{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 5-digit number, $16807=7^5$, is also a fifth power. Similarly, the 9-digit number, $134217728=8^9$, is a ninth power.\n",
    "\n",
    "How many _n_-digit positive integers exist which are also an *n*th power?\n",
    "\n",
    "$(\\log_{10} x) + 1 = n ; x = y ^ n$\n",
    "\n",
    "$log_y x = n = (\\log_{10} x) + 1$\n",
    "\n",
    "Possibly valid if $ log_y x \\le n = (\\log_{10} x) + 1$\n",
    "\n",
    "Possibly valid if $\\log_{10} y^n \\le n + 1$\n",
    "\n",
    "$n \\log_{10} y \\le n + 1$\n",
    "\n",
    "$\\log_{10} y \\le \\frac{n + 1}{n}$\n",
    "\n",
    "$y < 10$ as $10^n$ has $n+1$ digits.\n",
    "\n",
    "$y^n$ has $\\left \\lfloor \\log_{10} y^n \\right \\rfloor + 1 = \\left \\lfloor n \\log_{10} y \\right \\rfloor + 1$ digits.\n",
    "\n",
    "$\\left \\lfloor n \\log_{10} y \\right \\rfloor + 1 = n$\n",
    "\n",
    "$n \\left \\lfloor \\log_{10} y \\right \\rfloor = n - 1$\n",
    "\n",
    "$\\left \\lfloor \\log_{10} y \\right \\rfloor = \\frac{n - 1}{n}$\n",
    "\n",
    "$\\left \\lfloor \\log_{10} y \\right \\rfloor = 1 - \\frac{1}{n}$\n",
    "\n",
    "$\\frac{1}{n} = 1 - \\left \\lfloor \\log_{10} y \\right \\rfloor$\n",
    "\n",
    "${n} = \\frac{1}{1 - \\left \\lfloor \\log_{10} y \\right \\rfloor}$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "$ n \\log_{10} y  + 1 < n$\n",
    "\n",
    "$n \\log_{10} y< n - 1$\n",
    "\n",
    "$ \\log_{10} y < \\frac{n - 1}{n}$\n",
    "\n",
    "$\\log_{10} y < 1 - \\frac{1}{n}$\n",
    "\n",
    "$\\frac{1}{n} > 1 -  \\log_{10} y $\n",
    "\n",
    "${n} < \\frac{1}{1 -  \\log_{10} y}$\n",
    "\n",
    "${n} = \\left \\lfloor \\frac{1}{1 -  \\log_{10} y} \\right \\rfloor$\n",
    "\n",
    "| y | n | $y^n$ |\n",
    "|-----|\n",
    "| 1 | 1 | 1 |\n",
    "| 2 | 1 | 2 |\n",
    "| 3 | 1 | 3 |\n",
    "| 4 | 1 | 4 |\n",
    "| 4 | 2 | 16 |\n",
    "| 5 | 1 | 5 |\n",
    "| 5 | 2 | 25 |\n",
    "| 5 | 3 | 125 |\n",
    "| 6 | 1 | 6 |\n",
    "| 6 | 2 | 36 |\n",
    "| 6 | 3 | 216 |\n",
    "| 6 | 4 | 1296 |\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1, 1\n",
      "2, 1\n",
      "3, 1\n",
      "4, 2\n",
      "5, 3\n",
      "6, 4\n",
      "7, 6\n",
      "8, 10\n",
      "9, 21\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1..9"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1..9).each do |y|\n",
    "  puts \"#{y}, #{(1 / (1 - Math.log(y, 10))).floor}\"\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "49"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1..9).map {|y| (1 / (1 - Math.log(y, 10))).floor}.sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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