Done puzzle 63

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+{
+ "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": []
+  }
+ ],
+ "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
+}