156 lines
3.4 KiB
Plaintext
156 lines
3.4 KiB
Plaintext
{
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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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"# Square root convergents (Euler Problem 57)"
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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=57](https://projecteuler.net/problem=57)\n",
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"\n",
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"It is possible to show that the square root of two can be expressed as an infinite continued fraction.\n",
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"\n",
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"√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + ... ))) = 1.414213...\n",
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"\n",
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"By expanding this for the first four iterations, we get:\n",
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"\n",
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"1 + 1/2 = 3/2 = 1.5\n",
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"\n",
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"1 + 1/(2 + 1/2) = 7/5 = 1.4\n",
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"\n",
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"1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...\n",
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"\n",
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"1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...\n",
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"\n",
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"The next three expansions are 99/70, 239/169, and 577/408, but the eighth expansion, 1393/985, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.\n",
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"\n",
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"In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?"
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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_digit_count(n):\n",
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" c = 0\n",
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" while n:\n",
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" c += 1\n",
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" n //= 10\n",
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" return c\n",
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"\n",
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"assert(get_digit_count(1337) == 4)"
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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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"collapsed": true
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},
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"outputs": [],
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"source": [
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"def gcd(a, b):\n",
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" if b > a:\n",
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" a, b = b, a\n",
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" while a % b != 0:\n",
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" a, b = b, a % b\n",
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" return b\n",
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" \n",
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"assert(gcd(100, 35) == 5)\n",
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"\n",
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"def add_fractions(n1, d1, n2, d2):\n",
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" d = d1 * d2\n",
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" n1 = n1 * (d // d1)\n",
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" n2 = n2 * (d // d2)\n",
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" n = n1 + n2\n",
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" p = gcd(n, d)\n",
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" return (n // p, d // 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": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"def next_expension(n, d):\n",
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" n, d = add_fractions(1, 1, n, d)\n",
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" return add_fractions(1, 1, d, n)\n",
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"\n",
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"c = 0\n",
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"\n",
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"n, d = (3, 2)\n",
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"for i in range(1000):\n",
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" if get_digit_count(n) > get_digit_count(d):\n",
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" c += 1\n",
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" n, d = next_expension(n, 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": 4,
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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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"153\n"
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]
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}
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],
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"source": [
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"s = c\n",
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"print(s)\n",
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"assert(s == 153)"
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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": "Mon, 24 Dec 2018, 23:35",
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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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"square root",
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"gcd",
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"airplane",
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"fractions"
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]
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"nbformat": 4,
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}
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