From: Neil Smith Date: Wed, 24 Sep 2014 20:04:20 +0000 (+0100) Subject: Added pre-course exercise solutions X-Git-Url: https://git.njae.me.uk/?a=commitdiff_plain;h=2164c26a5c25c650aa0b624e04e70da79c2fa0b4;p=cas-master-teacher-training.git Added pre-course exercise solutions --- 2164c26a5c25c650aa0b624e04e70da79c2fa0b4 diff --git a/euler-11.ipynb b/euler-11.ipynb new file mode 100644 index 0000000..1e61c32 --- /dev/null +++ b/euler-11.ipynb @@ -0,0 +1,862 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:2513cb86760ebf975b3dce5f7884ee5e23c13f94e20cc9af8fc245f251455fda" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Project Euler problem 11\n", + "Find the largest product of four adjacent numbers in the grid." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Constants" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "ROWS = COLUMNS = 20\n", + "SECTION_LEN = 4" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 97 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "##Data structure\n", + "Convert the text of the numbers into a 2d array of integers.\n", + "\n", + "(Alterntive data structures include a 1d list of integers, or a dict with keys of (r, c) pairs.)" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "GRID_STRING = \"\"\"08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08\n", + "49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00\n", + "81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65\n", + "52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91\n", + "22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80\n", + "24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50\n", + "32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70\n", + "67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21\n", + "24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72\n", + "21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95\n", + "78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92\n", + "16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57\n", + "86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58\n", + "19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40\n", + "04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66\n", + "88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69\n", + "04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36\n", + "20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16\n", + "20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54\n", + "01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48\"\"\"\n", + "\n", + "GRID_LIST = [int(n) for n in GRID_STRING.split()]\n", + "GRID = [GRID_LIST[i:i+COLUMNS] for i in range(0, ROWS * COLUMNS, COLUMNS)]" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 98 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "for row in GRID:\n", + " print(row)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "[8, 2, 22, 97, 38, 15, 0, 40, 0, 75, 4, 5, 7, 78, 52, 12, 50, 77, 91, 8]\n", + "[49, 49, 99, 40, 17, 81, 18, 57, 60, 87, 17, 40, 98, 43, 69, 48, 4, 56, 62, 0]\n", + "[81, 49, 31, 73, 55, 79, 14, 29, 93, 71, 40, 67, 53, 88, 30, 3, 49, 13, 36, 65]\n", + "[52, 70, 95, 23, 4, 60, 11, 42, 69, 24, 68, 56, 1, 32, 56, 71, 37, 2, 36, 91]\n", + "[22, 31, 16, 71, 51, 67, 63, 89, 41, 92, 36, 54, 22, 40, 40, 28, 66, 33, 13, 80]\n", + "[24, 47, 32, 60, 99, 3, 45, 2, 44, 75, 33, 53, 78, 36, 84, 20, 35, 17, 12, 50]\n", + "[32, 98, 81, 28, 64, 23, 67, 10, 26, 38, 40, 67, 59, 54, 70, 66, 18, 38, 64, 70]\n", + "[67, 26, 20, 68, 2, 62, 12, 20, 95, 63, 94, 39, 63, 8, 40, 91, 66, 49, 94, 21]\n", + "[24, 55, 58, 5, 66, 73, 99, 26, 97, 17, 78, 78, 96, 83, 14, 88, 34, 89, 63, 72]\n", + "[21, 36, 23, 9, 75, 0, 76, 44, 20, 45, 35, 14, 0, 61, 33, 97, 34, 31, 33, 95]\n", + "[78, 17, 53, 28, 22, 75, 31, 67, 15, 94, 3, 80, 4, 62, 16, 14, 9, 53, 56, 92]\n", + "[16, 39, 5, 42, 96, 35, 31, 47, 55, 58, 88, 24, 0, 17, 54, 24, 36, 29, 85, 57]\n", + "[86, 56, 0, 48, 35, 71, 89, 7, 5, 44, 44, 37, 44, 60, 21, 58, 51, 54, 17, 58]\n", + "[19, 80, 81, 68, 5, 94, 47, 69, 28, 73, 92, 13, 86, 52, 17, 77, 4, 89, 55, 40]\n", + "[4, 52, 8, 83, 97, 35, 99, 16, 7, 97, 57, 32, 16, 26, 26, 79, 33, 27, 98, 66]\n", + "[88, 36, 68, 87, 57, 62, 20, 72, 3, 46, 33, 67, 46, 55, 12, 32, 63, 93, 53, 69]\n", + "[4, 42, 16, 73, 38, 25, 39, 11, 24, 94, 72, 18, 8, 46, 29, 32, 40, 62, 76, 36]\n", + "[20, 69, 36, 41, 72, 30, 23, 88, 34, 62, 99, 69, 82, 67, 59, 85, 74, 4, 36, 16]\n", + "[20, 73, 35, 29, 78, 31, 90, 1, 74, 31, 49, 71, 48, 86, 81, 16, 23, 57, 5, 54]\n", + "[1, 70, 54, 71, 83, 51, 54, 69, 16, 92, 33, 48, 61, 43, 52, 1, 89, 19, 67, 48]\n" + ] + } + ], + "prompt_number": 99 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#Directions\n", + "What lines do we examine? Each number paricipates in up to 8 \u00d7 4 = 32 lines (fewer near the edges), but we can use the fact that multipication is commutative to only examine four lines that start at a number.\n", + "\n", + "`directions` stores those directions, and how to move in the direction." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# Directions, as the pair (difference-in-row, difference-in-column)\n", + "DIRECTIONS = {'N': (-1, 0), 'NW': (-1, -1), 'W': (0, -1), 'SW': (1, -1)}" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 20 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Finding the right numbers\n", + "Given a starting position and a direction, find the right numbers.\n", + "\n", + "### Question:\n", + "Should we worry if the request goes out of the bounds of the grid?" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def numbers(row, column, direction):\n", + " nums = []\n", + " dr, dc = DIRECTIONS[direction]\n", + " for _ in range(SECTION_LEN):\n", + " nums.append(GRID[row][column])\n", + " row += dr\n", + " column += dc\n", + " return nums" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 71 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Test it." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "numbers(0, 3, 'W')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 72, + "text": [ + "[97, 22, 2, 8]" + ] + } + ], + "prompt_number": 72 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "numbers(3, 0, 'N')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 73, + "text": [ + "[52, 81, 49, 8]" + ] + } + ], + "prompt_number": 73 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "numbers(3, 3, 'NW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 74, + "text": [ + "[23, 31, 49, 8]" + ] + } + ], + "prompt_number": 74 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "numbers(3, 3, 'SW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 75, + "text": [ + "[23, 16, 47, 32]" + ] + } + ], + "prompt_number": 75 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Product of a list" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def product(ns):\n", + " p = 1\n", + " for n in ns:\n", + " p *= n\n", + " return p" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 25 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "product(numbers(0, 3, 'W'))" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 26, + "text": [ + "34144" + ] + } + ], + "prompt_number": 26 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "97 * 22 * 2 * 8" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 27, + "text": [ + "34144" + ] + } + ], + "prompt_number": 27 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## What directions don't take us out outside of the boundaries?" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def valid_direction_explicit(row, column, direction):\n", + " if direction == 'N' and row >= SECTION_LEN -1:\n", + " return True\n", + " elif direction == 'W' and column >= SECTION_LEN -1:\n", + " return True\n", + " elif direction == 'NW' and row >= SECTION_LEN -1 and column >= SECTION_LEN -1:\n", + " return True\n", + " elif direction == 'SW' and row + SECTION_LEN <= ROWS and column >= SECTION_LEN -1:\n", + " return True\n", + " else:\n", + " return False" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 63 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(0, 0, 'N')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 33, + "text": [ + "False" + ] + } + ], + "prompt_number": 33 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(5, 5, 'N')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 35, + "text": [ + "True" + ] + } + ], + "prompt_number": 35 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(5, 5, 'NW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 36, + "text": [ + "True" + ] + } + ], + "prompt_number": 36 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(5, 5, 'W')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 37, + "text": [ + "True" + ] + } + ], + "prompt_number": 37 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(5, 5, 'SW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 38, + "text": [ + "True" + ] + } + ], + "prompt_number": 38 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(17, 5, 'SW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 61, + "text": [ + "False" + ] + } + ], + "prompt_number": 61 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(16, 5, 'SW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 64, + "text": [ + "True" + ] + } + ], + "prompt_number": 64 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction_explicit(2, 2, 'NW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 60, + "text": [ + "False" + ] + } + ], + "prompt_number": 60 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def valid_direction(row, column, direction):\n", + " dr, dc = DIRECTIONS[direction]\n", + " end_row = row + dr * (SECTION_LEN -1)\n", + " end_col = column + dc * (SECTION_LEN -1)\n", + " if end_row >= 0 and end_row < ROWS and end_col >= 0 and end_col < COLUMNS:\n", + " return True\n", + " else:\n", + " return False" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 79 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(0, 0, 'N')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 80, + "text": [ + "False" + ] + } + ], + "prompt_number": 80 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(3, 3, 'N')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 81, + "text": [ + "True" + ] + } + ], + "prompt_number": 81 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(3, 3, 'NW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 82, + "text": [ + "True" + ] + } + ], + "prompt_number": 82 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(3, 3, 'W')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 83, + "text": [ + "True" + ] + } + ], + "prompt_number": 83 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(3, 3, 'SW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 84, + "text": [ + "True" + ] + } + ], + "prompt_number": 84 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(17, 17, 'N')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 85, + "text": [ + "True" + ] + } + ], + "prompt_number": 85 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(17, 17, 'NW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 86, + "text": [ + "True" + ] + } + ], + "prompt_number": 86 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(17, 17, 'W')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 87, + "text": [ + "True" + ] + } + ], + "prompt_number": 87 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "valid_direction(17, 17, 'SW')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 89, + "text": [ + "False" + ] + } + ], + "prompt_number": 89 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Now to solve the problem" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "best_product = 0\n", + "for row in range(ROWS):\n", + " for column in range(COLUMNS):\n", + " for direction in DIRECTIONS:\n", + " if valid_direction(row, column, direction):\n", + " this_product = product(numbers(row, column, direction))\n", + " if this_product > best_product:\n", + " best_product = this_product\n", + "best_product" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 94, + "text": [ + "70600674" + ] + } + ], + "prompt_number": 94 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "max(product(numbers(r, c, d)) \n", + " for r in range(ROWS) \n", + " for c in range(COLUMNS) \n", + " for d in DIRECTIONS \n", + " if valid_direction(r, c, d))" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 101, + "text": [ + "70600674" + ] + } + ], + "prompt_number": 101 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# All the code in one place" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "ROWS = COLUMNS = 20\n", + "SECTION_LEN = 4\n", + "\n", + "GRID_STRING = \"\"\"08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08\n", + "49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00\n", + "81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65\n", + "52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91\n", + "22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80\n", + "24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50\n", + "32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70\n", + "67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21\n", + "24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72\n", + "21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95\n", + "78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92\n", + "16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57\n", + "86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58\n", + "19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40\n", + "04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66\n", + "88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69\n", + "04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36\n", + "20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16\n", + "20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54\n", + "01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48\"\"\"\n", + "\n", + "GRID_LIST = [int(n) for n in GRID_STRING.split()]\n", + "GRID = [GRID_LIST[i:i+COLUMNS] for i in range(0, ROWS * COLUMNS, COLUMNS)]\n", + "\n", + "# Directions, as the pair (difference-in-row, difference-in-column)\n", + "DIRECTIONS = {'N': (-1, 0), 'NW': (-1, -1), 'W': (0, -1), 'SW': (1, -1)}\n", + "\n", + "def numbers(row, column, direction):\n", + " nums = []\n", + " dr, dc = DIRECTIONS[direction]\n", + " for _ in range(SECTION_LEN):\n", + " nums.append(GRID[row][column])\n", + " row += dr\n", + " column += dc\n", + " return nums\n", + "\n", + "def product(ns):\n", + " p = 1\n", + " for n in ns:\n", + " p *= n\n", + " return p\n", + "\n", + "def valid_direction(row, column, direction):\n", + " dr, dc = DIRECTIONS[direction]\n", + " end_row = row + dr * (SECTION_LEN -1)\n", + " end_col = column + dc * (SECTION_LEN -1)\n", + " if end_row >= 0 and end_row < ROWS and end_col >= 0 and end_col < COLUMNS:\n", + " return True\n", + " else:\n", + " return False\n", + "\n", + "max(product(numbers(r, c, d)) \n", + " for r in range(ROWS) \n", + " for c in range(COLUMNS) \n", + " for d in DIRECTIONS \n", + " if valid_direction(r, c, d)) " + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 102, + "text": [ + "70600674" + ] + } + ], + "prompt_number": 102 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [], + "language": "python", + "metadata": {}, + "outputs": [] + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/euler-14.ipynb b/euler-14.ipynb new file mode 100644 index 0000000..0128d75 --- /dev/null +++ b/euler-14.ipynb @@ -0,0 +1,1093 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:910fd3bd845f28d453d531005afb4e6d3ce4f040aa35835ad49dbb76f9e611fa" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Euler 14\n", + "\n", + "Which starting number, under one million, produces the longest Collatz chain?" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# We'll need this in a bit\n", + "import sys\n", + "sys.setrecursionlimit(1000000)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 79 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This is the obvious implementation of the problem: the length of the chain is one more than the length of the chain from the next number. \n", + "\n", + "For instance, given the chain:\n", + "\n", + " 13 \u2192 40 \u2192 20 \u2192 10 \u2192 5 \u2192 16 \u2192 8 \u2192 4 \u2192 2 \u2192 1\n", + "\n", + "we know that the length of chain starting at 13 is one more than the length of chain starting at 40, which is one more than the length of the chain starting at 20, ...\n", + "\n", + "Notice that we don't care about the values found in the chain, simply how long it is." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def collatz_length(n):\n", + " if n == 1:\n", + " return 1\n", + " elif n % 2 == 0:\n", + " return 1 + collatz_length(n // 2)\n", + " else:\n", + " return 1 + collatz_length(3 * n + 1)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 80 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Test it works" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_length(1)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 81, + "text": [ + "1" + ] + } + ], + "prompt_number": 81 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_length(2)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 82, + "text": [ + "2" + ] + } + ], + "prompt_number": 82 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_length(3)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 83, + "text": [ + "8" + ] + } + ], + "prompt_number": 83 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_length(13)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 84, + "text": [ + "10" + ] + } + ], + "prompt_number": 84 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "for i in range(1, 10):\n", + " print(i, collatz_length(i))" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "1 1\n", + "2 2\n", + "3 8\n", + "4 3\n", + "5 6\n", + "6 9\n", + "7 17\n", + "8 4\n", + "9 20\n" + ] + } + ], + "prompt_number": 85 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Performance\n", + "Let's find the longest chain starting from a number <= 10,000. We'll time how long it takes." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%%timeit\n", + "longest_start = 1\n", + "longest_chain = 0\n", + "for i in range(1, 10001):\n", + " this_chain = collatz_length(i)\n", + " if this_chain > longest_chain:\n", + " longest_start = i\n", + " longest_chain = this_chain\n", + "print(longest_start, '->', longest_chain)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "6171 -> 262\n", + "6171" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 262\n", + "6171" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 262\n", + "6171" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 262\n", + "1 loops, best of 3: 188 ms per loop\n" + ] + } + ], + "prompt_number": 86 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now try up to 1,000,000." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%%timeit\n", + "longest_start = 1\n", + "longest_chain = 0\n", + "for i in range(1, 1000001):\n", + " this_chain = collatz_length(i)\n", + " if this_chain > longest_chain:\n", + " longest_start = i\n", + " longest_chain = this_chain\n", + "print(longest_start, '->', longest_chain)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "837799 -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "1 loops, best of 3: 28.4 s per loop\n" + ] + } + ], + "prompt_number": 87 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Better performance\n", + "This is slow. Can we do better?\n", + "\n", + "Recall the sequence starting at 13:\n", + "\n", + "13 \u2192 40 \u2192 20 \u2192 10 \u2192 5 \u2192 16 \u2192 8 \u2192 4 \u2192 2 \u2192 1\n", + "\n", + "If we're finding the Collatz chain lengths for all numbers up to 13, we've just calculated the Collatz chain length of 10. We don't need to recalculate it. \n", + "\n", + "Instead, store the results in a cache and look them up if we have them." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_cache = {}\n", + "\n", + "def collatz_length_cache(n):\n", + " if n not in collatz_cache:\n", + " if n == 1:\n", + " collatz_cache[n] = 1\n", + " elif n % 2 == 0:\n", + " collatz_cache[n] = 1 + collatz_length_cache(n // 2)\n", + " else:\n", + " collatz_cache[n] = 1 + collatz_length_cache(3 * n + 1)\n", + " return collatz_cache[n]" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 88 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_length_cache(9)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 89, + "text": [ + "20" + ] + } + ], + "prompt_number": 89 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "collatz_cache" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 90, + "text": [ + "{1: 1,\n", + " 2: 2,\n", + " 34: 14,\n", + " 4: 3,\n", + " 5: 6,\n", + " 17: 13,\n", + " 40: 9,\n", + " 9: 20,\n", + " 10: 7,\n", + " 11: 15,\n", + " 13: 10,\n", + " 14: 18,\n", + " 16: 5,\n", + " 8: 4,\n", + " 20: 8,\n", + " 22: 16,\n", + " 7: 17,\n", + " 52: 12,\n", + " 26: 11,\n", + " 28: 19}" + ] + } + ], + "prompt_number": 90 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%%timeit\n", + "longest_start = 1\n", + "longest_chain = 0\n", + "collatz_cache = {}\n", + "for i in range(1, 10001):\n", + " this_chain = collatz_length_cache(i)\n", + " if this_chain > longest_chain:\n", + " longest_start = i\n", + " longest_chain = this_chain\n", + "print(longest_start, '->', longest_chain)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 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262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "6171 -> 262\n", + "100 loops, best of 3: 2.03 ms per loop\n" + ] + } + ], + "prompt_number": 93 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%%timeit\n", + "longest_start = 1\n", + "longest_chain = 0\n", + "collatz_cache = {}\n", + "for i in range(1, 1000001):\n", + " this_chain = collatz_length_cache(i)\n", + " if this_chain > longest_chain:\n", + " longest_start = i\n", + " longest_chain = this_chain\n", + "print(longest_start, '->', longest_chain)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "837799 -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "1 loops, best of 3: 248 ms per loop\n" + ] + } + ], + "prompt_number": 94 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#It's so useful, it's built in\n", + "Python 3.3 includes the `lru_cache` function decorator in the standard `functools` library. We can use that without faffing around with explicit caches." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "# Python 3.3 or higher:\n", + "from functools import lru_cache\n", + "\n", + "@lru_cache(maxsize=None)\n", + "def collatz_length_lru(n):\n", + " if n == 1:\n", + " return 1\n", + " elif n % 2 == 0:\n", + " return 1 + collatz_length_lru(n // 2)\n", + " else:\n", + " return 1 + collatz_length_lru(3 * n + 1)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 95 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%%timeit\n", + "longest_start = 1\n", + "longest_chain = 0\n", + "for i in range(1, 1000001):\n", + " this_chain = collatz_length_lru(i)\n", + " if this_chain > longest_chain:\n", + " longest_start = i\n", + " longest_chain = this_chain\n", + "print(longest_start, '->', longest_chain)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "stream": "stdout", + "text": [ + "837799 -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "837799" + ] + }, + { + "output_type": "stream", + "stream": "stdout", + "text": [ + " -> 525\n", + "1 loops, best of 3: 851 ms per loop\n" + ] + } + ], + "prompt_number": 96 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 96 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file