From d7c6c35666a0749c76f18c6b140eb6751b965a68 Mon Sep 17 00:00:00 2001
From: Peter Norvig <peter@norvig.com>
Date: Thu, 24 Aug 2017 23:17:01 -0700
Subject: [PATCH] Add files via upload

---
 Countdown.ipynb | 1973 ++++++++++++++++++++++++++++++++++++-----------
 1 file changed, 1525 insertions(+), 448 deletions(-)

diff --git a/Countdown.ipynb b/Countdown.ipynb
index 9fe91e7..031a901 100644
--- a/Countdown.ipynb
+++ b/Countdown.ipynb
@@ -4,7 +4,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -24,11 +23,11 @@
     "\n",
     "> You are allowed to use *only* the four basic arithmetical operations: +, -, &times;, ÷. But brackets (parentheses) can be used wherever needed. So, for example, the solution could begin\n",
     "\n",
-    "> (10 - 9) &times; (8 ...\n",
+    "> `(10 - 9) / (8` ...\n",
     "\n",
     "> or\n",
     "\n",
-    "> 10 + (9 &times; 8) ...\n",
+    "> `10 + (9 - 8)` ...\n",
     "\n",
     "Let's see if we can solve this puzzle, and some of the related ones from Alex's [first](http://www.theguardian.com/science/2016/jan/04/can-you-solve-it-complete-the-equation-10-9-8-7-6-5-4-3-2-1-2016) and [second](http://www.theguardian.com/science/2016/jan/04/did-you-solve-it-complete-the-equation-10-9-8-7-6-5-4-3-2-1-2016) post.  We'll start with a  simpler version of the puzzle."
    ]
@@ -37,7 +36,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -54,8 +52,6 @@
    "execution_count": 1,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -81,15 +77,13 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "That's small enough that we can just enumerate all the expressions and see if any evaluates to 2016.\n",
-    "We will generate each expression as a string and then evaluate it with `eval`. But we need to catch errors such as dividing by zero, so we'll define a wrapper function, `evaluate`:"
+    "The function `itertools.product` can enumerate all the ways of filling 9 blanks with one of the four operators:"
    ]
   },
   {
@@ -97,37 +91,42 @@
    "execution_count": 2,
    "metadata": {
     "button": false,
-    "collapsed": true,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "262144"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "from __future__ import division\n",
+    "import itertools\n",
     "\n",
-    "def evaluate(exp):\n",
-    "    \"eval exp, or return None if there is an arithmetic error.\"\n",
-    "    try:\n",
-    "        return eval(exp)\n",
-    "    except ArithmeticError:\n",
-    "        return None"
+    "operators = ('+', '-', '*', '/')\n",
+    "    \n",
+    "len(set(itertools.product(operators, repeat=9)))"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "We'll try each of the four operators in each of the nine blanks, evaluate each resulting expression, and collect the ones that evaluate to 2016:"
+    "So we can fill in the equation with each possible sequence of operations, and evaluate each string to see if it equals 2016. But we need to catch errors such as dividing by zero, so we'll define a wrapper function, `evaluate`, to do the `eval`:"
    ]
   },
   {
@@ -135,31 +134,6 @@
    "execution_count": 3,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [],
-   "source": [
-    "def solve_no_brackets(ops=('+', '-', '*', '/')):\n",
-    "    \"All solutions to the countdown puzzle (with no brackets).\"\n",
-    "    exps = ('10'+a+'9'+b+'8'+c+'7'+d+'6'+e+'5'+f+'4'+g+'3'+h+'2'+i+'1'\n",
-    "            for a in ops    for b in ops    for c in ops \n",
-    "            for d in ops    for e in ops    for f in ops \n",
-    "            for g in ops    for h in ops    for i in ops)\n",
-    "    return [exp for exp in exps if evaluate(exp) == 2016]"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 4,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -172,88 +146,52 @@
        "[]"
       ]
      },
-     "execution_count": 4,
+     "execution_count": 3,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "solve_no_brackets()"
+    "def evaluate(exp):\n",
+    "    \"eval exp, or return None if there is an arithmetic error.\"\n",
+    "    try:\n",
+    "        return eval(exp)\n",
+    "    except ArithmeticError:\n",
+    "        return None\n",
+    "\n",
+    "def solve_no_brackets(operators, target=2016):\n",
+    "    \"All solutions to the countdown puzzle (with no brackets).\"\n",
+    "    exps = ('10{}9{}8{}7{}6{}5{}4{}3{}2{}1'.format(*ops)\n",
+    "            for ops in itertools.product(operators, repeat=9))\n",
+    "    return [exp for exp in exps if evaluate(exp) == target]\n",
+    "\n",
+    "solve_no_brackets(operators)"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "Too bad; we did all that work and didn't find a solution. What years *can* we find solutions for? Let's modify `solve_no_brackets` to take a collection of target years, and return a dict of the form `{year: \"expression\"}` for each expression that evaluates to one of the target years:"
+    "Too bad; we did all that work and didn't find a solution. What years *can* we find solutions for? Let's modify `solve_no_brackets` to take a collection of target years rather than a single one, and return a dict of the form `{year: 'expression'}` for each expression that evaluates to one of the target years:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 4,
    "metadata": {
     "button": false,
-    "collapsed": true,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [],
-   "source": [
-    "def solve_no_brackets(targets, ops=('+', '-', '*', '/')):\n",
-    "    \"Solutions to the no-bracket countdown puzzle matching one of targets.\"\n",
-    "    exps = ('10'+a+'9'+b+'8'+c+'7'+d+'6'+e+'5'+f+'4'+g+'3'+h+'2'+i+'1'\n",
-    "            for a in ops    for b in ops    for c in ops \n",
-    "            for d in ops    for e in ops    for f in ops \n",
-    "            for g in ops    for h in ops    for i in ops)\n",
-    "    return {int(evaluate(exp)): exp\n",
-    "            for exp in exps if evaluate(exp) in targets}"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "button": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "source": [
-    "Let's try it:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "CPU times: user 12.7 s, sys: 265 ms, total: 13 s\n",
-      "Wall time: 18.9 s\n"
-     ]
-    },
     {
      "data": {
       "text/plain": [
@@ -268,34 +206,38 @@
        " 2019: '10*9*8*7/6/5*4*3+2+1'}"
       ]
      },
-     "execution_count": 6,
+     "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "%time solve_no_brackets(range(1900, 2100))"
+    "def solve_no_brackets(operators, targets):\n",
+    "    \"All solutions to the countdown puzzle (with no brackets).\"\n",
+    "    exps = ('10{}9{}8{}7{}6{}5{}4{}3{}2{}1'.format(*ops)\n",
+    "            for ops in itertools.product(operators, repeat=9))\n",
+    "    return {int(evaluate(exp)): exp for exp in exps if evaluate(exp) in targets}\n",
+    "\n",
+    "solve_no_brackets(operators, range(1900, 2100))"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "Interesting: in the 20th and 21st centuries, there are only two \"golden eras\" where the countdown equation works: the three year period centered on 1980, and the seven year period that is centered on 2016, but omits 2016. Note it takes about half a minute to do the computation."
+    "Interesting: in the 20th and 21st centuries, there are only two \"golden eras\" where the countdown equation works: the three year period centered on 1980, and the seven year period that is centered on 2016, but omits 2016. "
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -304,7 +246,7 @@
    "source": [
     "# Four Operators, With Brackets\n",
     "\n",
-    "Now let's return to the original puzzle, with the brackets. How many ways are there of bracketing an expression with 9 binary operators? I happen to remember that this is given by the [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number), and we can [look it up](http://www.wolframalpha.com/input/?i=9th+catalan+number) to find that there are 4862 diffferent bracketing. If we enumerated and evaluated all of them, 4862 times half a minute is [40 hours](https://www.google.com/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=4862+*+1%2F2+minute+in+hours). I'm impatient, so I'd like a faster approach. One possibility would be to switch to a faster compiled language like C++ or Go. But I'll concentrate on a better algorithm and stick with Python.\n",
+    "Now let's return to the original puzzle, with the brackets. How many ways are there of bracketing an expression with 9 binary operators? I happen to remember that this is given by the [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number), and we can [look it up](http://www.wolframalpha.com/input/?i=9th+catalan+number) to find that there are 4862 diffferent bracketing. If we enumerated and evaluated all of them, it would take about 4862 times longer than doing a single `solve_no_brackets`, which took about 6 seconds, so the estimated time would be about 8 hours. I'm impatient, so I'd like a faster approach. \n",
     "\n",
     "I'll use the idea of [dynamic programming](https://en.wikipedia.org/wiki/Dynamic_programming): break the problem down into simpler subparts, and compute an answer for each subpart, and store intermediate results in a table so we don't need to re-compute them when we need them again.\n",
     "\n",
@@ -317,11 +259,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 5,
    "metadata": {
     "button": false,
     "collapsed": true,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -339,11 +280,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 6,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
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@@ -364,7 +303,7 @@
        " ((10, 9, 8, 7, 6, 5, 4, 3, 2), (1,))]"
       ]
      },
-     "execution_count": 8,
+     "execution_count": 6,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -377,7 +316,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -413,11 +351,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 9,
+   "execution_count": 7,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -426,7 +362,6 @@
    "outputs": [],
    "source": [
     "from collections import defaultdict, Counter\n",
-    "import itertools\n",
     "\n",
     "EXPS = defaultdict(dict) # e.g., EXPS[(10, 9, 8)][27] == '((10+9)+8)'\n",
     "\n",
@@ -456,7 +391,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
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@@ -468,11 +402,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 8,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -506,7 +438,7 @@
        " 720: '((10*9)*8)'}"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -517,11 +449,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 9,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -534,7 +464,7 @@
        "'((10+9)+8)'"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 9,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -547,7 +477,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -561,11 +490,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 10,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -576,8 +503,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 1min, sys: 4 s, total: 1min 4s\n",
-      "Wall time: 1min 55s\n"
+      "CPU times: user 28.2 s, sys: 653 ms, total: 28.9 s\n",
+      "Wall time: 28.9 s\n"
      ]
     },
     {
@@ -586,7 +513,7 @@
        "'(((((((10+((9*8)*7))-6)-5)*4)+3)+2)-1)'"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -599,25 +526,22 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "We have an answer! And in a lot less than 40 hours, thanks to dynamic programming! \n",
+    "We have an answer! And in a lot less than 8 hours, thanks to dynamic programming! \n",
     "\n",
     "Removing unnecessry brackets, this equation is equivalent to:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 11,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -630,7 +554,7 @@
        "2016"
       ]
      },
-     "execution_count": 13,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -643,7 +567,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -674,11 +597,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 12,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -716,11 +637,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 13,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -754,7 +673,7 @@
        " 720: '((10*9)*8)'}"
       ]
      },
-     "execution_count": 15,
+     "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -766,11 +685,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 14,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
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@@ -804,7 +721,7 @@
        "         720: 2})"
       ]
      },
-     "execution_count": 16,
+     "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -815,11 +732,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 17,
+   "execution_count": 15,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
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@@ -832,7 +747,7 @@
        "2"
       ]
      },
-     "execution_count": 17,
+     "execution_count": 15,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -845,7 +760,6 @@
    "cell_type": "markdown",
    "metadata": {
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-    "deletable": true,
     "new_sheet": false,
     "run_control": {
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@@ -857,11 +771,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 18,
+   "execution_count": 16,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -872,8 +784,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "CPU times: user 2min 10s, sys: 5.36 s, total: 2min 16s\n",
-      "Wall time: 3min 4s\n"
+      "CPU times: user 1min 2s, sys: 973 ms, total: 1min 3s\n",
+      "Wall time: 1min 3s\n"
      ]
     },
     {
@@ -882,7 +794,7 @@
        "'(((((((10+((9*8)*7))-6)-5)*4)+3)+2)-1)'"
       ]
      },
-     "execution_count": 18,
+     "execution_count": 16,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -897,7 +809,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -911,11 +822,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 19,
+   "execution_count": 17,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -928,7 +837,7 @@
        "30066"
       ]
      },
-     "execution_count": 19,
+     "execution_count": 17,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -941,7 +850,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -952,196 +860,14 @@
     "\n",
     "**But I don't believe it.**\n",
     "\n",
-    "We'll see why in a bit, but first some trivia questions to answer:\n",
-    "\n",
-    "- How many distinct values can be made?"
+    "Why not? Because floating point division can have round-off errors.  For example:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 18,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "5789961"
-      ]
-     },
-     "execution_count": 20,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "len(expressions(c10))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "button": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "source": [
-    "- What's the smallest positive integer than cannot be made?"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 21,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "10843"
-      ]
-     },
-     "execution_count": 21,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "import itertools \n",
-    "\n",
-    "def unmakeable(numbers):\n",
-    "    \"Smallest positive integer than can't be made by numbers.\"\n",
-    "    return next(i for i in itertools.count(0) if i not in expressions(numbers))\n",
-    "\n",
-    "unmakeable(c10)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "button": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "source": [
-    "- What is the expected theoretical number of expressions? It is [Catalan(9)](http://www.wolframalpha.com/input/?i=9th+catalan+number) &times; 4<sup>9</sup>:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 22,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "1274544128"
-      ]
-     },
-     "execution_count": 22,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "4862 * 4 ** 9"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "button": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "source": [
-    "- How many expressions were actually entered in the `COUNTS` table?"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 23,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "1264876734"
-      ]
-     },
-     "execution_count": 23,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "sum(COUNTS[c10].values())"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {
-    "button": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "source": [
-    "So, about 1% of the 1.27 billion theoretically possible expressions do not appear in `COUNTS`; these are the `ZeroDivisionErrors`.\n",
-    "\n",
-    "# Dealing with Round-off Errors\n",
-    "\n",
-    "Why don't I believe the answer of  30,066 expressions? Because floating point division can have round-off errors.  For example:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 24,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1154,7 +880,7 @@
        "2015.9999999999998"
       ]
      },
-     "execution_count": 24,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1165,11 +891,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 19,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1182,7 +906,7 @@
        "False"
       ]
      },
-     "execution_count": 25,
+     "execution_count": 19,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1195,23 +919,22 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "So there might be perfectly good solutions that are hiding in the `EXPS` table under 2015.9999999999998 (or some similar number) when they should be exactly 2016. Let's find all the values that are very near to 2016:"
+    "# Dealing with Round-off Errors\n",
+    "\n",
+    "So there might be perfectly good solutions that are hiding in the `EXPS` table under `2015.9999999999998` (or some similar number) when they should be exactly `2016`. Let's find all the values that are very near to `2016`:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 20,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1234,21 +957,20 @@
        " 2016.0000000000023}"
       ]
      },
-     "execution_count": 26,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "{val for val in expressions(c10)\n",
-    " if abs(val - 2016) < 0.0001}"
+    " if abs(val - 2016) < 0.000001}"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1259,16 +981,15 @@
     "\n",
     "To find out for sure, let's re-do all the calculations using exact rational arithmetic, as provided by the `fractions.Fraction` data type. From experience I know that this will be an order of magnitude slower, so we're looking at maybe a 20 minute computation.\n",
     "\n",
-    "I'll replace the computation `L / R` with `divide(L, R)`, which calls `Fraction`. To mitigate the expense of this computation, I make two optimizations. First, `divide` replaces a whole fraction, such as `Fraction(6, 2)`, with an `int`, such as `3`. Second, I modify `expr` to *not* fill in the `EXPS[c10]` table, except for `EXPS[c10][2016]`. The rationale is that we don't need the other entries, and by not storing them, we save a lot of memory, and thus save garbage collection time."
+    "I'll replace the computation `L / R` with `divide(L, R)`, which calls `Fraction`. To mitigate the expense of this computation, I make two optimizations. First, `divide` replaces a whole fraction, such as `Fraction(6, 2)`, with an `int`, such as `3`. Second, I modify `expr` to *not* fill in  `EXPS[c10]`, except for `EXPS[c10][2016]`. The rationale is that we don't need the other entries, and by not storing them, we save a lot of memory, and thus save garbage collection time."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 21,
    "metadata": {
     "button": false,
     "collapsed": true,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1299,7 +1020,7 @@
     "def divide(L, R):\n",
     "    \"Exact rational division of L/R.\"\n",
     "    f = Fraction(L, R)\n",
-    "    return f.numerator if f.denominator == 1 else f\n",
+    "    return (f.numerator if f.denominator == 1 else f)\n",
     "\n",
     "def expr(numbers, value, exp, count):\n",
     "    \"Fill EXPS[numbers][val] with exp, and increment COUNTS.\"\n",
@@ -1313,7 +1034,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1325,37 +1045,39 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 22,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 13min 21s, sys: 6.11 s, total: 13min 28s\n",
+      "Wall time: 13min 33s\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "44499"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "clear()\n",
     "\n",
-    "%time expressions(c10)[2016]"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {
-    "button": false,
-    "collapsed": false,
-    "deletable": true,
-    "new_sheet": false,
-    "run_control": {
-     "read_only": false
-    }
-   },
-   "outputs": [],
-   "source": [
+    "%time expressions(c10)[2016]\n",
+    "\n",
     "COUNTS[c10][2016]"
    ]
   },
@@ -1363,21 +1085,151 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
    "source": [
-    "That did indeed take about ten times longer, but we now have an answer that I have more confidence in (but I wouldn't accept it as definitive until it was independently verified or I had an extensive tst suite). And of course, if you have a different definition of \"distinct solution,\" you will get a different answer."
+    "That did indeed take about ten times longer, but we now have an answer that I have more confidence in (but I wouldn't accept it as definitive until it was independently verified and had an extensive test suite). And of course, if you have a different definition of \"distinct solution,\" you will get a different answer."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "button": false,
+    "new_sheet": false,
+    "run_control": {
+     "read_only": false
+    }
+   },
+   "source": [
+    "There are some additional interesting questions we can answer:\n",
+    "\n",
+    "What's the smallest positive integer than *cannot* be made?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {
+    "button": false,
+    "new_sheet": false,
+    "run_control": {
+     "read_only": false
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "def unmakeable(numbers):\n",
+    "    \"Smallest positive integer than can't be made by numbers.\"\n",
+    "    return next(i for i in itertools.count(1) if i not in expressions(numbers))\n",
+    "\n",
+    "unmakeable(c10)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "button": false,
+    "new_sheet": false,
+    "run_control": {
+     "read_only": false
+    }
+   },
+   "source": [
+    "What is the expected theoretical number of expressions? \n",
+    "\n",
+    "It is [Catalan(9)](http://www.wolframalpha.com/input/?i=9th+catalan+number) &times; 4<sup>9</sup>:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {
+    "button": false,
+    "new_sheet": false,
+    "run_control": {
+     "read_only": false
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1274544128"
+      ]
+     },
+     "execution_count": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "4862 * 4 ** 9"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "button": false,
+    "new_sheet": false,
+    "run_control": {
+     "read_only": false
+    }
+   },
+   "source": [
+    "How many expressions were actually entered in the `COUNTS` table?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {
+    "button": false,
+    "new_sheet": false,
+    "run_control": {
+     "read_only": false
+    }
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "44499"
+      ]
+     },
+     "execution_count": 25,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "sum(COUNTS[c10].values())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "So, about 1% of the 1.27 billion theoretically possible expressions do not appear in `COUNTS`; these are the `ZeroDivisionErrors`."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1389,7 +1241,7 @@
     "Now let's turn to another of Alex's puzzles: making 2016 and other target values from a string of four or five `4`s,  with exponentiation allowed. Exponentiation is tricky for five reasons:\n",
     "\n",
     "- **Division by zero**: `(0 ** -1)` is the same as `(1 / 0)`, and gives a `ZeroDivisionError`.\n",
-    "- **Irrationals**: `(2 ** (1 / 2))` is an irrational number; so we can't do exact rational arithmetic.\n",
+    "- **Irrationals**: `(3 ** (1 / 2))` is an irrational number; so we can't do exact rational arithmetic.\n",
     "- **Imaginaries**: `(-1 ** (1 / 2))` is an imaginary number, but Python gives a `ValueError`.\n",
     "- **Overflow**: `(10. ** (9. ** 8.))`, as a `float`, gives a `OverflowError`.\n",
     "- **Finite memory**: [`(10 ** (9 ** (8 * 7)))`](http://www.wolframalpha.com/input/?i=10+%5E+9+%5E+56), as an `int`, gives an `OutOfMemoryError` (even if your memory was expanded to use every atom in the universe).\n",
@@ -1400,11 +1252,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 26,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1445,7 +1295,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1457,17 +1306,26 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 27,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "'(((4**4)-4)*(4+4))'"
+      ]
+     },
+     "execution_count": 27,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "clear()\n",
     "\n",
@@ -1478,7 +1336,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1490,11 +1347,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 28,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1513,7 +1368,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1525,17 +1379,35 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 29,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{0: '(((4+4)-4)-4)',\n",
+       " 1: '(((4+4)-4)/4)',\n",
+       " 2: '((4/(4+4))*4)',\n",
+       " 3: '(((4+4)+4)/4)',\n",
+       " 4: '(((4-4)*4)+4)',\n",
+       " 5: '(((4*4)+4)/4)',\n",
+       " 6: '(((4+4)/4)+4)',\n",
+       " 7: '((4-(4/4))+4)',\n",
+       " 8: '(((4+4)-4)+4)',\n",
+       " 9: '(((4/4)+4)+4)'}"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "makeable((4, 4, 4, 4))"
    ]
@@ -1544,7 +1416,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1558,17 +1429,38 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 30,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{0: '((((5-5)*5)-5)+5)',\n",
+       " 1: '(((5-(5*5))/5)+5)',\n",
+       " 2: '((((5+5)/5)-5)+5)',\n",
+       " 3: '((((5*5)-5)-5)/5)',\n",
+       " 4: '((((5-5)-5)/5)+5)',\n",
+       " 5: '((5/((5**5)**5))+5)',\n",
+       " 6: '((((5/5)+5)*5)/5)',\n",
+       " 7: '(((5/5)+(5/5))+5)',\n",
+       " 8: '((((5+5)+5)/5)+5)',\n",
+       " 9: '((((5+5)*5)-5)/5)',\n",
+       " 10: '((((5*5)-5)-5)-5)',\n",
+       " 11: '((((5+5)*5)+5)/5)',\n",
+       " 12: '((((5+5)/5)+5)+5)'}"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "ff = (5, 5, 5, 5, 5)\n",
     "\n",
@@ -1579,7 +1471,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1594,9 +1485,7 @@
     "\n",
     "- **digit concatenation**: `55`\n",
     "- **decimal point**: `.5`\n",
-    "- **unary operations**: \n",
-    "  - **factorial**: `5! = 120` \n",
-    "  - **square root**:  &radic; `5`\n",
+    "- **unary operations**: `5!`,  &radic; `5`\n",
     "\n",
     "\n",
     "We'll refactor `expressions` to call these three new subfunctions:\n",
@@ -1610,11 +1499,9 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 31,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1665,7 +1552,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1677,11 +1563,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 32,
    "metadata": {
     "button": false,
     "collapsed": true,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1700,7 +1585,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1712,17 +1596,190 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 33,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 48.3 s, sys: 2 s, total: 50.3 s\n",
+      "Wall time: 50.7 s\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "{0: '(5*(55-55))',\n",
+       " 1: '((55/55)**5)',\n",
+       " 2: '(55/(.5*55))',\n",
+       " 3: '√(5!-(555/5))',\n",
+       " 4: '(5-(55/55))',\n",
+       " 5: '(5.55-.55)',\n",
+       " 6: '(5+(55/55))',\n",
+       " 7: '(((5+55)/5)-5)',\n",
+       " 8: '(.5*(5+(55/5)))',\n",
+       " 9: '(5!-(555/5))',\n",
+       " 10: '(5!-(55+55))',\n",
+       " 11: '(5.5/(5.5-5))',\n",
+       " 12: '(5!/(55/5.5))',\n",
+       " 13: '((5+(5+55))/5)',\n",
+       " 14: '((5*5)-(55/5))',\n",
+       " 15: '(5+(55/5.5))',\n",
+       " 16: '(5+(5.5+5.5))',\n",
+       " 17: '(5+((5+55)/5))',\n",
+       " 18: '(5+((5!-55)/5))',\n",
+       " 19: '((5*5)-(.5+5.5))',\n",
+       " 20: '(55/(5*.55))',\n",
+       " 21: '(5+(5+(55/5)))',\n",
+       " 22: '((55+55)/5)',\n",
+       " 23: '((5+(55/.5))/5)',\n",
+       " 24: '(5-(55/55))!',\n",
+       " 25: '(55-(5+(5*5)))',\n",
+       " 26: '(55-(5+(5!/5)))',\n",
+       " 27: '(.5*(55-(5/5)))',\n",
+       " 28: '(.5*(.5+55.5))',\n",
+       " 29: '(5+(5!/(5.5-.5)))',\n",
+       " 30: '(5*((55/5)-5))',\n",
+       " 31: '(55-(5-(5/5))!)',\n",
+       " 32: '(55-((5!-5)/5))',\n",
+       " 33: '(.55*(5+55))',\n",
+       " 34: '((5!+(55-5))/5)',\n",
+       " 35: '(5+(55-(5*5)))',\n",
+       " 36: '((5*5)+(55/5))',\n",
+       " 37: '((5!+(5!-55))/5)',\n",
+       " 38: '(5+(.5*(5!*.55)))',\n",
+       " 39: '(((5*5)-5.5)/.5)',\n",
+       " 40: '(55-(5+(5+5)))',\n",
+       " 41: '(5!-((5!/5)+55))',\n",
+       " 42: '((5+5.5)/(.5*.5))',\n",
+       " 43: '(55-(5!/(5+5)))',\n",
+       " 44: '(55-(55/5))',\n",
+       " 45: '((5*5!)-555)',\n",
+       " 46: '(55-((5-.5)/.5))',\n",
+       " 47: '((5!/5)+((5!-5)/5))',\n",
+       " 48: '(5!/(5*(5.5-5)))',\n",
+       " 49: '(55-(.5+5.5))',\n",
+       " 50: '(55.5-5.5)',\n",
+       " 51: '(.5+(55.5-5))',\n",
+       " 52: '(55-(.5+(.5*5)))',\n",
+       " 53: '(55.5-(.5*5))',\n",
+       " 54: '(((5*55)-5)/5)',\n",
+       " 55: '(.5*(55+55))',\n",
+       " 56: '((5+(5*55))/5)',\n",
+       " 57: '(55+((5+5)/5))',\n",
+       " 58: '((.5*5)+55.5)',\n",
+       " 59: '(5+(55-(5/5)))',\n",
+       " 60: '(5+(55.5-.5))',\n",
+       " 61: '(5.5+55.5)',\n",
+       " 62: '((55-(5!/5))/.5)',\n",
+       " 63: '(5.5+(.5*(5!-5)))',\n",
+       " 64: '(5!-(.5+55.5))',\n",
+       " 65: '(.5+(5!-55.5))',\n",
+       " 66: '(55+(55/5))',\n",
+       " 67: '(55+(5!/(5+5)))',\n",
+       " 68: '(5.5+(.5*(5+5!)))',\n",
+       " 69: '(5!-((.5+(5*5))/.5))',\n",
+       " 70: '(5+(5+(5+55)))',\n",
+       " 71: '((.5*5!)+(55/5))',\n",
+       " 72: '((.5+5.5)!/(5+5))',\n",
+       " 73: '((5*5)+(5!/(.5*5)))',\n",
+       " 74: '((5!/5)+(55-5))',\n",
+       " 75: '((5*5)+(55-5))',\n",
+       " 76: '(5+(5+(5!*.55)))',\n",
+       " 77: '(5+(.5*(5!+(5!/5))))',\n",
+       " 78: '(55+((5!-5)/5))',\n",
+       " 79: '(55+(5-(5/5))!)',\n",
+       " 80: '(5*(5+(55/5)))',\n",
+       " 81: '((√.5+(5!/√(5-.5)))/√.5)',\n",
+       " 82: '(5!-(((5!/5)-5)/.5))',\n",
+       " 83: '((.5*5!)+((5!-5)/5))',\n",
+       " 84: '(5+((5!/5)+55))',\n",
+       " 85: '(5+((5*5)+55))',\n",
+       " 86: '((55/.5)-(5!/5))',\n",
+       " 87: '((555-5!)/5)',\n",
+       " 88: '(.5+(.5*(5!+55)))',\n",
+       " 89: '(5!-(55-(5!/5)))',\n",
+       " 90: '(5!-(55-(5*5)))',\n",
+       " 91: '((5*5)+(5!*.55))',\n",
+       " 92: '(5!-(.5+(.5*55)))',\n",
+       " 93: '(.5+(5!-(.5*55)))',\n",
+       " 94: '(5!-((5/5)+(5*5)))',\n",
+       " 95: '(((55-5)/.5)-5)',\n",
+       " 96: '(5!-(5!/(5.5-.5)))',\n",
+       " 97: '(5!-((5!/5)-(5/5)))',\n",
+       " 98: '(5!-(55/(.5*5)))',\n",
+       " 99: '((55-5.5)/.5)',\n",
+       " 100: '((55/.5)-(5+5))',\n",
+       " 101: '((55.5-5)/.5)',\n",
+       " 102: '(5+(5!-((5!-5)/5)))',\n",
+       " 103: '(55+(5!/(.5*5)))',\n",
+       " 104: '(5!-(5+(55/5)))',\n",
+       " 105: '(55+(55-5))',\n",
+       " 106: '((555/5)-5)',\n",
+       " 107: '(5!-((5!-55)/5))',\n",
+       " 108: '(5!-((5+55)/5))',\n",
+       " 109: '(5!-(5.5+5.5))',\n",
+       " 110: '((555-5)/5)',\n",
+       " 111: '(555/√(5*5))',\n",
+       " 112: '((5+555)/5)',\n",
+       " 113: '(5!-(5+((5+5)/5)))',\n",
+       " 114: '(5+(5!-(55/5)))',\n",
+       " 115: '(5+(55+55))',\n",
+       " 116: '(5+(555/5))',\n",
+       " 117: '(5!-(5.5-(.5*5)))',\n",
+       " 118: '(5!-(5!/(5+55)))',\n",
+       " 119: '(5!-(55/55))',\n",
+       " 120: '(5.55-.55)!',\n",
+       " 121: '(5!+(55/55))',\n",
+       " 122: '(5!+(5!/(5+55)))',\n",
+       " 123: '(5!+(5.5-(.5*5)))',\n",
+       " 124: '(5!+√(5+(55/5)))',\n",
+       " 125: '(.5*(5*(55-5)))',\n",
+       " 126: '(5!+((55/5)-5))',\n",
+       " 127: '((5!/(5/5.5))-5)',\n",
+       " 128: '(5!+(5.5+(.5*5)))',\n",
+       " 129: '((5!-55.5)/.5)',\n",
+       " 130: '(5!+(55/5.5))',\n",
+       " 131: '(5!+(5.5+5.5))',\n",
+       " 132: '(5!*(.55+.55))',\n",
+       " 133: '(5!+((5!-55)/5))',\n",
+       " 134: '((5!/5)+(55/.5))',\n",
+       " 135: '((5!+555)/5)',\n",
+       " 136: '(5+(5!+(55/5)))',\n",
+       " 137: '(5+(5!/(5/5.5)))',\n",
+       " 138: '(.5+(.5*(5*55)))',\n",
+       " 139: '(((.5+5.5)!/5)-5)',\n",
+       " 140: '(.5*(5+(5*55)))',\n",
+       " 141: '(5!+((5+5.5)/.5))',\n",
+       " 142: '(5!+(55/(.5*5)))',\n",
+       " 143: '(((.5+5.5)!-5)/5)',\n",
+       " 144: '(((55/5)-5)!/5)',\n",
+       " 145: '(5!+(.5*(55-5)))',\n",
+       " 146: '(5!+((5/5)+(5*5)))',\n",
+       " 147: '(5!+((.5*55)-.5))',\n",
+       " 148: '(.5+(5!+(.5*55)))',\n",
+       " 149: '(5+((.5+5.5)!/5))',\n",
+       " 150: '(5*(55-(5*5)))',\n",
+       " 151: '(5!+(55-(5!/5)))',\n",
+       " 152: '(5!+(((5+5)/5)**5))',\n",
+       " 153: '(5!+(.5*(5!*.55)))',\n",
+       " 154: '(5+(5+(5!+(5!/5))))',\n",
+       " 155: '(5*(55-(5!/5)))',\n",
+       " 156: '((5!+(5!*5.5))/5)'}"
+      ]
+     },
+     "execution_count": 33,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "clear()\n",
     "\n",
@@ -1733,7 +1790,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1747,7 +1803,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1777,11 +1832,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 34,
    "metadata": {
     "button": false,
     "collapsed": true,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1816,7 +1870,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1828,17 +1881,1034 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 35,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 2min 42s, sys: 2.81 s, total: 2min 45s\n",
+      "Wall time: 2min 46s\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "{0: '⌊.55555⌋',\n",
+       " 1: '⌈.55555⌉',\n",
+       " 2: '⌊5*.5555⌋',\n",
+       " 3: '⌈5*.5555⌉',\n",
+       " 4: '⌊5-.5555⌋',\n",
+       " 5: '⌊5.5555⌋',\n",
+       " 6: '⌈5.5555⌉',\n",
+       " 7: '⌈.5+5.555⌉',\n",
+       " 8: '⌈√55+.555⌉',\n",
+       " 9: '⌊55/5.55⌋',\n",
+       " 10: '⌊555/55⌋',\n",
+       " 11: '⌈555/55⌉',\n",
+       " 12: '⌈55.55/5⌉',\n",
+       " 13: '⌈√5*5.555⌉',\n",
+       " 14: '⌈√55/.555⌉',\n",
+       " 15: '(5+(55/5.5))',\n",
+       " 16: '(5+⌊55.5/5⌋)',\n",
+       " 17: '(5+⌈55.5/5⌉)',\n",
+       " 18: '⌊55/⌈5*.55⌉⌋',\n",
+       " 19: '⌈55/⌈5*.55⌉⌉',\n",
+       " 20: '(5*⌊5-.555⌋)',\n",
+       " 21: '⌊5!/5.555⌋',\n",
+       " 22: '⌈5!/5.555⌉',\n",
+       " 23: '⌈555/(5*5)⌉',\n",
+       " 24: '⌊5-.5555⌋!',\n",
+       " 25: '(5*⌊5.555⌋)',\n",
+       " 26: '(⌈.55*55⌉-5)',\n",
+       " 27: '⌊5*5.555⌋',\n",
+       " 28: '⌈5*5.555⌉',\n",
+       " 29: '(5+⌊5-.555⌋!)',\n",
+       " 30: '⌊55*.555⌋',\n",
+       " 31: '⌈55*.555⌉',\n",
+       " 32: '(5+⌊5*5.55⌋)',\n",
+       " 33: '(5+⌈5*5.55⌉)',\n",
+       " 34: '⌈⌈5.5⌉*5.55⌉',\n",
+       " 35: '(5+⌊.55*55⌋)',\n",
+       " 36: '(5+⌈.55*55⌉)',\n",
+       " 37: '⌊5*(55/√55)⌋',\n",
+       " 38: '⌊⌊√55⌋*5.55⌋',\n",
+       " 39: '⌊√.5*55.55⌋',\n",
+       " 40: '⌊5.5*√55.5⌋',\n",
+       " 41: '⌊√55*5.55⌋',\n",
+       " 42: '⌈√55*5.55⌉',\n",
+       " 43: '(55-⌊5+√55⌋)',\n",
+       " 44: '(55-(55/5))',\n",
+       " 45: '(5*⌊5/.555⌋)',\n",
+       " 46: '⌊5555/5!⌋',\n",
+       " 47: '⌈5555/5!⌉',\n",
+       " 48: '⌈55-√55.5⌉',\n",
+       " 49: '⌊55-5.55⌋',\n",
+       " 50: '⌊55.55-5⌋',\n",
+       " 51: '⌈55.55-5⌉',\n",
+       " 52: '(55-⌈5*.55⌉)',\n",
+       " 53: '⌊55.55-√5⌋',\n",
+       " 54: '⌊55-.555⌋',\n",
+       " 55: '⌊55.555⌋',\n",
+       " 56: '⌈55.555⌉',\n",
+       " 57: '⌈.5+55.55⌉',\n",
+       " 58: '⌈√5+55.55⌉',\n",
+       " 59: '(5+⌊55-.55⌋)',\n",
+       " 60: '⌊5+55.55⌋',\n",
+       " 61: '⌈5+55.55⌉',\n",
+       " 62: '⌊√55+55.5⌋',\n",
+       " 63: '⌈√55+55.5⌉',\n",
+       " 64: '⌊5!-55.55⌋',\n",
+       " 65: '⌈5!-55.55⌉',\n",
+       " 66: '⌊5!*.5555⌋',\n",
+       " 67: '⌈5!*.5555⌉',\n",
+       " 68: '(5+⌈55+√55⌉)',\n",
+       " 69: '⌊555/⌈√55⌉⌋',\n",
+       " 70: '⌈555/⌈√55⌉⌉',\n",
+       " 71: '(5+⌊5!*.555⌋)',\n",
+       " 72: '(5+⌈5!*.555⌉)',\n",
+       " 73: '(5!-⌊55-√55⌋)',\n",
+       " 74: '⌊555/√55⌋',\n",
+       " 75: '⌈555/√55⌉',\n",
+       " 76: '(55+⌊5!/5.5⌋)',\n",
+       " 77: '(55+⌈5!/5.5⌉)',\n",
+       " 78: '⌊55.55/√.5⌋',\n",
+       " 79: '⌊555/⌊√55⌋⌋',\n",
+       " 80: '⌊(5*5)+55.5⌋',\n",
+       " 81: '⌈(5*5)+55.5⌉',\n",
+       " 82: '(55+⌊.5*55⌋)',\n",
+       " 83: '(55+⌈.5*55⌉)',\n",
+       " 84: '(5+((5!/5)+55))',\n",
+       " 85: '(5+(55+(5*5)))',\n",
+       " 86: '⌊⌈5+55.5⌉/√.5⌋',\n",
+       " 87: '((555-5!)/5)',\n",
+       " 88: '⌈√5**5.555⌉',\n",
+       " 89: '(5!-⌈.55*55⌉)',\n",
+       " 90: '⌊(55-5)/.55⌋',\n",
+       " 91: '⌈(55-5)/.55⌉',\n",
+       " 92: '⌊555/⌈5.5⌉⌋',\n",
+       " 93: '⌈555/⌈5.5⌉⌉',\n",
+       " 94: '(⌊55/.55⌋-5)',\n",
+       " 95: '(⌈55/.55⌉-5)',\n",
+       " 96: '⌊.55*(5!+55)⌋',\n",
+       " 97: '⌈.55*(5!+55)⌉',\n",
+       " 98: '(⌊55-5.5⌋/.5)',\n",
+       " 99: '⌊55/.555⌋',\n",
+       " 100: '⌊555/5.5⌋',\n",
+       " 101: '⌈555/5.5⌉',\n",
+       " 102: '⌈⌈55.5⌉/.55⌉',\n",
+       " 103: '⌊⌊√5+55⌋/.55⌋',\n",
+       " 104: '(5+⌊55/.55⌋)',\n",
+       " 105: '(55+(55-5))',\n",
+       " 106: '((555/5)-5)',\n",
+       " 107: '(55+⌊55-√5⌋)',\n",
+       " 108: '(5!-⌈55.5/5⌉)',\n",
+       " 109: '(55+⌊55-.5⌋)',\n",
+       " 110: '⌊55+55.5⌋',\n",
+       " 111: '⌈55+55.5⌉',\n",
+       " 112: '⌈555.5/5⌉',\n",
+       " 113: '(55+⌈√5+55⌉)',\n",
+       " 114: '⌊5!-5.555⌋',\n",
+       " 115: '⌈5!-5.555⌉',\n",
+       " 116: '(5+(555/5))',\n",
+       " 117: '(5!-⌈5*.555⌉)',\n",
+       " 118: '(5!-⌈5.55/5⌉)',\n",
+       " 119: '⌊5!-.5555⌋',\n",
+       " 120: '⌊5.5555⌋!',\n",
+       " 121: '⌈5!+.5555⌉',\n",
+       " 122: '(5!+⌈5.55/5⌉)',\n",
+       " 123: '⌊555/(5-.5)⌋',\n",
+       " 124: '⌊√5*55.55⌋',\n",
+       " 125: '⌊5!+5.555⌋',\n",
+       " 126: '⌈5!+5.555⌉',\n",
+       " 127: '⌊⌊5**5.5⌋/55⌋',\n",
+       " 128: '⌈⌊5**5.5⌋/55⌉',\n",
+       " 129: '(5!+⌊5/.555⌋)',\n",
+       " 130: '⌊√5.5*55.5⌋',\n",
+       " 131: '⌈√5.5*55.5⌉',\n",
+       " 132: '(5!+⌈55.5/5⌉)',\n",
+       " 133: '⌈55*(√55-5)⌉',\n",
+       " 134: '⌊55*√⌈5.55⌉⌋',\n",
+       " 135: '(5*⌊5*5.55⌋)',\n",
+       " 136: '⌈√⌈5.5⌉*55.5⌉',\n",
+       " 137: '⌊5.5*⌈55/√5⌉⌋',\n",
+       " 138: '⌊5*(5*5.55)⌋',\n",
+       " 139: '⌈5*(5*5.55)⌉',\n",
+       " 140: '(5*⌈5*5.55⌉)',\n",
+       " 141: '(5!+⌊5!/5.55⌋)',\n",
+       " 142: '(5!+⌈5!/5.55⌉)',\n",
+       " 143: '(5+⌈.5*(5*55)⌉)',\n",
+       " 144: '(⌈5.555⌉!/5)',\n",
+       " 145: '(5+(5*⌈.5*55⌉))',\n",
+       " 146: '⌊55*(5-√5.5)⌋',\n",
+       " 147: '(5!+⌊5*5.55⌋)',\n",
+       " 148: '(5!+⌈5*5.55⌉)',\n",
+       " 149: '⌈5.5*⌊.5*55⌋⌉',\n",
+       " 150: '(5*⌊.55*55⌋)',\n",
+       " 151: '⌊.55*(5*55)⌋',\n",
+       " 152: '⌈.55*(5*55)⌉',\n",
+       " 153: '⌊55*(√5+.55)⌋',\n",
+       " 154: '(5.5*⌈.5*55⌉)',\n",
+       " 155: '(5*⌈.55*55⌉)',\n",
+       " 156: '⌈(55+55)/√.5⌉',\n",
+       " 157: '(⌊.5*555⌋-5!)',\n",
+       " 158: '(⌈.5*555⌉-5!)',\n",
+       " 159: '(5!+⌊√.5*55.5⌋)',\n",
+       " 160: '(5*(5+⌊.5*55⌋))',\n",
+       " 161: '⌊(.5*5)**5.55⌋',\n",
+       " 162: '⌈(.5*5)**5.55⌉',\n",
+       " 163: '(⌊5!/.55⌋-55)',\n",
+       " 164: '(⌈5!/.55⌉-55)',\n",
+       " 165: '(55*⌈5*.55⌉)',\n",
+       " 166: '⌊⌈.5*5⌉*55.5⌋',\n",
+       " 167: '⌈⌈.5*5⌉*55.5⌉',\n",
+       " 168: '(5!+⌈55-√55⌉)',\n",
+       " 169: '(5!+⌊55-5.5⌋)',\n",
+       " 170: '(5!+⌊55.5-5⌋)',\n",
+       " 171: '(5!+⌈55.5-5⌉)',\n",
+       " 172: '⌈5.555**⌈√5⌉⌉',\n",
+       " 173: '⌊55*√⌊5+5.5⌋⌋',\n",
+       " 174: '(5!+⌊55-.55⌋)',\n",
+       " 175: '⌊5!+55.55⌋',\n",
+       " 176: '⌈5!+55.55⌉',\n",
+       " 177: '(55+⌊√5*55⌋)',\n",
+       " 178: '(55+⌈√5*55⌉)',\n",
+       " 179: '⌊55*(5.5-√5)⌋',\n",
+       " 180: '(5+⌊5!+55.5⌋)',\n",
+       " 181: '(5+⌈5!+55.5⌉)',\n",
+       " 182: '(⌊5.5*55⌋-5!)',\n",
+       " 183: '(⌈5.5*55⌉-5!)',\n",
+       " 184: '(5!+⌊5!-55.5⌋)',\n",
+       " 185: '(555/⌈.5*5⌉)',\n",
+       " 186: '⌈555.5/⌈√5⌉⌉',\n",
+       " 187: '(5!+⌈5!*.555⌉)',\n",
+       " 188: '((⌈.5*5⌉**5)-55)',\n",
+       " 189: '(⌊√55⌋*⌊.5*55⌋)',\n",
+       " 190: '(5*⌈5*√55.5⌉)',\n",
+       " 191: '⌈55*√⌊5+√55⌋⌉',\n",
+       " 192: '(⌈5.55⌉/(.5**5))',\n",
+       " 193: '⌊5!**(.55+.55)⌋',\n",
+       " 194: '⌊(555-5!)/√5⌋',\n",
+       " 195: '⌈(555-5!)/√5⌉',\n",
+       " 196: '(.5*⌊√.5*555⌋)',\n",
+       " 197: '⌈.5*⌈√.5*555⌉⌉',\n",
+       " 198: '(⌊55/.55⌋/.5)',\n",
+       " 199: '⌊55/(.5*.55)⌋',\n",
+       " 200: '(⌈55/.55⌉/.5)',\n",
+       " 201: '⌈555/(5-√5)⌉',\n",
+       " 202: '⌊555/(.5+√5)⌋',\n",
+       " 203: '⌊.5*⌊55*√55⌋⌋',\n",
+       " 204: '(.5*⌈55*√55⌉)',\n",
+       " 205: '(5*⌈5.5*√55⌉)',\n",
+       " 206: '⌈⌈5!-√55⌉/.55⌉',\n",
+       " 207: '⌊(5!-5)/.555⌋',\n",
+       " 208: '⌈(5!-5)/.555⌉',\n",
+       " 209: '(5.5*⌈5*√55⌉)',\n",
+       " 210: '⌈⌈5!-5.5⌉/.55⌉',\n",
+       " 211: '(⌊5!/.555⌋-5)',\n",
+       " 212: '(⌈5!/.555⌉-5)',\n",
+       " 213: '⌈⌊5!/.55⌋-5.5⌉',\n",
+       " 214: '⌈⌈5!/.55⌉-5.5⌉',\n",
+       " 215: '⌊⌊5!/.555⌋-.5⌋',\n",
+       " 216: '⌊5!/.5555⌋',\n",
+       " 217: '⌈5!/.5555⌉',\n",
+       " 218: '⌈.5*(555-5!)⌉',\n",
+       " 219: '(5!+⌊55/.55⌋)',\n",
+       " 220: '((5*55)-55)',\n",
+       " 221: '(5+⌊5!/.555⌋)',\n",
+       " 222: '(555/(.5*5))',\n",
+       " 223: '⌊5.5+⌊5!/.55⌋⌋',\n",
+       " 224: '⌊5.5+⌈5!/.55⌉⌋',\n",
+       " 225: '⌊(5+5!)/.555⌋',\n",
+       " 226: '⌈(5+5!)/.555⌉',\n",
+       " 227: '⌈555/√⌈5.5⌉⌉',\n",
+       " 228: '(⌊5!-5.55⌋/.5)',\n",
+       " 229: '⌈(5!-5.55)/.5⌉',\n",
+       " 230: '(5!+(55+55))',\n",
+       " 231: '(5!+(555/5))',\n",
+       " 232: '⌈(5!+√55)/.55⌉',\n",
+       " 233: '⌈⌈5!+√55⌉/.55⌉',\n",
+       " 234: '(5!+⌊5!-5.55⌋)',\n",
+       " 235: '(5*⌊55-√55⌋)',\n",
+       " 236: '⌊555/√5.5⌋',\n",
+       " 237: '⌈555/√5.5⌉',\n",
+       " 238: '⌈5*(55-√55)⌉',\n",
+       " 239: '(5!+⌊5!-.555⌋)',\n",
+       " 240: '(5*⌈55-√55⌉)',\n",
+       " 241: '(5!+⌈5!+.555⌉)',\n",
+       " 242: '(⌈5!+.555⌉/.5)',\n",
+       " 243: '(⌈5*.555⌉**5)',\n",
+       " 244: '⌊55*(5-.55)⌋',\n",
+       " 245: '(5*⌊55-5.5⌋)',\n",
+       " 246: '⌈(555-5)/√5⌉',\n",
+       " 247: '⌊5*(55-5.5)⌋',\n",
+       " 248: '⌊555.5/√5⌋',\n",
+       " 249: '⌈555.5/√5⌉',\n",
+       " 250: '(5*⌊55.5-5⌋)',\n",
+       " 251: '⌈(5+555)/√5⌉',\n",
+       " 252: '⌊5*(55.5-5)⌋',\n",
+       " 253: '⌈5*(55.5-5)⌉',\n",
+       " 254: '(5+⌈555/√5⌉)',\n",
+       " 255: '(5*⌈55.5-5⌉)',\n",
+       " 256: '(⌊55*√5.5⌋/.5)',\n",
+       " 257: '⌊55/(.5/√5.5)⌋',\n",
+       " 258: '(⌈55*√5.5⌉/.5)',\n",
+       " 259: '⌊5!/(55.5/5!)⌋',\n",
+       " 260: '(5*⌊55-√5.5⌋)',\n",
+       " 261: '⌈(5*.55)**5.5⌉',\n",
+       " 262: '⌈555/√(5-.5)⌉',\n",
+       " 263: '⌊5*(55-√5.5)⌋',\n",
+       " 264: '⌈5*(55-√5.5)⌉',\n",
+       " 265: '(5*⌈55-√5.5⌉)',\n",
+       " 266: '⌊5*(55.5-√5)⌋',\n",
+       " 267: '⌊(5*55)-√55⌋',\n",
+       " 268: '⌈(5*55)-√55⌉',\n",
+       " 269: '⌊(5*55)-5.5⌋',\n",
+       " 270: '(5*⌊55-.55⌋)',\n",
+       " 271: '(⌈.5-5⌉+(5*55))',\n",
+       " 272: '(⌊.5*555⌋-5)',\n",
+       " 273: '(⌈.5*555⌉-5)',\n",
+       " 274: '⌊(5*55)-.55⌋',\n",
+       " 275: '(5*⌊55.55⌋)',\n",
+       " 276: '⌈.55+(5*55)⌉',\n",
+       " 277: '⌊5*55.55⌋',\n",
+       " 278: '⌈5*55.55⌉',\n",
+       " 279: '⌈.5+⌈.5*555⌉⌉',\n",
+       " 280: '(5*⌈55.55⌉)',\n",
+       " 281: '⌈5.5+(5*55)⌉',\n",
+       " 282: '(5+⌊.5*555⌋)',\n",
+       " 283: '(5+⌈.5*555⌉)',\n",
+       " 284: '⌊5*((5**5)/55)⌋',\n",
+       " 285: '(5*⌊√5+55.5⌋)',\n",
+       " 286: '(5.5*⌊55-√5⌋)',\n",
+       " 287: '(555-⌊√5*5!⌋)',\n",
+       " 288: '(⌈55*√55⌉-5!)',\n",
+       " 289: '⌈55*√(.5*55)⌉',\n",
+       " 290: '(5*⌈√5+55.5⌉)',\n",
+       " 291: '⌊5.5*⌈55-√5⌉⌋',\n",
+       " 292: '⌈5.5*⌈55-√5⌉⌉',\n",
+       " 293: '⌊5!*√⌈5.555⌉⌋',\n",
+       " 294: '⌈5!*√⌈5.555⌉⌉',\n",
+       " 295: '(5!+⌊5!+55.5⌋)',\n",
+       " 296: '(5!+⌈5!+55.5⌉)',\n",
+       " 297: '(⌊5.5*55⌋-5)',\n",
+       " 298: '(⌈5.5*55⌉-5)',\n",
+       " 299: '⌊5.5*(55-.5)⌋',\n",
+       " 300: '(5*⌊5+55.5⌋)',\n",
+       " 301: '⌊⌊5.5*55⌋-.5⌋',\n",
+       " 302: '⌊5*(5+55.5)⌋',\n",
+       " 303: '⌈5*(5+55.5)⌉',\n",
+       " 304: '⌈.5+⌈5.5*55⌉⌉',\n",
+       " 305: '⌊.55*555⌋',\n",
+       " 306: '⌈.55*555⌉',\n",
+       " 307: '(5+⌊5.5*55⌋)',\n",
+       " 308: '(5+⌈5.5*55⌉)',\n",
+       " 309: '(⌈(√.5+5)*55⌉-5)',\n",
+       " 310: '(5*⌊55+√55⌋)',\n",
+       " 311: '⌈(√5**5)*5.55⌉',\n",
+       " 312: '⌊5*(55+√55)⌋',\n",
+       " 313: '⌈5*(55+√55)⌉',\n",
+       " 314: '⌈5.5*⌊√5+55⌋⌉',\n",
+       " 315: '(5*⌈55+√55⌉)',\n",
+       " 316: '⌊√(55/5.5)**5⌋',\n",
+       " 317: '⌈√(55/5.5)**5⌉',\n",
+       " 318: '⌊(5!+55)/.55⌋',\n",
+       " 319: '⌈(5!+55)/.55⌉',\n",
+       " 320: '(5*⌊5!-55.5⌋)',\n",
+       " 321: '⌊⌈5.555⌉!/√5⌋',\n",
+       " 322: '⌊5*(5!-55.5)⌋',\n",
+       " 323: '⌈5*(5!-55.5)⌉',\n",
+       " 324: '(⌈5.5⌉*⌊55-.5⌋)',\n",
+       " 325: '(5*⌈5!-55.5⌉)',\n",
+       " 326: '⌈(5!/55)**√55⌉',\n",
+       " 327: '(⌈5.5⌉*(55-.5))',\n",
+       " 328: '⌈.5*(55+(5*5!))⌉',\n",
+       " 329: '⌊(55*⌈5.5⌉)-.5⌋',\n",
+       " 330: '(55*⌈5.55⌉)',\n",
+       " 331: '⌈.5+(55*⌈5.5⌉)⌉',\n",
+       " 332: '⌊.55*(5+(5*5!))⌋',\n",
+       " 333: '(⌈5.5⌉*55.5)',\n",
+       " 334: '⌈5*(5!*.555)⌉',\n",
+       " 335: '(5*⌈5!*.555⌉)',\n",
+       " 336: '(5!+⌊5!/.555⌋)',\n",
+       " 337: '⌊.5*(5!+555)⌋',\n",
+       " 338: '⌈.5*(5!+555)⌉',\n",
+       " 339: '⌊55*√⌈5*√55⌉⌋',\n",
+       " 340: '⌈55*√⌈5*√55⌉⌉',\n",
+       " 341: '⌊55*(√.5+5.5)⌋',\n",
+       " 342: '⌈55*(√.5+5.5)⌉',\n",
+       " 343: '⌊5*(.55*(5+5!))⌋',\n",
+       " 344: '⌈5*(.55*(5+5!))⌉',\n",
+       " 345: '(5*⌈.55*(5+5!)⌉)',\n",
+       " 346: '(⌊55*√(5+5)⌋/.5)',\n",
+       " 347: '⌊5!*(.55+√5.5)⌋',\n",
+       " 348: '(⌈5.5⌉*⌈√5+55⌉)',\n",
+       " 349: '(⌊(5-.55)**5⌋/5)',\n",
+       " 350: '(⌊5!+55.5⌋/.5)',\n",
+       " 351: '((5!+55.5)/.5)',\n",
+       " 352: '(⌈5!+55.5⌉/.5)',\n",
+       " 353: '⌈⌈555/√5⌉/√.5⌉',\n",
+       " 354: '⌊(.5*⌈5.5⌉!)-5.5⌋',\n",
+       " 355: '(5*(5+(5!*.55)))',\n",
+       " 356: '⌈(55/5)**√⌈5.5⌉⌉',\n",
+       " 357: '⌊5.5*(5!-55)⌋',\n",
+       " 358: '⌈5.5*(5!-55)⌉',\n",
+       " 359: '⌊.5*⌊⌈5.55⌉!-.5⌋⌋',\n",
+       " 360: '(.5*⌈5.555⌉!)',\n",
+       " 361: '⌊(√5+5)*(55-5)⌋',\n",
+       " 362: '⌊.5*(5+⌈5.55⌉!)⌋',\n",
+       " 363: '(5.5*(5!*.55))',\n",
+       " 364: '⌈5!*(.55*5.5)⌉',\n",
+       " 365: '(5+(.5*⌈5.55⌉!))',\n",
+       " 366: '(5!*(.55+(.5*5)))',\n",
+       " 367: '(5!+⌊55*(5-.5)⌋)',\n",
+       " 368: '(5!+⌊555/√5⌋)',\n",
+       " 369: '(5!+⌈555/√5⌉)',\n",
+       " 370: '⌊√55*(55-5)⌋',\n",
+       " 371: '⌈√55*(55-5)⌉',\n",
+       " 372: '(55+⌈√(5+5)**5⌉)',\n",
+       " 373: '⌊55*√⌈5.5**√5⌉⌋',\n",
+       " 374: '⌈55*√⌈5.5**√5⌉⌉',\n",
+       " 375: '(5*⌈(5+5)*√55⌉)',\n",
+       " 376: '(55+⌊⌈5.5⌉!/√5⌋)',\n",
+       " 377: '(55+⌈⌈5.5⌉!/√5⌉)',\n",
+       " 378: '(⌊√55⌋*⌊55-.5⌋)',\n",
+       " 379: '⌊5!*√(55/5.5)⌋',\n",
+       " 380: '⌊55*(√55-.5)⌋',\n",
+       " 381: '⌈55*(√55-.5)⌉',\n",
+       " 382: '⌈55**(√55/5)⌉',\n",
+       " 383: '(5!+⌊5*(55-√5)⌋)',\n",
+       " 384: '⌊(55*⌊√55⌋)-.5⌋',\n",
+       " 385: '(55*⌊√55.5⌋)',\n",
+       " 386: '(555-⌊5!/√.5⌋)',\n",
+       " 387: '(⌊√.5*555⌋-5)',\n",
+       " 388: '⌊⌊√55⌋*55.5⌋',\n",
+       " 389: '⌈⌊√55⌋*55.5⌉',\n",
+       " 390: '(5*⌊55.5/√.5⌋)',\n",
+       " 391: '⌊⌊√.5*555⌋-.5⌋',\n",
+       " 392: '⌊√.5*555.5⌋',\n",
+       " 393: '⌈√.5*555.5⌉',\n",
+       " 394: '⌈√.5*⌈555.5⌉⌉',\n",
+       " 395: '⌊√.5*(5+555)⌋',\n",
+       " 396: '⌊.55*⌈5.55⌉!⌋',\n",
+       " 397: '(5!+⌊.5*555⌋)',\n",
+       " 398: '(5!+⌈.5*555⌉)',\n",
+       " 399: '⌊⌈5.5⌉!*.555⌋',\n",
+       " 400: '⌈⌈5.5⌉!*.555⌉',\n",
+       " 401: '⌊(√5+5)*55.5⌋',\n",
+       " 402: '(⌊55*√55⌋-5)',\n",
+       " 403: '(⌈55*√55⌉-5)',\n",
+       " 404: '⌊√55*(55-.5)⌋',\n",
+       " 405: '⌈√55*(55-.5)⌉',\n",
+       " 406: '⌊⌊55*√55⌋-.5⌋',\n",
+       " 407: '⌊55*√⌊55.5⌋⌋',\n",
+       " 408: '⌈55*√⌊55.5⌋⌉',\n",
+       " 409: '⌊55*√55.5⌋',\n",
+       " 410: '⌈55*√55.5⌉',\n",
+       " 411: '⌊√55*55.5⌋',\n",
+       " 412: '⌈√55*55.5⌉',\n",
+       " 413: '(5+⌈55*√55⌉)',\n",
+       " 414: '⌈√55.5**⌈.5*5⌉⌉',\n",
+       " 415: '⌊√55*⌈55.5⌉⌋',\n",
+       " 416: '⌈√55*⌈55.5⌉⌉',\n",
+       " 417: '⌊5*(5**(5*.55))⌋',\n",
+       " 418: '⌊55*√⌈√5+55⌉⌋',\n",
+       " 419: '⌊(5**5)/√55.5⌋',\n",
+       " 420: '⌊⌈5*.55⌉**5.5⌋',\n",
+       " 421: '⌈⌈5*.55⌉**5.5⌉',\n",
+       " 422: '(5!+⌊5.5*55⌋)',\n",
+       " 423: '(5!+⌈5.5*55⌉)',\n",
+       " 424: '⌈5.5*⌊55/√.5⌋⌉',\n",
+       " 425: '⌊55*(√5+5.5)⌋',\n",
+       " 426: '⌊55*√(5+55)⌋',\n",
+       " 427: '⌈55*√(5+55)⌉',\n",
+       " 428: '⌊⌈5.5*55⌉/√.5⌋',\n",
+       " 429: '(5.5*⌈55/√.5⌉)',\n",
+       " 430: '(555-(5+5!))',\n",
+       " 431: '⌈√55*⌈√5+55⌉⌉',\n",
+       " 432: '(555-⌈√5+5!⌉)',\n",
+       " 433: '(555-⌊√5+5!⌋)',\n",
+       " 434: '(555-⌈.5+5!⌉)',\n",
+       " 435: '⌊555.5-5!⌋',\n",
+       " 436: '⌈555.5-5!⌉',\n",
+       " 437: '(⌊√5-5!⌋+555)',\n",
+       " 438: '(⌈√5-5!⌉+555)',\n",
+       " 439: '⌊(⌈√55⌉*55)-.5⌋',\n",
+       " 440: '(5+(555-5!))',\n",
+       " 441: '⌈.5+(⌈√55⌉*55)⌉',\n",
+       " 442: '(5+⌈5!/(.5*.55)⌉)',\n",
+       " 443: '⌊55*√(5!-55)⌋',\n",
+       " 444: '(⌈√55⌉*55.5)',\n",
+       " 445: '⌈√55*(5+55)⌉',\n",
+       " 446: '⌊55*√(5!*.55)⌋',\n",
+       " 447: '⌊⌈√5⌉**5.555⌋',\n",
+       " 448: '⌈⌈√5⌉**5.555⌉',\n",
+       " 449: '⌈55*(5+√(5+5))⌉',\n",
+       " 450: '(5!+(55*⌈5.5⌉))',\n",
+       " 451: '⌊5!*(⌈5.55⌉-√5)⌋',\n",
+       " 452: '⌊55*(√5+⌈5.5⌉)⌋',\n",
+       " 453: '⌈55*(√5+⌈5.5⌉)⌉',\n",
+       " 454: '(⌊(5+5!)/.55⌋/.5)',\n",
+       " 455: '(⌊√55⌋*(5!-55))',\n",
+       " 456: '⌊5.5*⌊√5**5.5⌋⌋',\n",
+       " 457: '⌈5.5*⌊√5**5.5⌋⌉',\n",
+       " 458: '(⌊5-.5⌋*(5!-5.5))',\n",
+       " 459: '⌊555/(.5+√.5)⌋',\n",
+       " 460: '⌈555/(.5+√.5)⌉',\n",
+       " 461: '⌈55*√⌊√5.5**5⌋⌉',\n",
+       " 462: '(5.5*⌈√5**5.5⌉)',\n",
+       " 463: '⌈5*(5!-(.5*55))⌉',\n",
+       " 464: '(⌈√55⌉*⌈√5+55⌉)',\n",
+       " 465: '(5*(5!-⌊.5*55⌋))',\n",
+       " 466: '⌊⌊√.5*5!⌋*5.55⌋',\n",
+       " 467: '⌊55*(.5+⌈√55⌉)⌋',\n",
+       " 468: '(5!*(5-(5.5/5)))',\n",
+       " 469: '⌊55*(5+(√.5*5))⌋',\n",
+       " 470: '(555-⌈√.5*5!⌉)',\n",
+       " 471: '(555-⌊√.5*5!⌋)',\n",
+       " 472: '⌈⌈√.5*5!⌉*5.55⌉',\n",
+       " 473: '(5.5*⌈5**(5-√5)⌉)',\n",
+       " 474: '(5!*(5-(.5+.55)))',\n",
+       " 475: '((5!*⌊5-.55⌋)-5)',\n",
+       " 476: '(55+⌈⌈√5⌉**5.5⌉)',\n",
+       " 477: '⌊√.5*(5!+555)⌋',\n",
+       " 478: '⌈√.5*(5!+555)⌉',\n",
+       " 479: '((5*5!)-⌈5!+.55⌉)',\n",
+       " 480: '(5!*⌊5-.555⌋)',\n",
+       " 481: '((5*5!)-⌊5!-.55⌋)',\n",
+       " 482: '⌊√55*(5!-55)⌋',\n",
+       " 483: '⌈√55*(5!-55)⌉',\n",
+       " 484: '⌈(√5*5!)/.555⌉',\n",
+       " 485: '⌊√5*⌈5!/.555⌉⌋',\n",
+       " 486: '⌈√5*⌈5!/.555⌉⌉',\n",
+       " 487: '⌊5!/(.55**√5.5)⌋',\n",
+       " 488: '⌊.55*⌊5!*√55⌋⌋',\n",
+       " 489: '⌊.55*⌈5!*√55⌉⌋',\n",
+       " 490: '⌈.55*⌈5!*√55⌉⌉',\n",
+       " 491: '⌈5!*((5/.55)-5)⌉',\n",
+       " 492: '(⌊5-.5⌋*⌈√5*55⌉)',\n",
+       " 493: '⌈(⌊5/.55⌋**5)/5!⌉',\n",
+       " 494: '(⌊55*(5-.5)⌋/.5)',\n",
+       " 495: '(5*⌊55/.55⌋)',\n",
+       " 496: '(⌊555/√5⌋/.5)',\n",
+       " 497: '⌈555/(.5*√5)⌉',\n",
+       " 498: '(⌈555/√5⌉/.5)',\n",
+       " 499: '⌊5*(55/.55)⌋',\n",
+       " 500: '(555-55)',\n",
+       " 501: '(555-⌈5!/√5⌉)',\n",
+       " 502: '(555-⌊5!/√5⌋)',\n",
+       " 503: '⌊⌊5.5**5⌋/(5+5)⌋',\n",
+       " 504: '⌈⌊5.5**5⌋/(5+5)⌉',\n",
+       " 505: '(5!+(55*⌊√55⌋))',\n",
+       " 506: '⌊(5-.5)*(5!-√55)⌋',\n",
+       " 507: '(⌊5*(5!+5.5)⌋-5!)',\n",
+       " 508: '(⌈5*(5!+5.5)⌉-5!)',\n",
+       " 509: '⌊√.5*⌈5.555⌉!⌋',\n",
+       " 510: '⌈√.5*⌈5.555⌉!⌉',\n",
+       " 511: '⌈⌊5!*√5.5⌋/.55⌉',\n",
+       " 512: '(5!+⌊√.5*555⌋)',\n",
+       " 513: '(5!+⌈√.5*555⌉)',\n",
+       " 514: '⌈√5.5*⌈5!/.55⌉⌉',\n",
+       " 515: '(5!+(5!+(5*55)))',\n",
+       " 516: '(5!+⌊.55*⌈5.5⌉!⌋)',\n",
+       " 517: '(5!+⌈.55*⌈5.5⌉!⌉)',\n",
+       " 518: '(555-⌈5**√5⌉)',\n",
+       " 519: '(555-⌊5**√5⌋)',\n",
+       " 520: '((5*(5!-5))-55)',\n",
+       " 521: '⌈(5**5)/⌈5.55⌉⌉',\n",
+       " 522: '⌊55*(5+(5-.5))⌋',\n",
+       " 523: '⌈55*(5+(5-.5))⌉',\n",
+       " 524: '(⌈5!*(5!/55)⌉/.5)',\n",
+       " 525: '(5*((55/.5)-5))',\n",
+       " 526: '(⌊5*(55-√5)⌋/.5)',\n",
+       " 527: '(5!+⌊55*√55⌋)',\n",
+       " 528: '(5!+⌈55*√55⌉)',\n",
+       " 529: '((5!*(5-.55))-5)',\n",
+       " 530: '(555-(5*5))',\n",
+       " 531: '(555-(5!/5))',\n",
+       " 532: '⌊5**(5-(5.5/5))⌋',\n",
+       " 533: '⌊5!*(5-.555)⌋',\n",
+       " 534: '⌈5!*(5-.555)⌉',\n",
+       " 535: '(55+(5!*⌊5-.5⌋))',\n",
+       " 536: '⌈(5-.5)*⌊5!-.55⌋⌉',\n",
+       " 537: '⌊5.5*((.5*5)**5)⌋',\n",
+       " 538: '⌈5.5*((.5*5)**5)⌉',\n",
+       " 539: '(5+(5!*(5-.55)))',\n",
+       " 540: '((5+5)*⌊55-.5⌋)',\n",
+       " 541: '⌈.55+(5!*(5-.5))⌉',\n",
+       " 542: '(⌊5*(5!-.5)⌋-55)',\n",
+       " 543: '(555-⌈√5*5⌉)',\n",
+       " 544: '(555-⌊√5*5⌋)',\n",
+       " 545: '(555-(5+5))',\n",
+       " 546: '((5!*5.55)-5!)',\n",
+       " 547: '⌊555-√55⌋',\n",
+       " 548: '⌈555-√55⌉',\n",
+       " 549: '⌊555-5.5⌋',\n",
+       " 550: '⌈555-5.5⌉',\n",
+       " 551: '⌈555.5-5⌉',\n",
+       " 552: '⌊555-√5.5⌋',\n",
+       " 553: '⌊555.5-√5⌋',\n",
+       " 554: '⌊555-.55⌋',\n",
+       " 555: '⌊555.55⌋',\n",
+       " 556: '⌈555.55⌉',\n",
+       " 557: '⌊√5+555.5⌋',\n",
+       " 558: '⌈√5+555.5⌉',\n",
+       " 559: '(5+⌊555-.5⌋)',\n",
+       " 560: '⌊5+555.5⌋',\n",
+       " 561: '⌈5+555.5⌉',\n",
+       " 562: '⌊√55+555⌋',\n",
+       " 563: '⌈√55+555⌉',\n",
+       " 564: '⌈(5**5)/5.55⌉',\n",
+       " 565: '(5+(5+555))',\n",
+       " 566: '(⌊√5*5⌋+555)',\n",
+       " 567: '(⌈√5*5⌉+555)',\n",
+       " 568: '(⌈5*(5!-5.5)⌉-5)',\n",
+       " 569: '⌊(5*(5!-5))-5.5⌋',\n",
+       " 570: '(5*⌊5!-5.55⌋)',\n",
+       " 571: '⌊√55*⌊55/√.5⌋⌋',\n",
+       " 572: '⌊5*(5!-5.55)⌋',\n",
+       " 573: '⌈5*(5!-5.55)⌉',\n",
+       " 574: '(5+⌈(5**5)/5.5⌉)',\n",
+       " 575: '(5*⌈5!-5.55⌉)',\n",
+       " 576: '⌈.55+(5*(5!-5))⌉',\n",
+       " 577: '⌊55*(5+5.5)⌋',\n",
+       " 578: '⌈55*(5+5.5)⌉',\n",
+       " 579: '((5!/5)+555)',\n",
+       " 580: '((5*5)+555)',\n",
+       " 581: '⌈5.5+(5*(5!-5))⌉',\n",
+       " 582: '⌈5!*(5.55-√.5)⌉',\n",
+       " 583: '(55/(5/⌊5!/√5⌋))',\n",
+       " 584: '⌈5*(5!-√(55/5))⌉',\n",
+       " 585: '(5*(5!-⌈5*.55⌉))',\n",
+       " 586: '⌊5*(5!-(5*.55))⌋',\n",
+       " 587: '((⌊√5⌋**5)+555)',\n",
+       " 588: '⌈5!*√⌊5-.555⌋!⌉',\n",
+       " 589: '((5*5!)-(55/5))',\n",
+       " 590: '⌈5!*(5-(5/55))⌉',\n",
+       " 591: '(⌊5**√5⌋+555)',\n",
+       " 592: '(⌈5**√5⌉+555)',\n",
+       " 593: '⌈(5*5!)-√55.5⌉',\n",
+       " 594: '⌊(5*5!)-5.55⌋',\n",
+       " 595: '(5*⌊5!-.555⌋)',\n",
+       " 596: '⌈√(5+5)**5.55⌉',\n",
+       " 597: '⌊5*(5!-.555)⌋',\n",
+       " 598: '⌈5*(5!-.555)⌉',\n",
+       " 599: '⌊(5*5!)-.555⌋',\n",
+       " 600: '(5*⌊5.555⌋!)',\n",
+       " 601: '⌈(5*5!)+.555⌉',\n",
+       " 602: '⌊5*(5!+.555)⌋',\n",
+       " 603: '⌈5*(5!+.555)⌉',\n",
+       " 604: '(⌊5.5*55⌋/.5)',\n",
+       " 605: '(55*(55/5))',\n",
+       " 606: '(⌈5.5*55⌉/.5)',\n",
+       " 607: '⌊(5*5!)+√55.5⌋',\n",
+       " 608: '(⌊5!/√5⌋+555)',\n",
+       " 609: '(⌈5!/√5⌉+555)',\n",
+       " 610: '(55+555)',\n",
+       " 611: '(⌈√5**5⌉+555)',\n",
+       " 612: '(5+⌊(5*5!)+√55⌋)',\n",
+       " 613: '⌊5*(5!+(5*.55))⌋',\n",
+       " 614: '(((5**5)-55)/5)',\n",
+       " 615: '((.5*5!)+555)',\n",
+       " 616: '(5.5*⌊5!-√55⌋)',\n",
+       " 617: '⌊5*(.5+⌈√5*55⌉)⌋',\n",
+       " 618: '⌈5*(.5+⌈√5*55⌉)⌉',\n",
+       " 619: '⌊√5*⌊.5*555⌋⌋',\n",
+       " 620: '(5*⌊√5*55.5⌋)',\n",
+       " 621: '(.5*⌈√5*555⌉)',\n",
+       " 622: '⌈√5*⌈.5*555⌉⌉',\n",
+       " 623: '⌊5!*√⌊5*5.55⌋⌋',\n",
+       " 624: '⌈5!*√⌊5*5.55⌋⌉',\n",
+       " 625: '(5**⌊5-.555⌋)',\n",
+       " 626: '(⌊5.5+(5**5)⌋/5)',\n",
+       " 627: '⌊5*(5!+5.55)⌋',\n",
+       " 628: '⌈5*(5!+5.55)⌉',\n",
+       " 629: '⌊5.5*(5!-5.5)⌋',\n",
+       " 630: '(5*⌈5!+5.55⌉)',\n",
+       " 631: '⌈5.5+(5*(5+5!))⌉',\n",
+       " 632: '⌊5.5*⌈5!-5.5⌉⌋',\n",
+       " 633: '⌈5.5*⌈5!-5.5⌉⌉',\n",
+       " 634: '⌊5!*√⌈5*5.55⌉⌋',\n",
+       " 635: '(5*⌊5!+√55.5⌋)',\n",
+       " 636: '(((5**5)+55)/5)',\n",
+       " 637: '⌊5*(5!+√55.5)⌋',\n",
+       " 638: '⌊(5!-5)*5.55⌋',\n",
+       " 639: '⌈(5!-5)*5.55⌉',\n",
+       " 640: '(5*⌊55*√5.5⌋)',\n",
+       " 641: '⌈.5+(5*⌈5!+√55⌉)⌉',\n",
+       " 642: '(5+⌊5*(5!+√55)⌋)',\n",
+       " 643: '⌊5.5*⌊5!-√5.5⌋⌋',\n",
+       " 644: '⌊5*(55*√5.5)⌋',\n",
+       " 645: '(5*⌈55*√5.5⌉)',\n",
+       " 646: '⌈5*(5!+(5/.55))⌉',\n",
+       " 647: '⌊5.5*(5!-√5.5)⌋',\n",
+       " 648: '(5!*(⌊.5*55⌋/5))',\n",
+       " 649: '⌊⌊5!-√5⌋*5.55⌋',\n",
+       " 650: '((5+5)*(5!-55))',\n",
+       " 651: '(⌈√5⌉*⌈5!/.555⌉)',\n",
+       " 652: '⌊(5!*5.5)-√55⌋',\n",
+       " 653: '⌊(5!-√5)*5.55⌋',\n",
+       " 654: '⌊5.5*⌊5!-.55⌋⌋',\n",
+       " 655: '⌊(5*5!)+55.5⌋',\n",
+       " 656: '⌈(5*5!)+55.5⌉',\n",
+       " 657: '⌈5.5*(5!-.55)⌉',\n",
+       " 658: '⌈5!*√⌊.55*55⌋⌉',\n",
+       " 659: '⌊(5!*5.5)-.55⌋',\n",
+       " 660: '(5.5*⌊5.55⌋!)',\n",
+       " 661: '((5!*5.55)-5)',\n",
+       " 662: '⌊(5!-√.5)*5.55⌋',\n",
+       " 663: '⌊5.5*(5!+.55)⌋',\n",
+       " 664: '⌊⌈5.5⌉!-55.5⌋',\n",
+       " 665: '(⌈5.55⌉!-55)',\n",
+       " 666: '⌊5!*5.555⌋',\n",
+       " 667: '⌈5!*5.555⌉',\n",
+       " 668: '⌊(.5+5!)*5.55⌋',\n",
+       " 669: '⌈(.5+5!)*5.55⌉',\n",
+       " 670: '(5!+(555-5))',\n",
+       " 671: '(5+(5!*5.55))',\n",
+       " 672: '(5!+⌊555-√5⌋)',\n",
+       " 673: '(5!+⌈555-√5⌉)',\n",
+       " 674: '(5!+⌊555-.5⌋)',\n",
+       " 675: '⌊5!+555.5⌋',\n",
+       " 676: '⌈5!+555.5⌉',\n",
+       " 677: '(5!+⌊√5+555⌋)',\n",
+       " 678: '(5!+⌈√5+555⌉)',\n",
+       " 679: '⌈(5.5**5)/√55⌉',\n",
+       " 680: '(5+(5!+555))',\n",
+       " 681: '⌈5!*(5+(5/√55))⌉',\n",
+       " 682: '⌊55*(5+√55)⌋',\n",
+       " 683: '⌈55*(5+√55)⌉',\n",
+       " 684: '(⌈5.5⌉*⌊5!-5.5⌋)',\n",
+       " 685: '(5*⌊.5*(5*55)⌋)',\n",
+       " 686: '⌊(√5-⌈.5⌉)*555⌋',\n",
+       " 687: '⌊5.5*⌊5!+5.5⌋⌋',\n",
+       " 688: '⌈5.5*⌊5!+5.5⌋⌉',\n",
+       " 689: '(5!+⌈(5**5)/5.5⌉)',\n",
+       " 690: '⌊5.5*(5!+5.5)⌋',\n",
+       " 691: '⌈5.5*(5!+5.5)⌉',\n",
+       " 692: '(5+⌊5.5*(5+5!)⌋)',\n",
+       " 693: '⌊(5+5!)*5.55⌋',\n",
+       " 694: '⌈(5+5!)*5.55⌉',\n",
+       " 695: '(⌈5.55⌉!-(5*5))',\n",
+       " 696: '(⌈5.55⌉!-(5!/5))',\n",
+       " 697: '(⌈(.5*√55)**5⌉-5)',\n",
+       " 698: '⌊5.5*⌊5!+√55⌋⌋',\n",
+       " 699: '⌈5.5*⌊5!+√55⌋⌉',\n",
+       " 700: '(5*(5*⌈.5*55⌉))',\n",
+       " 701: '⌈5.5*(5!+√55)⌉',\n",
+       " 702: '⌊(5**5)/(5-.55)⌋',\n",
+       " 703: '⌈(5**5)/(5-.55)⌉',\n",
+       " 704: '(5.5*⌈5!+√55⌉)',\n",
+       " 705: '(5*(5!+⌊5!/5.5⌋))',\n",
+       " 706: '⌊(5!-√5)*⌈5.55⌉⌋',\n",
+       " 707: '(⌈5.5⌉!-⌈5+√55⌉)',\n",
+       " 708: '(⌈5!-√5⌉*⌈5.55⌉)',\n",
+       " 709: '(⌈5.5⌉!-(55/5))',\n",
+       " 710: '(⌈5.55⌉!-(5+5))',\n",
+       " 711: '(⌈5.5⌉!-⌊5/.55⌋)',\n",
+       " 712: '⌊⌈5.55⌉!-√55⌋',\n",
+       " 713: '⌈⌈5.55⌉!-√55⌉',\n",
+       " 714: '⌊⌈5.55⌉!-5.5⌋',\n",
+       " 715: '(⌈5.555⌉!-5)',\n",
+       " 716: '(⌈5.5⌉!-⌊5-.55⌋)',\n",
+       " 717: '⌊⌈5.555⌉!-√5⌋',\n",
+       " 718: '⌈⌈5.555⌉!-√5⌉',\n",
+       " 719: '⌊⌈5.555⌉!-.5⌋',\n",
+       " 720: '⌈5.5555⌉!',\n",
+       " 721: '⌈.5+⌈5.555⌉!⌉',\n",
+       " 722: '⌊√5+⌈5.555⌉!⌋',\n",
+       " 723: '⌈√5+⌈5.555⌉!⌉',\n",
+       " 724: '(⌊5!/√.5⌋+555)',\n",
+       " 725: '(5+⌈5.555⌉!)',\n",
+       " 726: '⌈5.5+⌈5.55⌉!⌉',\n",
+       " 727: '⌊√55+⌈5.55⌉!⌋',\n",
+       " 728: '⌈√55+⌈5.55⌉!⌉',\n",
+       " 729: '(⌈√5⌉**⌈5.555⌉)',\n",
+       " 730: '(5+(5+⌈5.55⌉!))',\n",
+       " 731: '(⌈5.5⌉!+(55/5))',\n",
+       " 732: '(5!*(5+(5.5/5)))',\n",
+       " 733: '(5+⌈⌈5.5⌉!+√55⌉)',\n",
+       " 734: '(5!+⌊√5*(5*55)⌋)',\n",
+       " 735: '(5*(5!+⌊.5*55⌋))',\n",
+       " 736: '⌊√5.5**√(5+55)⌋',\n",
+       " 737: '⌊5*(5!+(.5*55))⌋',\n",
+       " 738: '(⌈5.5⌉*⌈√5*55⌉)',\n",
+       " 739: '⌊5!*√⌈5*√55.5⌉⌋',\n",
+       " 740: '(5*(5!+⌈.5*55⌉))',\n",
+       " 741: '(⌈5.5⌉!+⌊5!/5.5⌋)',\n",
+       " 742: '(⌈5.5⌉!+⌈5!/5.5⌉)',\n",
+       " 743: '⌊⌊5!**√5⌋/(5+55)⌋',\n",
+       " 744: '((5!/5)+⌈5.55⌉!)',\n",
+       " 745: '((5*5)+⌈5.55⌉!)',\n",
+       " 746: '(5!+((5+(5**5))/5))',\n",
+       " 747: '(⌈5.5⌉!+⌊.5*55⌋)',\n",
+       " 748: '(⌈5.5⌉!+⌈.5*55⌉)',\n",
+       " 749: '⌈5!*(√5+⌊5-.55⌋)⌉',\n",
+       " 750: '((5+5!)*⌈5.55⌉)',\n",
+       " 751: '⌈5!*(√.5+5.55)⌉',\n",
+       " 752: '(5!+⌊5.5*(5!-5)⌋)',\n",
+       " 753: '(⌈5.5⌉*(5!+5.5))',\n",
+       " 754: '(5!+⌊5!*√⌈.5*55⌉⌋)',\n",
+       " 755: '((5*(5!+55))-5!)',\n",
+       " 756: '(⌈5.5⌉*⌈5!+5.5⌉)',\n",
+       " 757: '(⌈5.5⌉!+⌊5*√55⌋)',\n",
+       " 758: '(⌈5.5⌉!+⌈5*√55⌉)',\n",
+       " 759: '⌈5!*√⌊5.5*√55⌋⌉',\n",
+       " 760: '(5*⌊55*(5-√5)⌋)',\n",
+       " 761: '⌈5*(55*(5-√5))⌉',\n",
+       " 762: '(⌈5.5⌉*⌊5!+√55⌋)',\n",
+       " 763: '⌈5*(5!**(.5+.55))⌉',\n",
+       " 764: '⌊⌈5.5⌉*(5!+√55)⌋',\n",
+       " 765: '(5*⌈55*(5-√5)⌉)',\n",
+       " 766: '⌈(⌈√5⌉**5.5)/.55⌉',\n",
+       " 767: '⌈((5*5)-5.5)**√5⌉',\n",
+       " 768: '⌊√⌈5.55⌉**√55⌋',\n",
+       " 769: '⌊55**√(5*.55)⌋',\n",
+       " 770: '(55*⌊√55/.5⌋)',\n",
+       " 771: '⌈(5*5.55)**⌊√5⌋⌉',\n",
+       " 772: '(55+⌊⌈5.5⌉!-√5⌋)',\n",
+       " 773: '(⌊5!*√55.5⌋-5!)',\n",
+       " 774: '(55+⌊⌈5.5⌉!-.5⌋)',\n",
+       " 775: '(55+⌈5.55⌉!)',\n",
+       " 776: '⌈⌈5.5⌉!+55.5⌉',\n",
+       " 777: '⌊(555-5)/√.5⌋',\n",
+       " 778: '⌈(555-5)/√.5⌉',\n",
+       " 779: '(⌊555/√.5⌋-5)',\n",
+       " 780: '(⌈555/√.5⌉-5)',\n",
+       " 781: '⌊(5**5)/⌊5-.55⌋⌋',\n",
+       " 782: '⌈(5**5)/⌊5-.55⌋⌉',\n",
+       " 783: '⌊⌊555/√.5⌋-.5⌋',\n",
+       " 784: '⌊√⌊.5*5⌋*555⌋',\n",
+       " 785: '⌊555.5/√.5⌋',\n",
+       " 786: '⌈555.5/√.5⌉',\n",
+       " 787: '⌈⌈555.5⌉/√.5⌉',\n",
+       " 788: '⌈(5-.5)*(5!+55)⌉',\n",
+       " 789: '(5+⌊555/√.5⌋)',\n",
+       " 790: '(5+⌈555/√.5⌉)',\n",
+       " 791: '⌊(5+555)/√.5⌋',\n",
+       " 792: '⌈(5+555)/√.5⌉',\n",
+       " 793: '⌈⌈5.5⌉!*(5.5/5)⌉',\n",
+       " 794: '(⌈.55*⌈5.5⌉!⌉/.5)',\n",
+       " 795: '(5!+(5!+555))',\n",
+       " 796: '⌈55**(5/⌈.5*5⌉)⌉',\n",
+       " 797: '⌊5.5*(5!+(5*5))⌋',\n",
+       " 798: '((⌈√5⌉**5)+555)',\n",
+       " 799: '⌈5!*(5+√(5*.55))⌉',\n",
+       " 800: '(5*((⌈√5⌉*55)-5))',\n",
+       " 801: '⌊⌊√55⌋*(5!-5.5)⌋',\n",
+       " 802: '⌈⌊√55⌋*(5!-5.5)⌉',\n",
+       " 803: '⌊⌊5!**√5⌋/55.5⌋',\n",
+       " 804: '⌈⌊5!**√5⌋/55.5⌉',\n",
+       " 805: '((5!-5)*⌊√55.5⌋)',\n",
+       " 806: '⌊(5-.5)**(5-.55)⌋',\n",
+       " 807: '(5!+⌊5.5*(5+5!)⌋)',\n",
+       " 808: '(5!+⌈5.5*(5+5!)⌉)',\n",
+       " 809: '(⌊5!*√(5+55)⌋-5!)',\n",
+       " 810: '(⌈√5⌉*((5*55)-5))',\n",
+       " 811: '⌈(5+(5!**√5))/55⌉',\n",
+       " 812: '⌈(5*⌊5-.55⌋)**√5⌉',\n",
+       " 813: '(⌊5!*(√5+5)⌋-55)',\n",
+       " 814: '(⌊55*√55⌋/.5)',\n",
+       " 815: '⌊55/(.5/√55)⌋',\n",
+       " 816: '(⌈55*√55⌉/.5)',\n",
+       " 817: '⌊5**(5/(⌈5.5⌉/5))⌋',\n",
+       " 818: '((5*5!)+⌊5!/.55⌋)',\n",
+       " 819: '((5*5!)+⌈5!/.55⌉)',\n",
+       " 820: '((⌈√5⌉*(5*55))-5)',\n",
+       " 821: '⌊5!/(√.5**5.55)⌋',\n",
+       " 822: '⌊5!*√⌊55-√55⌋⌋',\n",
+       " 823: '(⌊√5*5!⌋+555)',\n",
+       " 824: '(⌈√5*5!⌉+555)',\n",
+       " 825: '(55*(5+(5+5)))',\n",
+       " 826: '⌊(5*5!)**(.5+.55)⌋',\n",
+       " 827: '⌊(5-.55)**(5-.5)⌋',\n",
+       " 828: '⌈(5-.55)**(5-.5)⌉',\n",
+       " 829: '⌈(√5+5)*(5!-5.5)⌉',\n",
+       " 830: '(5*⌊⌈√5⌉*55.5⌋)',\n",
+       " 831: '(⌈√5⌉*⌊.5*555⌋)',\n",
+       " 832: '⌊(.5+⌈.5⌉)*555⌋',\n",
+       " 833: '⌈(.5+⌈.5⌉)*555⌉',\n",
+       " 834: '(⌊5!*√55⌋-55)',\n",
+       " 835: '(⌈5!*√55⌉-55)',\n",
+       " 836: '⌊⌊√55⌋*(5!-.55)⌋',\n",
+       " 837: '⌈⌊√55⌋*(5!-.55)⌉',\n",
+       " 838: '⌊(5.5**5)/⌈5.5⌉⌋',\n",
+       " 839: '⌈(5.5**5)/⌈5.5⌉⌉',\n",
+       " 840: '(5!+⌈5.555⌉!)',\n",
+       " 841: '(5!+⌈.5+⌈5.55⌉!⌉)',\n",
+       " 842: '(⌈5.5⌉!+⌊√5*55⌋)',\n",
+       " 843: '(⌈5.5⌉!+⌈√5*55⌉)',\n",
+       " 844: '⌊5!*√(55-5.5)⌋',\n",
+       " 845: '⌊√55*⌊5!-5.5⌋⌋',\n",
+       " 846: '⌈√55*⌊5!-5.5⌋⌉',\n",
+       " 847: '(⌊√55⌋*⌈5!+.55⌉)',\n",
+       " 848: '⌊5!*√⌊55.5-5⌋⌋',\n",
+       " 849: '⌊√55*(5!-5.5)⌋',\n",
+       " 850: '(5*(5!+(55-5)))',\n",
+       " 851: '⌊⌊(5!-5)*√55⌋-.5⌋',\n",
+       " 852: '⌊5!*√(55.5-5)⌋',\n",
+       " 853: '⌈5!*√(55.5-5)⌉',\n",
+       " 854: '(5+⌈5!*√(55-5)⌉)',\n",
+       " 855: '(5*⌈5.55**⌈√5⌉⌉)',\n",
+       " 856: '⌊(5!-5)*√55.5⌋',\n",
+       " 857: '⌈(5!-5)*√55.5⌉',\n",
+       " 858: '(5+⌈(5!-5)*√55⌉)',\n",
+       " 859: '⌊55**(√5-.55)⌋',\n",
+       " 860: '⌈55**(√5-.55)⌉',\n",
+       " 861: '⌊5!*(5+(5!/55))⌋',\n",
+       " 862: '⌈5!*(5+(5!/55))⌉',\n",
+       " 863: '⌈5**(⌊5!/5.5⌋/5)⌉',\n",
+       " 864: '(⌊.5*55⌋/(.5**5))',\n",
+       " 865: '(5*⌊55*√(5+5)⌋)',\n",
+       " 866: '⌈5!*√⌊55-√5.5⌋⌉',\n",
+       " 867: '⌊√55*⌊5!-√5.5⌋⌋',\n",
+       " 868: '⌈√55*⌊5!-√5.5⌋⌉',\n",
+       " 869: '⌊5*(55*√(5+5))⌋',\n",
+       " 870: '((555-5!)/.5)',\n",
+       " 871: '⌊5!**√⌈5.55/5⌉⌋',\n",
+       " 872: '⌊5*(5!+(55-.5))⌋',\n",
+       " 873: '⌈5*(5!+(55-.5))⌉',\n",
+       " 874: '⌊(5*(5!+55))-.5⌋',\n",
+       " 875: '(5*⌊5!+55.5⌋)',\n",
+       " 876: '⌈.5+(5*(5!+55))⌉',\n",
+       " 877: '⌊5*(5!+55.5)⌋',\n",
+       " 878: '⌈5*(5!+55.5)⌉',\n",
+       " 879: '⌈√5*⌈√.5*555⌉⌉',\n",
+       " 880: '(5*⌈5!+55.5⌉)',\n",
+       " 881: '⌊5!*√⌊55-.55⌋⌋',\n",
+       " 882: '⌊√55*⌊5!-.55⌋⌋',\n",
+       " 883: '⌊⌊5!*√55⌋-5.5⌋',\n",
+       " 884: '⌊⌈5!*√55⌉-5.5⌋',\n",
+       " 885: '⌈⌈5!*√55⌉-5.5⌉',\n",
+       " 886: '⌈√55*(5!-.55)⌉',\n",
+       " 887: '⌈⌊5!-.5⌋*√55.5⌉',\n",
+       " 888: '(⌊5!*√55.5⌋-5)',\n",
+       " 889: '⌊5!*(55/√55)⌋',\n",
+       " 890: '⌈5!*(55/√55)⌉',\n",
+       " 891: '⌈.55+(5!*√55)⌉',\n",
+       " 892: '⌊⌊5!*√55.5⌋-.5⌋',\n",
+       " 893: '⌊⌊5.5⌋!*√55.5⌋',\n",
+       " 894: '⌊5.5+⌊5!*√55⌋⌋',\n",
+       " 895: '⌈5.5+⌊5!*√55⌋⌉',\n",
+       " 896: '⌈5.5+⌈5!*√55⌉⌉',\n",
+       " 897: '⌊5!*√⌈55.55⌉⌋',\n",
+       " 898: '⌈5!*√⌈55.55⌉⌉',\n",
+       " 899: '(5+⌈5!*√55.5⌉)',\n",
+       " 900: '(5*(5+(5!+55)))',\n",
+       " 901: '⌊⌈.5+5!⌉*√55.5⌋',\n",
+       " 902: '⌊(5-(5.5/5))**5⌋',\n",
+       " 903: '⌈(5-(5.5/5))**5⌉',\n",
+       " 904: '(5!+⌊555/√.5⌋)',\n",
+       " 905: '(5!+⌈555/√.5⌉)',\n",
+       " 906: '(⌈√5⌉*⌊5.5*55⌋)',\n",
+       " 907: '⌊⌈√5⌉*(5.5*55)⌋',\n",
+       " 908: '⌈⌈√5⌉*(5.5*55)⌉',\n",
+       " 909: '(⌈√5⌉*⌈5.5*55⌉)',\n",
+       " 910: '⌊√5*⌊55*√55⌋⌋',\n",
+       " 911: '⌈√5*⌊55*√55⌋⌉',\n",
+       " 912: '⌊√5*⌈55*√55⌉⌋',\n",
+       " 913: '⌈√5*⌈55*√55⌉⌉',\n",
+       " 914: '⌊⌊5.5**5⌋/5.5⌋',\n",
+       " 915: '⌊5.5**⌊5-.55⌋⌋',\n",
+       " 916: '⌈5.5**⌊5-.55⌋⌉',\n",
+       " 917: '⌈⌈√5+5!⌉*√55.5⌉',\n",
+       " 918: '⌊5.5*⌈5.5**⌈√5⌉⌉⌋',\n",
+       " 919: '⌈5.5*⌈5.5**⌈√5⌉⌉⌉',\n",
+       " 920: '(5+⌊5.5**⌊5-.5⌋⌋)',\n",
+       " 921: '⌊(.5*5)**√55.5⌋',\n",
+       " 922: '⌈(.5*5)**√55.5⌉',\n",
+       " 923: '(55+⌊5!*(√5+5)⌋)',\n",
+       " 924: '(55+⌈5!*(√5+5)⌉)',\n",
+       " 925: '(5*(555/⌈√5⌉))',\n",
+       " 926: '⌊⌊(5+5!)*√55⌋-.5⌋',\n",
+       " 927: '⌊√55*⌊5!+5.5⌋⌋',\n",
+       " 928: '⌈√55*⌊5!+5.5⌋⌉',\n",
+       " 929: '⌊5!*√⌊5+55.5⌋⌋',\n",
+       " 930: '⌈5!*√⌊5+55.5⌋⌉',\n",
+       " 931: '⌊(5+5!)*√55.5⌋',\n",
+       " 932: '⌈(5+5!)*√55.5⌉',\n",
+       " 933: '⌊5!*√(5+55.5)⌋',\n",
+       " 934: '⌈5!*√(5+55.5)⌉',\n",
+       " 935: '(55*⌈5!/√55⌉)',\n",
+       " 936: '⌈(5+5!)*√⌈55.5⌉⌉',\n",
+       " 937: '⌊5!*√⌈5+55.5⌉⌋',\n",
+       " 938: '⌈5!*√⌈5+55.5⌉⌉',\n",
+       " 939: '⌊5.5/(.5**√55)⌋',\n",
+       " 940: '⌈5.5/(.5**√55)⌉',\n",
+       " 941: '⌊√55*⌊5!+√55⌋⌋',\n",
+       " 942: '⌊⌊5**5.5⌋/√55⌋',\n",
+       " 943: '⌈⌊5**5.5⌋/√55⌉',\n",
+       " 944: '(55+⌊5!*√55⌋)',\n",
+       " 945: '(55+⌈5!*√55⌉)',\n",
+       " 946: '(5.5*⌊(5+5)**√5⌋)',\n",
+       " 947: '⌊(√.5+⌈.5⌉)*555⌋',\n",
+       " 948: '⌊5.55**⌊5-.5⌋⌋',\n",
+       " 949: '⌈5.55**⌊5-.5⌋⌉',\n",
+       " 950: '(5*(5*⌈5*√55⌉))',\n",
+       " 951: '⌊5.5*⌈(5+5)**√5⌉⌋',\n",
+       " 952: '⌊5!*√⌈55+√55⌉⌋',\n",
+       " 953: '⌈5!*√⌈55+√55⌉⌉',\n",
+       " 954: '⌊(5!+555)/√.5⌋',\n",
+       " 955: '⌊5!*(.55+√55)⌋',\n",
+       " 956: '⌈5!*(.55+√55)⌉',\n",
+       " 957: '⌊5!*(.5+√⌈55.5⌉)⌋',\n",
+       " 958: '⌈5!*(.5+√⌈55.5⌉)⌉',\n",
+       " 959: '⌊(5!*⌈√55⌉)-.55⌋',\n",
+       " 960: '(5!*⌈55/√55⌉)',\n",
+       " 961: '⌊√⌈.5*5⌉*555⌋',\n",
+       " 962: '⌊5.5*(5!+55)⌋',\n",
+       " 963: '⌈5.5*(5!+55)⌉',\n",
+       " 964: '⌈(√5-.5)*555⌉',\n",
+       " 965: '⌊(5!/5.55)**√5⌋',\n",
+       " 966: '⌈(5!/5.55)**√5⌉',\n",
+       " 967: '⌊5!*√⌈5!-55.5⌉⌋',\n",
+       " 968: '⌈5!*√⌈5!-55.5⌉⌉',\n",
+       " 969: '((⌊5-.5⌋**5)-55)',\n",
+       " 970: '(5*⌈5!**(5.5/5)⌉)',\n",
+       " 971: '(⌈5*(5!/.55)⌉-5!)',\n",
+       " 972: '⌊√5*(555-5!)⌋',\n",
+       " 973: '⌈√5*(555-5!)⌉',\n",
+       " 974: '⌊5!*√⌊5!*.555⌋⌋',\n",
+       " 975: '⌈5!*√⌊5!*.555⌋⌉',\n",
+       " 976: '(⌈√55⌉*⌊√5*55⌋)',\n",
+       " 977: '⌈(5+5)*((.5*5)**5)⌉',\n",
+       " 978: '⌊5!*(√.5+√55.5)⌋',\n",
+       " 979: '⌊5!*(5+√⌊5+5.5⌋)⌋',\n",
+       " 980: '⌈5!*(5+√⌊5+5.5⌋)⌉',\n",
+       " 981: '((5-.5)*⌊5!/.55⌋)',\n",
+       " 982: '⌊5!*√⌈5!*.555⌉⌋',\n",
+       " 983: '⌈5!*√⌈5!*.555⌉⌉',\n",
+       " 984: '(⌈√55⌉*⌈√5*55⌉)',\n",
+       " 985: '⌊(5-.5)*⌈5!/.55⌉⌋',\n",
+       " 986: '⌈(5-.5)*⌈5!/.55⌉⌉',\n",
+       " 987: '⌊√⌈√5*5⌉**5.55⌋',\n",
+       " 988: '⌊5!*(√5+⌈5.55⌉)⌋',\n",
+       " 989: '⌈5!*(√5+⌈5.55⌉)⌉',\n",
+       " 990: '((555/.5)-5!)',\n",
+       " 991: '⌊5!*(5-(√5-5.5))⌋',\n",
+       " 992: '⌈5!*(5-(√5-5.5))⌉',\n",
+       " 993: '⌈√5*⌊⌈√5⌉**5.55⌋⌉',\n",
+       " 994: '⌊5!*(5*√(5*.55))⌋',\n",
+       " 995: '(⌈5.5⌉!+(5*55))',\n",
+       " 996: '⌊⌈5**(5-√.5)⌉-5.5⌋',\n",
+       " 997: '⌊5!*(5+√(55/5))⌋',\n",
+       " 998: '⌊⌊5**5.5⌋/⌊√55⌋⌋',\n",
+       " 999: '⌈⌊5**5.5⌋/⌊√55⌋⌉',\n",
+       " ...}"
+      ]
+     },
+     "execution_count": 35,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "clear()\n",
     "\n",
@@ -1849,7 +2919,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1861,17 +2930,26 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 36,
    "metadata": {
     "button": false,
-    "collapsed": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
     }
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "23308"
+      ]
+     },
+     "execution_count": 36,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
     "unmakeable(ff)"
    ]
@@ -1880,7 +2958,6 @@
    "cell_type": "markdown",
    "metadata": {
     "button": false,
-    "deletable": true,
     "new_sheet": false,
     "run_control": {
      "read_only": false
@@ -1911,9 +2988,9 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.1"
+   "version": "3.5.3"
   }
  },
  "nbformat": 4,
- "nbformat_minor": 0
+ "nbformat_minor": 1
 }