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   "source": [
    "## CS 201: Collatz\n",
    "\n",
    "<p>\n",
    "<script language=\"JavaScript\">\n",
    "    document.write(\"Last modified: \" + document.lastModified)\n",
    "</script>\n",
    "    \n",
    "See <a target=ee href=\"collatz.rkt\">collatz.rkt</a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Collatz Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "(define (collatz n)\n",
    "  (if (= (modulo n 2) 0) (/ n 2)\n",
    "      (+ 1 (* n 3))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try it out."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<code>34</code>"
      ],
      "text/plain": [
       "34"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(collatz 11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<code>17</code>"
      ],
      "text/plain": [
       "17"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(collatz 34)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<code>52</code>"
      ],
      "text/plain": [
       "52"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(collatz 17)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Recursion\n",
    "\n",
    "We can have <code>collatz</code> call its own result.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<code>52</code>"
      ],
      "text/plain": [
       "52"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(collatz (collatz (collatz 11)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having a function call itself is known as <b>recursion</b>. We can generate the series based on Collatz using recursion.\n",
    "\n",
    "The function <code>c-series</code> calls itself until the result converges to 1.  The function uses <code>let</code> to introduce a local variable <code>next</code>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "(define (c-series n)\n",
    "  (print n)\n",
    "  (newline)\n",
    "  (if (equal? n 1) 'done\n",
    "      (let ((next (collatz n)))\n",
    "    (c-series next))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11\n",
      "34\n",
      "17\n",
      "52\n",
      "26\n",
      "13\n",
      "40\n",
      "20\n",
      "10\n",
      "5\n",
      "16\n",
      "8\n",
      "4\n",
      "2\n",
      "1\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<code>'done</code>"
      ],
      "text/plain": [
       "'done"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(c-series 11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can generate the series based on Collatz using recursion as above, but without the \n",
    "temporary variable <code>next</code>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "(define (c-series2 n)\n",
    "  (print n)\n",
    "  (newline)\n",
    "  (if (equal? n 1) 'done\n",
    "      (c-series2 (collatz n))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "11\n",
      "34\n",
      "17\n",
      "52\n",
      "26\n",
      "13\n",
      "40\n",
      "20\n",
      "10\n",
      "5\n",
      "16\n",
      "8\n",
      "4\n",
      "2\n",
      "1\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<code>'done</code>"
      ],
      "text/plain": [
       "'done"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(c-series2 11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Trace\n",
    "\n",
    "We can trace the execution of the recursive functions using the <code>trace</code> function, which prints out each call to the given function.\n",
    "\n",
    "When writing recursive code, you will often find it useful to use <code>trace</code> to see what is going on underneath the hood."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "(require racket/trace)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "(trace c-series)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ">(c-series 11)\n",
      "11\n",
      ">(c-series 34)\n",
      "34\n",
      ">(c-series 17)\n",
      "17\n",
      ">(c-series 52)\n",
      "52\n",
      ">(c-series 26)\n",
      "26\n",
      ">(c-series 13)\n",
      "13\n",
      ">(c-series 40)\n",
      "40\n",
      ">(c-series 20)\n",
      "20\n",
      ">(c-series 10)\n",
      "10\n",
      ">(c-series 5)\n",
      "5\n",
      ">(c-series 16)\n",
      "16\n",
      ">(c-series 8)\n",
      "8\n",
      ">(c-series 4)\n",
      "4\n",
      ">(c-series 2)\n",
      "2\n",
      ">(c-series 1)\n",
      "1\n",
      "<'done\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<code>'done</code>"
      ],
      "text/plain": [
       "'done"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(c-series 11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define a <code>c-series3</code> which does not print out results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "(define (c-series3 n)\n",
    "  (if (equal? n 1) 'done\n",
    "      (c-series3 (collatz n))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<code>'done</code>"
      ],
      "text/plain": [
       "'done"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(c-series3 11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Not really too useful, unless we trace it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "(trace c-series3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ">(c-series3 11)\n",
      ">(c-series3 34)\n",
      ">(c-series3 17)\n",
      ">(c-series3 52)\n",
      ">(c-series3 26)\n",
      ">(c-series3 13)\n",
      ">(c-series3 40)\n",
      ">(c-series3 20)\n",
      ">(c-series3 10)\n",
      ">(c-series3 5)\n",
      ">(c-series3 16)\n",
      ">(c-series3 8)\n",
      ">(c-series3 4)\n",
      ">(c-series3 2)\n",
      ">(c-series3 1)\n",
      "<'done\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<code>'done</code>"
      ],
      "text/plain": [
       "'done"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(c-series3 11)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "End of Collatz notebook."
   ]
  }
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