{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sub-string divisibility (Euler Problem 43)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "[https://projecteuler.net/problem=43](https://projecteuler.net/problem=43)\n",
    "\n",
    "The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property.\n",
    "\n",
    "Let $d_1$ be the 1st digit, $d_2$ be the 2nd digit, and so on. In this way, we note the following:\n",
    "\n",
    "$d_2d_3d_4$=406 is divisible by 2\n",
    "\n",
    "$d_3d_4d_5$=063 is divisible by 3\n",
    "\n",
    "$d_4d_5d_6$=635 is divisible by 5\n",
    "\n",
    "$d_5d_6d_7=357$ is divisible by 7\n",
    "\n",
    "$d_6d_7d_8=572$ is divisible by 11\n",
    "\n",
    "$d_7d_8d_9=728$ is divisible by 13\n",
    "\n",
    "$d_8d_9d_{10} =289$ is divisible by 17\n",
    "\n",
    "Find the sum of all 0 to 9 pandigital numbers with this property."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We write a function that tests for the sub-string divisibility property. Then we simply brute force."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def is_sub_string_divisible(s):\n",
    "    s = \"\".join(s)\n",
    "    if int(s[1:4]) % 2 != 0:\n",
    "        return False\n",
    "    if int(s[2:5]) % 3 != 0:\n",
    "        return False\n",
    "    if int(s[3:6]) % 5 != 0:\n",
    "        return False\n",
    "    if int(s[4:7]) % 7 != 0:\n",
    "        return False\n",
    "    if int(s[5:8]) % 11 != 0:\n",
    "        return False\n",
    "    if int(s[6:9]) % 13 != 0:\n",
    "        return False\n",
    "    if int(s[7:10]) % 17 != 0:\n",
    "        return False\n",
    "    return True\n",
    "\n",
    "assert(is_sub_string_divisible(\"1406357289\"))\n",
    "assert(is_sub_string_divisible(\"1406357298\") ==  False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "import itertools\n",
    "\n",
    "s = sum([int(\"\".join(p)) for p in itertools.permutations(\"0123456789\") if is_sub_string_divisible(p)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "16695334890\n"
     ]
    }
   ],
   "source": [
    "print(s)\n",
    "assert(s == 16695334890)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I just want to write my own permutations functions to see that I am able to do it quickly. We pick each element once and combine it with the permutations for the remaining items:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "def permutations(items):\n",
    "    items = list(items)\n",
    "    if not items:\n",
    "        return [[]]\n",
    "    perms = []\n",
    "    for item in items:\n",
    "        items_new = list(items)\n",
    "        items_new.remove(item)\n",
    "        permutations_new = permutations(items_new)\n",
    "        for perm in permutations_new:\n",
    "            perms.append([item] + perm)\n",
    "    return perms\n",
    "\n",
    "\n",
    "assert(permutations([]) == [[]])\n",
    "assert(permutations([1]) == [[1]])\n",
    "assert(permutations([1,2]) == [[1,2], [2,1]])\n",
    "assert(permutations([1,2,3]) == [[1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1]])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "completion_date": "Fri, 21 Dec 2018, 16:28",
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   "language": "python3.6",
   "name": "python3"
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  "language_info": {
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.5"
  },
  "tags": [
   "pandigital",
   "divisibility",
   "permutations"
  ]
 },
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}