From: Neil Smith Date: Fri, 5 May 2017 16:44:32 +0000 (+0100) Subject: Done puzzle 64 X-Git-Url: https://git.njae.me.uk/?a=commitdiff_plain;h=refs%2Fheads%2Fmaster;p=project-euler.git Done puzzle 64 --- diff --git a/euler64.ipynb b/euler64.ipynb new file mode 100644 index 0000000..7d8d1dc --- /dev/null +++ b/euler64.ipynb @@ -0,0 +1,470 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$\\sqrt{23} = 4 + \\sqrt{23} - 4 = \\frac{1}{\\frac{1}{\\sqrt{23}-4}}$\n", + "\n", + "$\\frac{1}{\\sqrt{23}-4} = \\frac{\\sqrt{23}+4}{(\\sqrt{23}-4)(\\sqrt{23}+4)} = \\frac{\\sqrt{23}+4}{23 - 16}$\n", + "\n", + "$$\n", + "\\begin{align}\n", + "\\frac{1}{\\sqrt{23}-4} & = \\frac{\\sqrt{23}+4}{(\\sqrt{23}-4)(\\sqrt{23}+4)} \\\\\n", + " & =\\frac{\\sqrt{23}+4}{23 - 16} \\\\\n", + " & =\\frac{\\sqrt{23}+4}{7} , \\left \\lfloor \\frac{\\sqrt{23} + 4}{7} \\right \\rfloor = 1\\\\\n", + " & =1 + \\frac{\\sqrt{23} + 4 -7}{7} \\\\\n", + " & =1 + \\frac{\\sqrt{23} -3}{7} \\\\\n", + " & =1 + \\frac{\\sqrt{23} -3}{7} \n", + "\\end{align}\n", + "$$\n", + "\n", + "\n", + "$$\n", + "\\begin{align}\n", + "1 + \\frac{\\sqrt{23} -3}{7} & = 1 + \\frac{1}{\\frac{7}{\\sqrt{23} -3}} \\\\\n", + "\\frac{7}{\\sqrt{23} -3} & =\\frac{7(\\sqrt{23}+3)}{(\\sqrt{23} -3)(\\sqrt{23}+3)}\\\\\n", + " & =\\frac{7(\\sqrt{23}+3)}{23 - 9} \\\\\n", + " & =\\frac{7(\\sqrt{23}+3)}{14}\\\\\n", + " & =\\frac{\\sqrt{23} + 3}{2}, \\left \\lfloor \\frac{\\sqrt{23} + 3}{2} \\right \\rfloor = 3\\\\\n", + " & = 3 + \\frac{\\sqrt{23} + 3}{2} - \\frac{6}{2}\\\\\n", + " & = 3 + \\frac{\\sqrt{23} - 3}{2}\\\\\n", + "\\end{align}\n", + "$$\n", + "\n", + "$$\n", + "\\begin{align}\n", + "\\frac{k}{\\sqrt{n} - a} \n", + " &= \\frac{k \\left( \\sqrt{n} + a \\right)}{\\left( \\sqrt{n} - a \\right) \\left( \\sqrt{n} + a \\right)} \\\\\n", + " & = \\frac{k \\left( \\sqrt{n} + a \\right)}{n - a^2} \\\\\n", + " & =\\left \\lfloor \\frac{k \\left( \\sqrt{n} + a \\right) }{n - a^2} \\right \\rfloor + \n", + " \\frac{k \\left( \\sqrt{n} + a \\right)}{n - a^2} - \\left \\lfloor \\frac{k \\left( \\sqrt{n} + a \\right) }{n - a^2} \\right \\rfloor \\\\\n", + " & =\\left \\lfloor \\frac{k \\left( \\sqrt{n} + a \\right) }{n - a^2} \\right \\rfloor + \n", + " \\frac{\\sqrt{n} + a}{\\frac{n - a^2}{k}} - \\left \\lfloor \\frac{k \\left( \\sqrt{n} + a \\right) }{n - a^2} \\right \\rfloor \\\\\n", + " & =\\left \\lfloor \\frac{k \\left( \\sqrt{n} + a \\right) }{n - a^2} \\right \\rfloor + \n", + " \\frac{\\sqrt{n} + a - \\left \\lfloor \\frac{k \\left( \\sqrt{n} + a \\right) }{n - a^2} \\right \\rfloor \\left( \\frac{n - a^2}{k} \\right)}{\\frac{n - a^2}{k}}\\\\\n", + " & = \\mathrm{lead} + \\frac{\\sqrt{n} + \\mathrm{trail}}{\\mathrm{denom}}\n", + "\\end{align}\n", + "$$" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + ":continued" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def continued(n, k, a)\n", + " lead = ((k * (Math.sqrt(n) + a)) / (n - a ** 2)).floor\n", + " denom = (n - a ** 2) / k\n", + " trail = a - lead * denom\n", + " [lead, denom, trail]\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, 7, -3]" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 1, Math.sqrt(23).floor)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, 7, -3]" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 1, 4)" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[3, 2, -3]" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 7, 3)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, 7, -4]" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 2, 3)" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[8, 1, -4]" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 7, 4)" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, 7, -3]" + ] + }, + "execution_count": 7, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 1, 4)" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[3, 2, -3]" + ] + }, + "execution_count": 8, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continued(23, 7, 3)" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + ":continue_start" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def continue_start(n)\n", + " continued(n, 1, Math.sqrt(n).floor)\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[1, 7, -3]" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continue_start(23)" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + ":continue_until_repeat" + ] + }, + "execution_count": 11, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def continue_until_repeat(n)\n", + " a_n = [Math.sqrt(n).floor]\n", + " states = []\n", + " l, d, t = continue_start(n)\n", + " while !states.include?([d, t])\n", + " a_n << l\n", + " states << [d, t]\n", + " l, d, t = continued(n, d, -1 * t)\n", + " end\n", + " [a_n, states, states.length - states.index([d, t])]\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[[4, 1, 3, 1, 8], [[7, -3], [2, -3], [7, -4], [1, -4]], 4]" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continue_until_repeat 23" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[[3, 1, 1, 1, 1, 6], [[4, -1], [3, -2], [3, -1], [4, -3], [1, -3]], 5]" + ] + }, + "execution_count": 13, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "continue_until_repeat 13" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "4" + ] + }, + "execution_count": 14, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Math.sqrt(8).floor ** 2" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + ":repeat_period_length" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "def repeat_period_length(n)\n", + " a_n, states, period_length = continue_until_repeat(n)\n", + " period_length\n", + "end" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "5" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "repeat_period_length 13" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "4" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "odd_counts = 0\n", + "(1..13).each do |i|\n", + " if (Math.sqrt(i).floor ** 2) != i\n", + " # puts \"#{i} -> #{repeat_period_length i}\"\n", + " odd_counts += 1 if repeat_period_length(i) % 2 == 1\n", + " end\n", + "end\n", + "odd_counts" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1322" + ] + }, + "execution_count": 21, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "odd_counts = 0\n", + "(1..(10 ** 4)).each do |i|\n", + " if (Math.sqrt(i).floor ** 2) != i\n", + " # puts \"#{i} -> #{repeat_period_length i}\"\n", + " odd_counts += 1 if repeat_period_length(i) % 2 == 1\n", + " end\n", + "end\n", + "odd_counts" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Ruby 2.4.0", + "language": "ruby", + "name": "ruby" + }, + "language_info": { + "file_extension": ".rb", + "mimetype": "application/x-ruby", + "name": "ruby", + "version": "2.4.0" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}