sbu_projects/ams_595/PyAssignment1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework on Control Flow and Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(100 points total)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Conditional Statements\n",
    "\n",
    "(15 points)\n",
    "Translate the following MATLAB code into Python using `if-elif-else`. \n",
    "```\n",
    "n = input('Enter a number: ');\n",
    "\n",
    "switch n\n",
    "    case -1\n",
    "        disp('negative one')\n",
    "    case 0\n",
    "        disp('zero')\n",
    "    case 1\n",
    "        disp('positive one')\n",
    "    otherwise\n",
    "        disp('other value')\n",
    "end\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Enter a number: 1\n",
      "positive one\n"
     ]
    }
   ],
   "source": [
    "n = int(input(\"Enter a number: \"))\n",
    "if (n == -1):\n",
    "    print('negative one')\n",
    "elif (n == 0):\n",
    "    print('zero')\n",
    "elif (n == 1):\n",
    "    print('positive one')\n",
    "else:\n",
    "    print('other value')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. While Loops\n",
    "\n",
    "(15 points) Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:\n",
    "\n",
    "```1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...```\n",
    "\n",
    "Write a while-loop to construct a list of the numbers in the Fibonacci sequence whose values do not exceed 4,000,000."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578]\n"
     ]
    }
   ],
   "source": [
    "fib_list = []\n",
    "x = 1 # initial values\n",
    "y = 0\n",
    "while ((x + y) <= 4000000):\n",
    "    z = x + y\n",
    "    fib_list.append(z) # add z to the fibonacci list if it is < 4000000\n",
    "    y = x # change initial values \n",
    "    x = z\n",
    "print(fib_list)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. List Comprehension\n",
    "\n",
    "(20 points) A palindrome is a word, phrase, number, or other sequence of characters which reads the same backwards or forwards. \n",
    "\n",
    "For example, the number 121 is a palindrome, the number 123 is not. \n",
    "\n",
    "Use list comprehension to generate a list of palindromes made from the product of two 3-digit numbers. \n",
    "\n",
    "That is, each member of your list is of the form $x$, where $x$ is a palindrome and $x = n\\cdot m$ for some 3 digit numbers $n$ and $m$. Your list should contain all such possibilities. \n",
    "\n",
    "Finally, use this list to determine the largest palindrome that is the product of two 3-digit numbers. \n",
    "\n",
    "Hint: if `x` and `y` are two integers, you can use `str(x*y) == str(y*x)[::-1]` to determine whether `x * y` is a palindrome."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
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      "906609\n"
     ]
    }
   ],
   "source": [
    "pal_list = []\n",
    "pal_list = [x*y for x in range(100,1000) for y in range(100,1000) if (str(x*y) == str(y*x)[::-1])]\n",
    "# the above line checks which numbers x*y are palindromes for x = 100-999 and y = 100-999\n",
    "print(pal_list)\n",
    "pal_list.sort() # sorts list of palindromes from smallest to largest\n",
    "print(pal_list[-1]) # prints last value in pal_list, which is the largest palindrome"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Functions\n",
    "\n",
    "2a. (10 points) Write a function that determines if a number is prime."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def isprime(n):\n",
    "    if (n > 1):\n",
    "        for i in range(2,n):\n",
    "            if (n % i) == 0: # if n is divisible by i, then it is not prime\n",
    "                return False\n",
    "        else:\n",
    "            return True\n",
    "    else:\n",
    "        return False # if n < 1, then it is not prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n",
      "False\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "# test codes\n",
    "print(isprime(2))\n",
    "print(isprime(10))\n",
    "print(isprime(17))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "2b. (10 points) Write a function that finds all the factors of a number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def factorize(n):\n",
    "    answer = []\n",
    "    for i in range(1,n+1):\n",
    "        if (n % i) == 0: # if n is divisible by i, then i is a factor of n\n",
    "            answer.append(i) # add i to the list of factors of n\n",
    "    return answer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2]\n",
      "[1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72]\n",
      "[1, 2, 4, 7, 14, 28, 49, 98, 196]\n"
     ]
    }
   ],
   "source": [
    "# test codes\n",
    "print(factorize(2))\n",
    "print(factorize(72))\n",
    "print(factorize(196))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2c. (10 points) Write a function that finds the prime factorization of a number."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prime_factorize(n):\n",
    "    answer = []\n",
    "    i = 2\n",
    "    while (i * i) <= n: \n",
    "        if n % i:\n",
    "            i += 1 # i is not a prime factor\n",
    "        else:\n",
    "            n //= i\n",
    "            answer.append(i) # i is a prime factor, therefore add it to list of prime factors\n",
    "    if n > 1:\n",
    "        answer.append(n) # add n to list of prime factors\n",
    "    return answer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2]\n",
      "[2, 2, 2, 3, 3]\n",
      "[2, 2, 7, 7]\n"
     ]
    }
   ],
   "source": [
    "# test codes\n",
    "print(prime_factorize(2))\n",
    "print(prime_factorize(72))\n",
    "print(prime_factorize(196))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Recursive Functions\n",
    "(20 points) The following function uses recursion to generate the $n$th row of Pascal's triangle:\n",
    "```\n",
    "              1\n",
    "           1     1\n",
    "        1     2     1\n",
    "      1    3     3     1\n",
    "   1    4     6     4    1\n",
    "1     5    10    10    5    1\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 5, 10, 10, 5, 1]\n"
     ]
    }
   ],
   "source": [
    "def pascal(n):\n",
    "    if n == 1:\n",
    "        return [1]\n",
    "    else:\n",
    "        p_line = pascal(n-1)\n",
    "        line = [ p_line[i]+p_line[i+1] for i in range(len(p_line)-1)]\n",
    "        line.insert(0,1)\n",
    "        line.append(1)\n",
    "    return line\n",
    "\n",
    "print(pascal(6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rewrite the above function to use iteration (using only `while` or `for` loops) instead of recursion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 5, 10, 10, 5, 1]\n"
     ]
    }
   ],
   "source": [
    "def pascal(n):\n",
    "    n_list = []\n",
    "    for i in range(1,n+1):\n",
    "        line = 1 # the start of a row is always 1\n",
    "        for j in range(1,i+1):\n",
    "            if i == n:\n",
    "                n_list.append(line) # add line to n_list\n",
    "            line = line * (i-j) // j # to save the nth row of n_list\n",
    "    return n_list\n",
    "\n",
    "print(pascal(6))"
   ]
  }
 ],
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