697064507995236
37774242535411291684276865538926205024910326572967
23701913275725675285653248258265463092207058596522
29798860272258331913126375147341994889534765745501
18495701454879288984856827726077713721403798879715
38298203783031473527721580348144513491373226651381
34829543829199918180278916522431027392251122869539
40957953066405232632538044100059654939159879593635
29746152185502371307642255121183693803580388584903
41698116222072977186158236678424689157993532961922
62467957194401269043877107275048102390895523597457
23189706772547915061505504953922979530901129967519
86188088225875314529584099251203829009407770775672
11306739708304724483816533873502340845647058077308
82959174767140363198008187129011875491310547126581
97623331044818386269515456334926366572897563400500
42846280183517070527831839425882145521227251250327
55121603546981200581762165212827652751691296897789
32238195734329339946437501907836945765883352399886
75506164965184775180738168837861091527357929701337
62177842752192623401942399639168044983993173312731
32924185707147349566916674687634660915035914677504
99518671430235219628894890102423325116913619626622
73267460800591547471830798392868535206946944540724
76841822524674417161514036427982273348055556214818
97142617910342598647204516893989422179826088076852
87783646182799346313767754307809363333018982642090
10848802521674670883215120185883543223812876952786
71329612474782464538636993009049310363619763878039
62184073572399794223406235393808339651327408011116
66627891981488087797941876876144230030984490851411
60661826293682836764744779239180335110989069790714
85786944089552990653640447425576083659976645795096
66024396409905389607120198219976047599490197230297
64913982680032973156037120041377903785566085089252
16730939319872750275468906903707539413042652315011
94809377245048795150954100921645863754710598436791
78639167021187492431995700641917969777599028300699
15368713711936614952811305876380278410754449733078
40789923115535562561142322423255033685442488917353
44889911501440648020369068063960672322193204149535
41503128880339536053299340368006977710650566631954
81234880673210146739058568557934581403627822703280
82616570773948327592232845941706525094512325230608
22918802058777319719839450180888072429661980811197
77158542502016545090413245809786882778948721859617
72107838435069186155435662884062257473692284509516
20849603980134001723930671666823555245252804609722
53503534226472524250874054075591789781264330331690 |
Only need to add the first 11 digits.
With the number in a string n,
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2
3
4
| >>> n = n.splitlines()
>>> n = [int(x[0:11]) for x in n]
>>> str(sum(n))[0:10]
'5537376230' |
Tags: problem13, project euler, python, sum of digits
Posted in sum of digits | No Comments »
February 4th, 2009
n! means n × (n − 1) × ... × 3 × 2 × 1
Find the sum of the digits in the number 100!
Two liner building off of problem 16:
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3
4
| >>> n = reduce(mul, range(1,101))
>>> sum([int(x) for x in str(n)])
648
>>> |
Tags: factorial, problem1, project euler, python, sum of digits
Posted in sum of digits | No Comments »
February 4th, 2009
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
A little sieve of Eratosthenes…
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| # sieve of Eratosthenes
def sieve(n):
primes = range(2,n)
x = 0
while primes[x] < n**0.5:
primes = [y for y in primes if y==primes[x] or y%primes[x]]
x = x+1
return primes
>>> sum(sieve(2000000))
142913828922L |
Tags: primes, problem10, project euler, sieve of eratosthenes
Posted in primes | 1 Comment »
February 4th, 2009
A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a^(2) + b^(2) = c^(2)
For example, 3^(2) + 4^(2) = 9 + 16 = 25 = 5^(2).
There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.
Solved mathematically with a little help from the python interpreter and this site:
a= n^2 -m^2; b = 2mn; c = n^2 + m^2
a + b + c = 1000
(n^2 -m^2) + 2mn + (n^2 + m^2) = 1000
2n^2 + 2mn = 1000
2n(n+m) = 1000
n(n+m) = 500
n^2 + mn = 500
m < n
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| >>> for i in range(1,31):
if not (500 - i**2) % i:
print i
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20 |
A few moments of thought reveals that only m=20 satisfies the inequality m < n.
n = 20; m = 5;
a = 200; b = 375; c = 425;
abc = 31875000
Tags: problem1, project euler, pythagorean triplets
Posted in pythagorean triplets | No Comments »
February 3rd, 2009
2^(15) = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the number 2^(1000)?
Another very simple solution in python. I almost used reduce(add,...). Silly!
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| >>> sum([int(x) for x in str(2**1000)])
1366
>>> |
Tags: problem16, project euler, python, sum of digits
Posted in sum of digits | 1 Comment »
February 3rd, 2009
Find the greatest product of five consecutive digits in the 1000-digit number.
73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
Simple little one…
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| def problem8(n):
nums = [int(x) for x in n]
prods = [reduce(mul,nums[i:i+5]) for i in range(len(nums)-5)]
return max(prods) |
Tags: problem8, project euler, python, sum of digits
Posted in sum of digits | No Comments »
February 3rd, 2009
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.
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| def is_palindrome(x):
x = str(x)
return x == x[::-1]
>>> ans = 0
for x in range(100,1000):
for y in range(x,1000):
prod = x*y
if is_palindrome(prod):
ans = max([prod,ans])
>>> ans
906609 |
Tags: palindrome, problem4, project euler
Posted in palindrome | No Comments »
February 3rd, 2009
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 10001st prime number?
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| import math
def naive_test_prime(x):
if not x % 2 and x is not 2:
return False
sqrx = int(math.ceil(x**0.5))
for y in range(3, sqrx + 1, 2):
if not x % y:
return False
return True
def find_nth_prime(n):
k = 1
primes = [2,3]
while len(primes) <= n:
a = 6*k - 1
b = 6*k + 1
if naive_test_prime(a):
primes.append(a)
if naive_test_prime(b):
primes.append(b)
k = k + 1
return primes[n-1]
>> find_nth_prime(10001)
>> 104743 |
Tags: 6k+/-1, primes, problem7, project euler
Posted in primes | No Comments »
February 3rd, 2009
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
Was originally going to go another route, but range/xrange (and lists in general) in python are limited to int size.
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| def gpf(x):
prime_factors = []
c = 3
while c <= x:
if not x % c:
x = x / c
prime_factors.append(c)
c = c + 2
return max(prime_factors)
>>> gpf(600851475143)
6857
>>> |
Tags: composites, factors, primes, problem3, project euler
Posted in primes | No Comments »
February 3rd, 2009
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 number that is evenly divisible by all of the numbers from 1 to 20?
Brute force with a few obvious optimizations (using only the upper half of the range of divisors; starting from the lowest possible solution 20*19).
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| def multiples(x,y):
while True:
yield x*y
y = y+1
for m in multiples(20,19):
solution = None
for n in range(11,20):
if m % n:
break
else:
print m
break |
Tags: generator, lcm, multiples, problem5, project euler, python
Posted in multiples | No Comments »
January 29th, 2009
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.
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| sum(range(1,101)) ** 2 - sum( [x**2 for x in range(1,101)] ) |
Of course there is a simpler way to do this using formulae, but that’s no fun.
Tags: problem 6, project euler, python, squares, sum
Posted in squares | No Comments »
January 29th, 2009
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, …
Find the sum of all the even-valued terms in the sequence which do not exceed four million.
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| # use a generator for fibonacci series
def fibonacci(x):
a, b = 1, 2
while a < x:
yield a
a, b = b, a+b
for x in fibonacci(4000000):
if not x % 2:
y += x
y
4613732 |
Tags: fibonacci, generator, project euler, python
Posted in fibonacci | 1 Comment »