{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$f(n) = 1 + \\frac{1}{1 + f(n-1)}$\n",
    "\n",
    "$f(1) = \\frac{3}{2}$\n",
    "\n",
    "$f(2) = 1 + \\frac{1}{1 + f(1)} = 1 + \\frac{1}{1 + \\frac{3}{2}} = 1 + \\frac{1}{\\frac{5}{2}} = 1 + \\frac{2}{5} = \\frac{7}{5}$\n",
    "\n",
    "$f(3) = 1 + \\frac{1}{1 + f(2)} = 1 + \\frac{1}{1 + \\frac{7}{5}} = 1 + \\frac{1}{\\frac{12}{5}} = 1 + \\frac{5}{12} = \\frac{17}{12}$\n",
    "\n",
    "$f(n) = 1 + \\frac{1}{1 + \\frac{n}{d}} = 1 + \\frac{1}{\\frac{n+d}{d}} = 1 + \\frac{d}{n+d} = \\frac{n+2d}{n+d}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "153"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "count = 0\n",
    "n, d = 3, 2\n",
    "(1..1000).each do\n",
    "  # puts \"#{n}, #{d}, #{n.to_s.length > d.to_s.length}\"\n",
    "  count += 1 if n.to_s.length > d.to_s.length\n",
    "  n, d = n + 2 * d, n + d\n",
    "end\n",
    "count"
   ]
  },
  {
   "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
}