{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import print_function, division\n",
    "\n",
    "from itertools import starmap, product\n",
    "\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def obtainable(coins):\n",
    "    \"\"\"Find the set of values obtainable with the given coins.\"\"\"\n",
    "    combos = set(coins)\n",
    "    combos.update(map(sum, product(coins, coins)))\n",
    "    return combos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "coins = [1, 2, 4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{1, 2, 3, 4, 5, 6, 8}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "obtainable(coins)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "values = set(range(1, 21))\n",
    "    \n",
    "def unobtainable(coins):\n",
    "    \"\"\"Find values that can't be obtained with the given coins.\"\"\"\n",
    "    return values - obtainable(coins)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "unobtainable(coins)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def possible_new_coin(coins, value):\n",
    "    \"\"\"Generate new coins that would make the given value obtainable.\"\"\"\n",
    "    if value % 2 == 0:\n",
    "        yield value // 2\n",
    "        \n",
    "    for coin in coins:\n",
    "        if value > coin:\n",
    "            yield value - coin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{3, 5, 6}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(possible_new_coin(coins, 7))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{5, 7, 8}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(possible_new_coin(coins, 9))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{5, 6, 8, 9}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(possible_new_coin(coins, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class Combinator:\n",
    "    def __init__(self, values):\n",
    "        self.values = values\n",
    "        \n",
    "        combo = frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90})\n",
    "        self.combos = {combo}\n",
    "        self.smallest = len(combo)\n",
    "\n",
    "    def add_combo(self, coins):\n",
    "        \"\"\"Add a combo to the cache and update `smallest`.\"\"\"\n",
    "        self.combos.add(coins)\n",
    "        n = len(coins)\n",
    "        if n < self.smallest:\n",
    "            self.smallest = n\n",
    "\n",
    "    def unobtainable(self, coins):\n",
    "        \"\"\"Find values that can't be obtained with the given coins.\"\"\"\n",
    "        return self.values - obtainable(coins)\n",
    "\n",
    "    def consider(self, coins):\n",
    "        \"\"\"Starting with `coins`, search for a minimal combo.\"\"\"\n",
    "        \n",
    "        # if we've already seen better, don't bother\n",
    "        if len(coins) > self.smallest:\n",
    "            return\n",
    "        \n",
    "        # find unobtainable values\n",
    "        bad_values = unobtainable(coins)\n",
    "        \n",
    "        # if all values are obtainable, we're set\n",
    "        if len(bad_values) == 0:\n",
    "            self.add_combo(coins)\n",
    "            return\n",
    "    \n",
    "        # otherwise, consider possible coins we could add\n",
    "        for value in bad_values:\n",
    "            for new_coin in possible_new_coin(coins, value):\n",
    "                self.consider(coins | {new_coin})\n",
    "                \n",
    "    def print_winners(self):\n",
    "        \"\"\"Print the best combinations.\"\"\"\n",
    "        for combo in self.combos:\n",
    "            if len(combo) == self.smallest:\n",
    "                print(sorted(combo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "frozenset({1})"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "frozenset((1,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "values = set(range(1, 21))\n",
    "combinator = Combinator(values)\n",
    "combinator.consider(frozenset((1,)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "combinator.smallest"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 3, 5, 6, 13, 14]\n",
      "[1, 3, 4, 8, 9, 11]\n",
      "[1, 3, 5, 7, 9, 10]\n",
      "[1, 3, 4, 9, 11, 16]\n",
      "[1, 2, 5, 8, 9, 10]\n"
     ]
    }
   ],
   "source": [
    "combinator.print_winners()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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