source-code/sympy/triple_six_puzzle.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Triple-six puzzle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the following puzzle:\n",
    "\n",
    "  * $0 \\, 0 \\, 0 = 6$\n",
    "  * $1 \\, 1 \\, 1 = 6$\n",
    "  * $2 \\, 2 \\, 2 = 6$\n",
    "  * ...\n",
    "  * $10 \\, 10 \\, 10 = 6$\n",
    "\n",
    "Insert operators `+`, `-`, `*`, `/`, square root and factorial such that the equations hold, e.g., $(1 + 1 + 1)! = 6$.  For most, multiplle solutions are possible, e.g., trivially, $(\\sqrt{1} + \\sqrt{1} + \\sqrt{1})! = 6$.  A few less trivial examples would be $3! + 3 - 3 = 6$, $3!*3/3 = 6$, or $3*3 - 3 = 6$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use the sympy library to tackle the puzzle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:28:51.419876Z",
     "start_time": "2018-08-09T16:28:51.389840Z"
    }
   },
   "outputs": [],
   "source": [
    "import itertools\n",
    "import sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Representing expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by defining the binary operators `+`, `-`, `*` and `/` based on sympy primitives `sympy.Add`, `sympy.Mul`, and `sympy.Pow`.  For each operator, we set `evaluate` to `False` to ensure that we can create an expression that isn't symplified automatically, i.e., $x + x$ should not be simplified to $2x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.209547Z",
     "start_time": "2018-08-09T16:09:38.178666Z"
    }
   },
   "outputs": [],
   "source": [
    "def Add(x, y):\n",
    "    return sympy.Add(x, y, evaluate=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.242426Z",
     "start_time": "2018-08-09T16:09:38.226686Z"
    }
   },
   "outputs": [],
   "source": [
    "def Sub(x, y):\n",
    "    return Add(x, -y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.294289Z",
     "start_time": "2018-08-09T16:09:38.267306Z"
    }
   },
   "outputs": [],
   "source": [
    "def Mul(x, y):\n",
    "    return sympy.Mul(x, y, evaluate=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.321073Z",
     "start_time": "2018-08-09T16:09:38.312036Z"
    }
   },
   "outputs": [],
   "source": [
    "def Div(x, y):\n",
    "    return Mul(x, sympy.Pow(y, -1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:45:28.318244Z",
     "start_time": "2018-08-09T16:45:28.244440Z"
    }
   },
   "outputs": [],
   "source": [
    "binary_operators = [Add, Sub, Mul, Div]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we define the unary operators square root and factorial, again based on their sympy implementation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.389585Z",
     "start_time": "2018-08-09T16:09:38.372076Z"
    }
   },
   "outputs": [],
   "source": [
    "def Sqrt(x):\n",
    "    return sympy.sqrt(x, evaluate=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T07:35:20.973711Z",
     "start_time": "2018-08-10T07:35:20.905067Z"
    }
   },
   "outputs": [],
   "source": [
    "def Fac(x):\n",
    "    return sympy.factorial(x, evaluate=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T07:35:23.528622Z",
     "start_time": "2018-08-10T07:35:23.514876Z"
    }
   },
   "outputs": [],
   "source": [
    "unary_operators = [Sqrt, Fac]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using these operators, we can build experssions in a symbol `x`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.521338Z",
     "start_time": "2018-08-09T16:09:38.493631Z"
    }
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.645648Z",
     "start_time": "2018-08-09T16:09:38.532260Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-x + x*x"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = Sub(Mul(x, x), x)\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substituing 3 for `x` yields 6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.712011Z",
     "start_time": "2018-08-09T16:09:38.666757Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can construct more elaborate expressions as well.  This one will evalutate to 6 for `x` is 3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.759318Z",
     "start_time": "2018-08-09T16:09:38.731646Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x/x)*factorial(x)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = Mul(Fac(x), Div(x, x))\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.804162Z",
     "start_time": "2018-08-09T16:09:38.780220Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(x, 3) == 6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Counting expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have a way to represent expressions, we need a way to generate all relevant expressions systematically.\n",
    "\n",
    "Unfortunately, the number of mathematical expressions satifying the equations is infinite.  This is easy to proof since, e.g.,$1! = 1$, so $(1 + 1 + 1)! = (1! + 1! + 1!)! = (1!! + 1!! + 1!!)! = \\cdots = 6$.\n",
    "\n",
    "This problem can be circumvented by limiting the number of consecutive unary operations to some number, say 2.  This would yield $x$, $x!$, $\\sqrt{x}$, $x!!$, $\\sqrt{x}!$, $\\sqrt{x!}$, $\\sqrt{\\sqrt{x}}$.  In general, if the number of consecutive uniary opreations is $d_{unary}$, then the number of combinations is $c_{unary} = 2^{d_{unary} + 1} - 1$, so 7 for $n = 2$.\n",
    "\n",
    "If we consider only a single binary operator, the number of such expression would be $7 \\times 4 \\times 7 \\times 7$.  Now we can add the second binary operator.  The first expression will be either the first, or the second operated, so that gives rise to a factor of 2.  For the other operand, there are again 7 possibilities, and we can apply 7 combinations of unary operators.  In total, there are $7 \\times 7 \\times 4 \\times 7 \\times 2 \\times 7 \\times 4 \\times 7$ expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.854918Z",
     "start_time": "2018-08-09T16:09:38.816079Z"
    }
   },
   "outputs": [],
   "source": [
    "d_unary = sympy.Symbol('d_unary')\n",
    "n_unary = sympy.Symbol('n_unary')\n",
    "c_unary = n_unary**(d_unary + 1) - 1\n",
    "n_binary = sympy.Symbol('n_binary')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:38.988526Z",
     "start_time": "2018-08-09T16:09:38.865873Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "537824"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nr_expressions = c_unary*c_unary*n_binary*c_unary*2*c_unary*n_binary*c_unary\n",
    "nr_expressions.subs(((n_unary, 2), (n_binary, 4), (d_unary, 2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These expressions need to be evaluated for each number from 0 to 10 which gives a quite impressive number of evaluations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:39.030327Z",
     "start_time": "2018-08-09T16:09:38.997598Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "22*n_binary**2*(n_unary**(d_unary + 1) - 1)**5"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nr_evaluations = 11*nr_expressions\n",
    "nr_evaluations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obviously, many expressions will be equivalent due to the properties of the mathematical operations, e.g., associativity, $a + (b + c) = (a + b) + c = a + b + c$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also worth pointing out that this will not (reasonably) work.  One of the expressions that will be generated is $((10!! \\times 10!!)!! \\times 10!!)!!$, and you don't even want to think about the number of digits required to represent that number, given that $100!$ has 158 digits!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:09:39.078204Z",
     "start_time": "2018-08-09T16:09:39.042810Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factorial(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since $x!!$ will not yield any solutions that are non-trial, we will exclude this case.  That means that the total number of expressinos is reduced to $6 \\times 6 \\times 4 \\times 6 \\times 2 \\times 6 \\times 4 \\times 6$ expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T08:39:03.091552Z",
     "start_time": "2018-08-10T08:39:03.078810Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The total number of expressions is 248832\n"
     ]
    }
   ],
   "source": [
    "print(f'The total number of expressions is {6*6*4*6*2*6*4*6}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generating expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we have all ingredients to construct expressions.  We can write a generator for that purpose.  We define a helper function to apply a list of function, e.g., `[f, g]` to an argument `x` to result in `f(g(x))`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-09T16:58:41.427033Z",
     "start_time": "2018-08-09T16:58:41.401566Z"
    }
   },
   "outputs": [],
   "source": [
    "def compose(func_list, x):\n",
    "    expr = x\n",
    "    for func in func_list:\n",
    "        expr = func(expr)\n",
    "    return expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need a function that will generate all expressions, and we implement it as a generator using `yield`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T08:16:56.146219Z",
     "start_time": "2018-08-10T08:16:56.059423Z"
    }
   },
   "outputs": [],
   "source": [
    "def expression_trees():\n",
    "    u_ops = [[], [Sqrt], [Fac], [Sqrt, Sqrt], [Sqrt, Fac], [Fac, Sqrt]]\n",
    "    x = sympy.Symbol('x')\n",
    "    for u_ops_1 in u_ops:\n",
    "        operand_11 = compose(u_ops_1, x)\n",
    "        for u_ops_2 in u_ops:\n",
    "            operand_12 = compose(u_ops_2, x)\n",
    "            for b_op_1 in binary_operators:\n",
    "                for u_ops_3 in u_ops:\n",
    "                    operand_21 = compose(u_ops_3, b_op_1(operand_11, operand_12))\n",
    "                    for u_ops_4 in u_ops:\n",
    "                        operand_22 = compose(u_ops_4, x)\n",
    "                        for b_op_2 in binary_operators:\n",
    "                            for u_ops_5 in u_ops:\n",
    "                                yield compose(u_ops_5, b_op_2(operand_21, operand_22))\n",
    "                                yield compose(u_ops_5, b_op_2(operand_22, operand_21))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's verify that we really generate the expected number of expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T08:35:05.968230Z",
     "start_time": "2018-08-10T08:34:19.247907Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "248832 expression generated\n"
     ]
    }
   ],
   "source": [
    "counter = 0\n",
    "for _ in expression_trees():\n",
    "    counter += 1\n",
    "print(f'{counter} expression generated')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, the following function will solve the puzzle for a given value of $x$.  It will print all expressions that evaluate to 6, and return the total number of such expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T08:19:27.954557Z",
     "start_time": "2018-08-10T08:19:27.922834Z"
    }
   },
   "outputs": [],
   "source": [
    "def solve_puzzle(value):\n",
    "    counter = 0\n",
    "    for i, expr in enumerate(expression_trees()):\n",
    "        try:\n",
    "            if expr.subs(x, value) == 6:\n",
    "                counter += 1\n",
    "                print(str(expr).replace('x', str(value)))\n",
    "        except ValueError:\n",
    "            pass\n",
    "    print(f'{counter} solutions for {value:d}')\n",
    "    return counter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T08:20:49.938581Z",
     "start_time": "2018-08-10T08:19:29.762773Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "factorial(factorial(0) + factorial(0) + factorial(0))\n",
      "factorial(factorial(0) + factorial(0) + factorial(0))\n",
      "factorial(factorial(0) + factorial(0) + factorial(sqrt(0)))\n",
      "factorial(factorial(0) + factorial(0) + factorial(sqrt(0)))\n",
      "factorial(factorial(0) + factorial(0) + sqrt(factorial(0)))\n",
      "factorial(factorial(0) + factorial(0) + sqrt(factorial(0)))\n",
      "factorial(factorial(0) + factorial(factorial(0) + factorial(0)))\n",
      "factorial(factorial(0) + factorial(factorial(0) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(factorial(0) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(factorial(0) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(factorial(0) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(factorial(0) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(0) + factorial(0))\n",
      "factorial(factorial(sqrt(0)) + factorial(0) + factorial(0))\n",
      "factorial(factorial(sqrt(0)) + factorial(0) + factorial(sqrt(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(0) + factorial(sqrt(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(0) + sqrt(factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(0) + sqrt(factorial(0)))\n",
      "factorial(factorial(0) + factorial(factorial(sqrt(0)) + factorial(0)))\n",
      "factorial(factorial(0) + factorial(factorial(sqrt(0)) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(factorial(sqrt(0)) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(factorial(sqrt(0)) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(factorial(sqrt(0)) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(factorial(sqrt(0)) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(0) + factorial(0))\n",
      "factorial(sqrt(factorial(0)) + factorial(0) + factorial(0))\n",
      "factorial(sqrt(factorial(0)) + factorial(0) + factorial(sqrt(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(0) + factorial(sqrt(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(0) + sqrt(factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(0) + sqrt(factorial(0)))\n",
      "factorial(factorial(0) + factorial(sqrt(factorial(0)) + factorial(0)))\n",
      "factorial(factorial(0) + factorial(sqrt(factorial(0)) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(sqrt(factorial(0)) + factorial(0)))\n",
      "factorial(factorial(sqrt(0)) + factorial(sqrt(factorial(0)) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(sqrt(factorial(0)) + factorial(0)))\n",
      "factorial(sqrt(factorial(0)) + factorial(sqrt(factorial(0)) + factorial(0)))\n"
     ]
    },
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    243\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 244\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    245\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'finite'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    243\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 244\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    245\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'zero'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    243\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 244\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    245\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'real'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyError\u001b[0m                                  Traceback (most recent call last)",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    243\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 244\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    245\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mKeyError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyError\u001b[0m: 'integer'",
      "\nDuring handling of the above exception, another exception occurred:\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-77-51f93fe7fea8>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msolve_puzzle\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m<ipython-input-76-6b459ba78ff8>\u001b[0m in \u001b[0;36msolve_puzzle\u001b[0;34m(value)\u001b[0m\n\u001b[1;32m      3\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexpr\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpression_trees\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m             \u001b[0;32mif\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m6\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      6\u001b[0m                 \u001b[0mcounter\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      7\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mreplace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mvalue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36msubs\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    928\u001b[0m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    929\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msequence\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 930\u001b[0;31m                 \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    931\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasic\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    932\u001b[0m                     \u001b[0;32mbreak\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36m_subs\u001b[0;34m(self, old, new, **hints)\u001b[0m\n\u001b[1;32m   1042\u001b[0m         \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_subs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1043\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mrv\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1044\u001b[0;31m             \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfallback\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mold\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnew\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1045\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1046\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36mfallback\u001b[0;34m(self, old, new)\u001b[0m\n\u001b[1;32m   1019\u001b[0m                     \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0marg\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1020\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mhit\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1021\u001b[0;31m                 \u001b[0mrv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1022\u001b[0m                 \u001b[0mhack2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'hack2'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1023\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mhack2\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Mul\u001b[0m\u001b[0;34m:\u001b[0m  \u001b[0;31m# 2-arg hack\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/operations.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m     43\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     44\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 45\u001b[0;31m         \u001b[0mc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mflatten\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     46\u001b[0m         \u001b[0mis_commutative\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_part\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_commutative\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36mflatten\u001b[0;34m(cls, seq)\u001b[0m\n\u001b[1;32m    597\u001b[0m             \u001b[0;31m# we know for sure the result will be 0 except the multiplicand\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    598\u001b[0m             \u001b[0;31m# is infinity\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 599\u001b[0;31m             \u001b[0;32mif\u001b[0m \u001b[0many\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_finite\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;32mFalse\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mc\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mc_part\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    600\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mNaN\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    601\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m<genexpr>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    597\u001b[0m             \u001b[0;31m# we know for sure the result will be 0 except the multiplicand\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    598\u001b[0m             \u001b[0;31m# is infinity\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 599\u001b[0;31m             \u001b[0;32mif\u001b[0m \u001b[0many\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_finite\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;32mFalse\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mc\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mc_part\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    600\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mNaN\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    601\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    246\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    247\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 248\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    249\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    250\u001b[0m     \u001b[0mgetit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mas_property\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    301\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    302\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mhandler_map\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 303\u001b[0;31m             \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpk\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    304\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    305\u001b[0m             \u001b[0;31m# we might have found the value of fact\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    301\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    302\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mhandler_map\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 303\u001b[0;31m             \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpk\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    304\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    305\u001b[0m             \u001b[0;31m# we might have found the value of fact\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    289\u001b[0m         \u001b[0;32mpass\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    290\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 291\u001b[0;31m         \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    292\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    293\u001b[0m             \u001b[0massumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdeduce_all_facts\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36m_eval_is_algebraic\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1200\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1201\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1202\u001b[0;31m         \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_zero\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0m_is_one\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1203\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1204\u001b[0m         \u001b[0;32melif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_rational\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36m_is_one\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1195\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0m_is_one\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1196\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1197\u001b[0;31m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_zero\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1198\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1199\u001b[0m                 \u001b[0;31m# when the operation is not allowed\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    246\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    247\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 248\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    249\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    250\u001b[0m     \u001b[0mgetit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mas_property\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    289\u001b[0m         \u001b[0;32mpass\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    290\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 291\u001b[0;31m         \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    292\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    293\u001b[0m             \u001b[0massumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdeduce_all_facts\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/add.py\u001b[0m in \u001b[0;36m_eval_is_zero\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    544\u001b[0m             \u001b[0;32melif\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_imaginary\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    545\u001b[0m                 \u001b[0mim\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 546\u001b[0;31m             \u001b[0;32melif\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mImaginaryUnit\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_real\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    547\u001b[0m                 \u001b[0mim_or_z\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    548\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    246\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    247\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 248\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    249\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    250\u001b[0m     \u001b[0mgetit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mas_property\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    301\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    302\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mhandler_map\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 303\u001b[0;31m             \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpk\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    304\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    305\u001b[0m             \u001b[0;31m# we might have found the value of fact\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    301\u001b[0m             \u001b[0;32mcontinue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    302\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mhandler_map\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 303\u001b[0;31m             \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpk\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    304\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    305\u001b[0m             \u001b[0;31m# we might have found the value of fact\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    289\u001b[0m         \u001b[0;32mpass\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    290\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 291\u001b[0;31m         \u001b[0ma\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mevaluate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    292\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    293\u001b[0m             \u001b[0massumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdeduce_all_facts\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m_eval_is_even\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1359\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1360\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_eval_is_even\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1361\u001b[0;31m         \u001b[0mis_integer\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_integer\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1362\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1363\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mis_integer\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36mgetit\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    246\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    247\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_assumptions\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdefault_assumptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcopy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 248\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0m_ask\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    249\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    250\u001b[0m     \u001b[0mgetit\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mas_property\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/site-packages/sympy/core/assumptions.py\u001b[0m in \u001b[0;36m_ask\u001b[0;34m(fact, obj)\u001b[0m\n\u001b[1;32m    296\u001b[0m     \u001b[0;31m# Try assumption's prerequisites\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    297\u001b[0m     \u001b[0mprereq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_assume_rules\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mprereq\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mfact\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 298\u001b[0;31m     \u001b[0mshuffle\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprereq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    299\u001b[0m     \u001b[0;32mfor\u001b[0m \u001b[0mpk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mprereq\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    300\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mpk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0massumptions\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/random.py\u001b[0m in \u001b[0;36mshuffle\u001b[0;34m(self, x, random)\u001b[0m\n\u001b[1;32m    274\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mreversed\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    275\u001b[0m                 \u001b[0;31m# pick an element in x[:i+1] with which to exchange x[i]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 276\u001b[0;31m                 \u001b[0mj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrandbelow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    277\u001b[0m                 \u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    278\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py36/lib/python3.6/random.py\u001b[0m in \u001b[0;36m_randbelow\u001b[0;34m(self, n, int, maxsize, type, Method, BuiltinMethod)\u001b[0m\n\u001b[1;32m    229\u001b[0m         \u001b[0;31m# Only call self.getrandbits if the original random() builtin method\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    230\u001b[0m         \u001b[0;31m# has not been overridden or if a new getrandbits() was supplied.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 231\u001b[0;31m         \u001b[0;32mif\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrandom\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mBuiltinMethod\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgetrandbits\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0mMethod\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    232\u001b[0m             \u001b[0mk\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbit_length\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# don't use (n-1) here because n can be 1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    233\u001b[0m             \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetrandbits\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m          \u001b[0;31m# 0 <= r < 2**k\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "solve_puzzle(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since it takes considerable time to generate all expressions, it is more efficient to do that only once, and evaluate it for all $x$ values at once."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-08-10T08:19:27.954557Z",
     "start_time": "2018-08-10T08:19:27.922834Z"
    }
   },
   "outputs": [],
   "source": [
    "def solve_puzzle_all_values(max_value=10):\n",
    "    counter = 0\n",
    "    for i, expr in enumerate(expression_trees()):\n",
    "        for value in range(max_value + 1):\n",
    "            try:\n",
    "                if expr.subs(x, value) == 6:\n",
    "                    counter += 1\n",
    "                    print(str(expr).replace('x', str(value)))\n",
    "            except ValueError:\n",
    "                pass\n",
    "    print(f'{counter} solutions for 0 up to {max_value:d}')\n",
    "    return counter"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}