Solved 42 and 43.
parent
ab4840204c
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dd8208ce12
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@ -20,6 +20,49 @@
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"Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words? (The file is saved in the same directory as this notebook file as EulerProblem042.txt)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"def get_words():\n",
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" with open(\"EulerProblem042.txt\", \"r\") as f:\n",
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" s = f.read()\n",
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" words = [w.strip('\"') for w in s.split(\",\")]\n",
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" return words\n",
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"\n",
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"def calculate_word_value(word):\n",
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" word = word.upper()\n",
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" return sum([ord(letter) - 64 for letter in word])\n",
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"\n",
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"assert(calculate_word_value(\"sky\") == 55)\n",
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"\n",
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"def get_triangle_numbers(n):\n",
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" return {n * (n + 1) // 2 for n in range(1, 101)}"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"162\n"
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]
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}
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],
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"source": [
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"triangle_numbers = get_triangle_numbers(100)\n",
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"\n",
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"s = len([word for word in get_words() if calculate_word_value(word) in triangle_numbers])\n",
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"assert(s == 162)\n",
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"print(s)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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@ -31,7 +74,7 @@
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}
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],
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"metadata": {
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"completion_date": "",
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"completion_date": "Thu, 20 Dec 2018, 16:14",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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@ -13,6 +13,8 @@
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"collapsed": true
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},
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"source": [
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"[https://projecteuler.net/problem=43](https://projecteuler.net/problem=43)\n",
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"\n",
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"The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.\n",
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"\n",
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"Let $d_1$ be the 1st digit, $d_2$ be the 2nd digit, and so on. In this way, we note the following:\n",
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@ -34,6 +36,103 @@
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"Find the sum of all 0 to 9 pandigital numbers with this property."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We write a function that tests for the sub-string divisibility property. Then we simply brute force."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [],
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"source": [
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"def is_sub_string_divisible(s):\n",
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" s = \"\".join(s)\n",
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" if int(s[1:4]) % 2 != 0:\n",
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" return False\n",
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" if int(s[2:5]) % 3 != 0:\n",
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" return False\n",
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" if int(s[3:6]) % 5 != 0:\n",
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" return False\n",
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" if int(s[4:7]) % 7 != 0:\n",
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" return False\n",
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" if int(s[5:8]) % 11 != 0:\n",
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" return False\n",
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" if int(s[6:9]) % 13 != 0:\n",
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" return False\n",
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" if int(s[7:10]) % 17 != 0:\n",
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" return False\n",
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" return True\n",
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"\n",
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"assert(is_sub_string_divisible(\"1406357289\"))\n",
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"assert(is_sub_string_divisible(\"1406357298\") == False)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [],
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"source": [
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"import itertools\n",
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"\n",
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"s = sum([int(\"\".join(p)) for p in itertools.permutations(\"0123456789\") if is_sub_string_divisible(p)])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"16695334890\n"
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]
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}
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],
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"source": [
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"print(s)\n",
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"assert(s == 16695334890)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"I just want to write my own permutations functions to see that I am able to do it quickly. We pick each element once and combine it with the permutations for the remaining items:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 32,
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"metadata": {},
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"outputs": [],
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"source": [
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"def permutations(items):\n",
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" items = list(items)\n",
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" if not items:\n",
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" return [[]]\n",
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" perms = []\n",
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" for item in items:\n",
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" items_new = list(items)\n",
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" items_new.remove(item)\n",
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" permutations_new = permutations(items_new)\n",
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" for perm in permutations_new:\n",
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" perms.append([item] + perm)\n",
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" return perms\n",
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"\n",
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"\n",
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"assert(permutations([]) == [[]])\n",
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"assert(permutations([1]) == [[1]])\n",
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"assert(permutations([1,2]) == [[1,2], [2,1]])\n",
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"assert(permutations([1,2,3]) == [[1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1]])\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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@ -45,7 +144,7 @@
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}
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],
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"metadata": {
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"completion_date": "",
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"completion_date": "Fri, 21 Dec 2018, 16:28",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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@ -65,7 +164,8 @@
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},
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"tags": [
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"pandigital",
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"divisibility"
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"divisibility",
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"permutations"
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]
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},
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"nbformat": 4,
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@ -0,0 +1,60 @@
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Pentagon numbers (Euler Problem 44)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"collapsed": true
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},
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"source": [
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"[https://projecteuler.net/problem=44](https://projecteuler.net/problem=44)\n",
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"\n",
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"Pentagonal numbers are generated by the formula, $P_n=n(3n−1)/2$. The first ten pentagonal numbers are:\n",
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"\n",
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" 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...\n",
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"\n",
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"It can be seen that $P_4 + P_7 = 22 + 70 = 92 = P_8$. However, their difference, 70 − 22 = 48, is not pentagonal.\n",
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"\n",
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"Find the pair of pentagonal numbers, $P_j$ and $P_k$, for which their sum and difference are pentagonal and $D = |P_k − P_j|$ is minimised; what is the value of D?"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"completion_date": "",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.5"
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},
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"tags": []
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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@ -11789,6 +11789,56 @@ div#notebook {
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [1]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_words</span><span class="p">():</span>
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<span class="k">with</span> <span class="nb">open</span><span class="p">(</span><span class="s2">"EulerProblem042.txt"</span><span class="p">,</span> <span class="s2">"r"</span><span class="p">)</span> <span class="k">as</span> <span class="n">f</span><span class="p">:</span>
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<span class="n">s</span> <span class="o">=</span> <span class="n">f</span><span class="o">.</span><span class="n">read</span><span class="p">()</span>
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<span class="n">words</span> <span class="o">=</span> <span class="p">[</span><span class="n">w</span><span class="o">.</span><span class="n">strip</span><span class="p">(</span><span class="s1">'"'</span><span class="p">)</span> <span class="k">for</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">s</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s2">","</span><span class="p">)]</span>
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<span class="k">return</span> <span class="n">words</span>
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<span class="k">def</span> <span class="nf">calculate_word_value</span><span class="p">(</span><span class="n">word</span><span class="p">):</span>
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<span class="n">word</span> <span class="o">=</span> <span class="n">word</span><span class="o">.</span><span class="n">upper</span><span class="p">()</span>
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<span class="k">return</span> <span class="nb">sum</span><span class="p">([</span><span class="nb">ord</span><span class="p">(</span><span class="n">letter</span><span class="p">)</span> <span class="o">-</span> <span class="mi">64</span> <span class="k">for</span> <span class="n">letter</span> <span class="ow">in</span> <span class="n">word</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">calculate_word_value</span><span class="p">(</span><span class="s2">"sky"</span><span class="p">)</span> <span class="o">==</span> <span class="mi">55</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">get_triangle_numbers</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<span class="k">return</span> <span class="p">{</span><span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">101</span><span class="p">)}</span>
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</pre></div>
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</div>
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</div>
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</div>
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [2]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">triangle_numbers</span> <span class="o">=</span> <span class="n">get_triangle_numbers</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
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<span class="n">s</span> <span class="o">=</span> <span class="nb">len</span><span class="p">([</span><span class="n">word</span> <span class="k">for</span> <span class="n">word</span> <span class="ow">in</span> <span class="n">get_words</span><span class="p">()</span> <span class="k">if</span> <span class="n">calculate_word_value</span><span class="p">(</span><span class="n">word</span><span class="p">)</span> <span class="ow">in</span> <span class="n">triangle_numbers</span><span class="p">])</span>
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<span class="k">assert</span><span class="p">(</span><span class="n">s</span> <span class="o">==</span> <span class="mi">162</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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</div>
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<div class="output_wrapper">
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<div class="output">
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<div class="output_area">
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<div class="prompt"></div>
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<div class="output_subarea output_stream output_stdout output_text">
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<pre>162
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</pre>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [ ]:</div>
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<div class="inner_cell">
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<div class="input_area">
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@ -663,10 +663,10 @@
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</tr>
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<tr class="table-warning">
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<tr>
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<td><a href="EulerProblem042.html">Problem 042</a></td>
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<td></td>
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<td>Thu, 20 Dec 2018, 16:14</td>
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<td>
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<kbd>triangle</kbd>
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