From b7e7314e99dbe4f055a68072f29bc7440317df00 Mon Sep 17 00:00:00 2001 From: Felix Martin Date: Tue, 6 Feb 2018 19:23:59 +0100 Subject: [PATCH] Solved 22 in ipython. --- ipython/EulerProblem020.html | 254 +++++++++++++ ipython/EulerProblem021.html | 655 ++++++++++++++++++++++++++++++++++ ipython/EulerProblem021.ipynb | 28 +- ipython/EulerProblem022.html | 298 ++++++++++++++++ ipython/EulerProblem022.ipynb | 126 +++++++ ipython/EulerProblem022.txt | 1 + ipython/EulerProblem023.html | 210 +++++++++++ ipython/EulerProblem023.ipynb | 54 +++ 8 files changed, 1615 insertions(+), 11 deletions(-) create mode 100644 ipython/EulerProblem020.html create mode 100644 ipython/EulerProblem021.html create mode 100644 ipython/EulerProblem022.html create mode 100644 ipython/EulerProblem022.ipynb create mode 100644 ipython/EulerProblem022.txt create mode 100644 ipython/EulerProblem023.html create mode 100644 ipython/EulerProblem023.ipynb diff --git a/ipython/EulerProblem020.html b/ipython/EulerProblem020.html new file mode 100644 index 0000000..d782993 --- /dev/null +++ b/ipython/EulerProblem020.html @@ -0,0 +1,254 @@ + + + + + +EulerProblem020 + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+
+
+
+
+

Euler Problem 20

n! means n × (n − 1) × ... × 3 × 2 × 1

+

For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800, +and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.

+

Find the sum of the digits in the number 100!

+ +
+
+
+
+
+
+
+
+

Write our own factorial implementation and get the solution.

+ +
+
+
+
+
+
In [1]:
+
+
+
def factorial(n):
+    assert(n > 0)
+    if n == 1:
+        return 1
+    else:
+        return n * factorial(n - 1)
+
+ +
+
+
+ +
+
+
+
In [2]:
+
+
+
f100 = factorial(100)
+print(sum(map(int, str(f100))))
+assert(sum(map(int, str(f100))) == 648)
+
+ +
+
+
+ +
+
+ + +
+
+
648
+
+
+
+ +
+
+ +
+
+
+ + diff --git a/ipython/EulerProblem021.html b/ipython/EulerProblem021.html new file mode 100644 index 0000000..9079b67 --- /dev/null +++ b/ipython/EulerProblem021.html @@ -0,0 +1,655 @@ + + + + + +EulerProblem021 + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+
+
+
+
+

Euler Problem 21

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.

+ +
+
+
+
+
+
+
+
+

We reuse the sieve of eratosthenes (which we called get_primes_smaller previously). What annoys me about the implementation is that we do not stop scanning for prospects once the current prime squared is greater than the limit. Let's test if it returns the same result and compare the execution time.

+ +
+
+
+
+
+
In [1]:
+
+
+
def get_primes_smaller(limit):
+    primes = []
+    prospects = [n for n in range(2, limit)]
+    while prospects:
+        p = prospects[0]
+        prospects = [x for x in prospects if x % p != 0]
+        primes.append(p)
+    return primes
+
+def sieve_of_eratosthenes(limit):
+    primes = []
+    prospects = [n for n in range(2, limit)]
+    while prospects:
+        p = prospects[0]
+        prospects = [x for x in prospects if x % p != 0]
+        primes.append(p)
+        if p * p > limit:
+            break
+    primes += prospects
+    return primes    
+
+import timeit
+import functools
+print(timeit.timeit(functools.partial(get_primes_smaller, 10000), number=100))
+print(timeit.timeit(functools.partial(sieve_of_eratosthenes, 10000), number=100))
+assert(get_primes_smaller(10000) == sieve_of_eratosthenes(10000))
+
+ +
+
+
+ +
+
+ + +
+
+
5.062845234464119
+0.3188800166435408
+
+
+
+ +
+
+ +
+
+
+
+
+
+

Okay, this difference is sick actually. Interesting to see what the difference between O(sqrt(n)) and O(n) can mean. Now we can use the primes to do the factorization. I will again apply some optimizations and see how it turns out

+ +
+
+
+
+
+
In [2]:
+
+
+
def get_prime_factors_old(number):
+    prime_generator = sieve_of_eratosthenes(number // 2)
+    remainder = number
+    factors = []
+    for p in prime_generator:
+        while remainder % p == 0:
+            remainder //= p
+            factors.append(p)
+        if remainder == 1 or p * p > number:
+            break
+    if remainder != 1:
+        factors.append(remainder)
+    return factors
+
+def get_prime_factors(number):
+    remainder = number
+    factors = []
+    for p in sieve_of_eratosthenes(number // 2):
+        while remainder % p == 0:
+            remainder //= p
+            factors.append(p)
+        if remainder == 1:
+            break
+        if p > remainder or p * p > number:
+            factors.append(remainder)
+            break
+    return factors
+
+import timeit
+import functools
+print(timeit.timeit(functools.partial(get_prime_factors_old, 1000000), number=10))
+print(timeit.timeit(functools.partial(get_prime_factors, 1000000), number=10))
+assert(get_prime_factors_old(100000) == get_prime_factors(100000))
+
+ +
+
+
+ +
+
+ + +
+
+
6.370271180404346
+5.785088868397899
+
+
+
+ +
+
+ +
+
+
+
+
+
+

Okay, actually no big difference. Kind of expected if you check how similar the implementations are. I though stopping if p is greater than the remainder may be a common case.

+

Now we can use the prime factors to get all factors. For this each prime number can be used 0 to n times where n is the number how often the prime occurs in the prime factorization. For example, all possible fators for the primes $2, 2, 3$ are $2^0 \times 3^0, 2^1 \times 3^0, 2^2 \times 3^0, 2^0 \times 3^1, 2^1 \times 3^1, 2^2 \times 3^1$ which results in $1, 2, 4, 3, 6, 12$.

+

So we proceed in the following way. Once we have the prime factors we group then. Then we calculate the factors for each group. For example for $2,2$ the factors would be $1, 2, 4$ as explained in the previous paragraph. Then we calculate the combinations of all factor groups.

+ +
+
+
+
+
+
In [3]:
+
+
+
def group(xs):
+    from functools import reduce
+    rs = []
+    def f(xss, x):
+        if xss and x in xss[-1]:
+            xss[-1].append(x)
+        else:
+            xss.append([x])
+        return xss
+    return reduce(f, xs, [])
+
+assert(group([1, 2, 2, 2, 4, 4, 5]) == [[1], [2, 2, 2], [4, 4], [5]])
+
+def group(xs):
+    r = []
+    for x in xs:
+        if r and r[-1][0] == x:
+            r[-1].append(x)
+        else:
+            r.append([x])
+    return r
+
+assert(group([1, 2, 2, 2, 4, 4, 5]) == [[1], [2, 2, 2], [4, 4], [5]])
+
+ +
+
+
+ +
+
+
+
+
+
+

As you can see I had a fancy version with reduce first, but the loop is way more readable. We iterate over the input list. If the current item is in the last sublist of the return list we add it to the list. Otherwise we append a new sublist with the item.

+

Next we write a function that calculates the resulting factors for a factor group.

+ +
+
+
+
+
+
In [4]:
+
+
+
def prime_group_to_factors(group):
+    f = group[0]
+    n = len(group)
+    return [f**i for i in range(n + 1)]
+
+assert(prime_group_to_factors([2, 2, 2]) == [1, 2, 4, 8])
+
+ +
+
+
+ +
+
+
+
+
+
+

No we only have to get all factors by calculating the combination of two factor lists.

+ +
+
+
+
+
+
In [5]:
+
+
+
def combine_factors(xs, ys):
+    return [x * y for y in ys for x in xs]
+
+assert(combine_factors([1, 2, 4], [1, 3]) == [1, 2, 4, 3, 6, 12])
+
+ +
+
+
+ +
+
+
+
+
+
+

Now actually we want to combine an arbitrary number of factor lists.

+ +
+
+
+
+
+
In [6]:
+
+
+
def combine_factors(xss):
+    ys = [1]
+    for xs in xss:
+        ys = [x * y for y in ys for x in xs]
+    return ys
+
+ +
+
+
+ +
+
+
+
In [7]:
+
+
+
def get_divisors(n):
+    prime_factors = get_prime_factors(n)
+    prime_groups = group(prime_factors)
+    factor_groups = list(map(prime_group_to_factors, prime_groups))
+    divisors = combine_factors(factor_groups)
+    divisors.sort()
+    return divisors
+    
+assert(get_divisors(220) == [1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220])
+assert(get_divisors(13) == [1, 13])
+assert(get_divisors(26) == [1, 2, 13, 26])
+
+ +
+
+
+ +
+
+
+
+
+
+

We can finally start to get the solution. Remember that we only want to get the sum of proper divisors which are the divisors withouth the number itself.

+ +
+
+
+
+
+
In [8]:
+
+
+
def sum_of_proper_divisors(number):
+    return sum(get_divisors(number)[:-1])
+
+assert(sum_of_proper_divisors(220) == 284)
+assert(sum_of_proper_divisors(284) == 220)
+
+def is_amicable(number):
+    s = sum_of_proper_divisors(number)
+    if s == number:
+        return False
+    return number == sum_of_proper_divisors(s)
+
+assert(is_amicable(220))
+
+ +
+
+
+ +
+
+
+
+
+
+

The if statement is very import because a number pair $a, b$ is only amicable if $a \neq b$.

+ +
+
+
+
+
+
In [9]:
+
+
+
amicable_till_10000 = [i for i in range(1, 10000) if is_amicable(i)]
+assert(sum(amicable_till_10000) == 31626)
+print(sum(amicable_till_10000))
+
+ +
+
+
+ +
+
+ + +
+
+
31626
+
+
+
+ +
+
+ +
+
+
+
+
+
+

I found some brute force solutions in the Euler forum and they claim to have a sick performance. I want to compare them.

+ +
+
+
+
+
+
In [10]:
+
+
+
def brute_force():
+    return sum([i for i in range(1, 10000) if is_amicable(i)])
+
+import timeit
+print(timeit.timeit(brute_force, number=10))
+
+ +
+
+
+ +
+
+ + +
+
+
108.68903765424807
+
+
+
+ +
+
+ +
+
+
+
In [14]:
+
+
+
def sum_of_proper_divisors(n):
+    from math import sqrt
+    s = 1
+    for i in range(2, int(sqrt(n)) + 1):
+        if n % i == 0:
+            s += i
+            s += (n // i)
+    return s
+
+assert(sum_of_proper_divisors(220) == 284)
+       
+def is_amicable(number):
+    s = sum_of_proper_divisors(number)
+    if s == number:
+        return False
+    return number == sum_of_proper_divisors(s)
+
+def brute_force():
+    return sum([i for i in range(1, 10000) if is_amicable(i)])   
+    
+import timeit
+print(timeit.timeit(brute_force, number=10))
+
+ +
+
+
+ +
+
+ + +
+
+
0.7862764837699387
+
+
+
+ +
+
+ +
+
+
+
+
+
+

This is actually really embarrassing. Especially I was aware of the sqrt trick all the way but did not manage to put it into use properly. Another example where greate engineer will leave you far behind compared to great thinking.

+ +
+
+
+
+
+ + diff --git a/ipython/EulerProblem021.ipynb b/ipython/EulerProblem021.ipynb index 5741416..5aef2a4 100644 --- a/ipython/EulerProblem021.ipynb +++ b/ipython/EulerProblem021.ipynb @@ -24,7 +24,9 @@ { "cell_type": "code", "execution_count": 1, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [ { "name": "stdout", @@ -74,7 +76,9 @@ { "cell_type": "code", "execution_count": 2, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [ { "name": "stdout", @@ -294,7 +298,9 @@ { "cell_type": "code", "execution_count": 9, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [ { "name": "stdout", @@ -320,7 +326,9 @@ { "cell_type": "code", "execution_count": 10, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [ { "name": "stdout", @@ -341,7 +349,9 @@ { "cell_type": "code", "execution_count": 14, - "metadata": {}, + "metadata": { + "collapsed": false + }, "outputs": [ { "name": "stdout", @@ -401,16 +411,12 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.6.3" + "version": "3.5.4" }, "tags": [ "amicable", - "factors", - "combinations", - "reduce", - "prime", "sieve of eratosthenes", - "partial", + "factors", "timeit" ] }, diff --git a/ipython/EulerProblem022.html b/ipython/EulerProblem022.html new file mode 100644 index 0000000..7b7e050 --- /dev/null +++ b/ipython/EulerProblem022.html @@ -0,0 +1,298 @@ + + + + + +EulerProblem022 + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+
+
+
+
+

Euler Problem 22

Using names.txt (saved as EulerProblem022.txt in the same directory as this notebook), 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?

+ +
+
+
+
+
+
+
+
+

Okay, this should be straight forward.

+ +
+
+
+
+
+
In [ ]:
+
+
+
with open('EulerProblem022.txt', 'r') as f:
+    names = f.read().split(',')
+
+def get_score_for_name(name):
+    return sum([ord(c) - ord('A') + 1 for c in name if not c == '"'])
+
+assert(get_score_for_name('COLIN') == 53)
+names.sort()
+s = sum([(i + 1) * get_score_for_name(name) for i, name in enumerate(names)])
+assert(s == 871198282)
+print(s)
+
+ +
+
+
+ +
+
+
+
+
+
+

I think nothing to explain here. The only question is what was if we hadn't the Python sort function to sort into alphabetical order, then we would have to write our own compare function and use it with what ever sorting algorithm.

+ +
+
+
+
+
+
In [19]:
+
+
+
def compare(a, b):
+    try:
+        for i in range(len(a)):
+            if a[i] < b[i]:
+                return '{} before {}'.format(a, b)
+            elif a[i] > b[i]:
+                return '{} before {}'.format(b, a)
+    except IndexError:
+        pass
+    if len(a) < len(b):
+        return '{} before {}'.format(a, b)
+    elif len(a) > len(b):
+        return '{} before {}'.format(b, a)
+    else:
+        return '{} is {}'.format(b, a)
+
+print(compare('Felix', 'Arnold'))
+print(compare('Felix', 'Felixb'))
+print(compare('Felixb', 'Felix'))
+print(compare('Felix', 'Felix'))
+
+ +
+
+
+ +
+
+ + +
+
+
Arnold before Felix
+Felix before Felixb
+Felix before Felixb
+Felix is Felix
+
+
+
+ +
+
+ +
+
+
+
+
+
+

Obviously, the algorithm would return True/False or 0/1/-1 for real sorting.

+ +
+
+
+
+
+ + diff --git a/ipython/EulerProblem022.ipynb b/ipython/EulerProblem022.ipynb new file mode 100644 index 0000000..b346ee0 --- /dev/null +++ b/ipython/EulerProblem022.ipynb @@ -0,0 +1,126 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Euler Problem 22\n", + "\n", + "Using names.txt (saved as EulerProblem022.txt in the same directory as this notebook), 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.\n", + "\n", + "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.\n", + "\n", + "What is the total of all the name scores in the file?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Okay, this should be straight forward." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "with open('EulerProblem022.txt', 'r') as f:\n", + " names = f.read().split(',')\n", + "\n", + "def get_score_for_name(name):\n", + " return sum([ord(c) - ord('A') + 1 for c in name if not c == '\"'])\n", + "\n", + "assert(get_score_for_name('COLIN') == 53)\n", + "names.sort()\n", + "s = sum([(i + 1) * get_score_for_name(name) for i, name in enumerate(names)])\n", + "assert(s == 871198282)\n", + "print(s)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "I think nothing to explain here. The only question is what was if we hadn't the Python sort function to sort into alphabetical order, then we would have to write our own compare function and use it with what ever sorting algorithm." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Arnold before Felix\n", + "Felix before Felixb\n", + "Felix before Felixb\n", + "Felix is Felix\n" + ] + } + ], + "source": [ + "def compare(a, b):\n", + " try:\n", + " for i in range(len(a)):\n", + " if a[i] < b[i]:\n", + " return '{} before {}'.format(a, b)\n", + " elif a[i] > b[i]:\n", + " return '{} before {}'.format(b, a)\n", + " except IndexError:\n", + " pass\n", + " if len(a) < len(b):\n", + " return '{} before {}'.format(a, b)\n", + " elif len(a) > len(b):\n", + " return '{} before {}'.format(b, a)\n", + " else:\n", + " return '{} is {}'.format(b, a)\n", + "\n", + "print(compare('Felix', 'Arnold'))\n", + "print(compare('Felix', 'Felixb'))\n", + "print(compare('Felixb', 'Felix'))\n", + "print(compare('Felix', 'Felix'))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Obviously, the algorithm would return True/False or 0/1/-1 for real sorting." + ] + } + ], + "metadata": { + "completion_date": "Fri, 5 Sep 2014, 15:24", + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.5.4" + }, + "tags": [ + "sorting", + "lexicographical order" + ] + }, + "nbformat": 4, + "nbformat_minor": 0 +} diff --git a/ipython/EulerProblem022.txt b/ipython/EulerProblem022.txt new file mode 100644 index 0000000..7b8986b --- /dev/null +++ b/ipython/EulerProblem022.txt @@ -0,0 +1 @@ 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Euler Problem 23

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

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A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

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As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

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Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

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+ + diff --git a/ipython/EulerProblem023.ipynb b/ipython/EulerProblem023.ipynb new file mode 100644 index 0000000..20848be --- /dev/null +++ b/ipython/EulerProblem023.ipynb @@ -0,0 +1,54 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Euler Problem 23\n", + "\n", + "A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.\n", + "\n", + "A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.\n", + "\n", + "As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.\n", + "\n", + "Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "completion_date": "Thu, 5 Nov 2015, 14:48", + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.5.4" + }, + "tags": [ + "perfect number", + "abundant" + ] + }, + "nbformat": 4, + "nbformat_minor": 0 +}