20150224_RunnersParadox.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from pylab import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1067a35c0>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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       "nRfG/j3wOeA0Se8EbgcOWuXzeHGWmVl7lG7vmJnZALnom5m1iIu+mVmLuOibmbWIi76ZWYu46JuZ\n",
       "tYiLvplZi7jom5m1yP8HrLqlLrC+4bQAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10659a7b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = linspace(0, 10)\n",
    "plot(x, sin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's create an array of 100 runners. Each runner is characterized by his speed, which we draw in the interval from 1 to 20 (km per hour)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "runners = random_integers(1, 20, size=(100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "runners.sort()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this distribution of speeds:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<Container object of 100 artists>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1071f2080>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 255,
       "width": 373
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "bar(range(runners.size), runners)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each runner propagates at his running speed. So while everyone starts at the same position, soon the faster runners will overtake the slower runners. This can be illustrated by the following figure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def propagate(runners, t):\n",
    "    distance = runners * t\n",
    "    return distance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "6kzvAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1080959e8>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 281,
       "width": 424
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "distance = zeros_like(runners, dtype=float32)\n",
    "total_time = 0\n",
    "for ind, t in enumerate([0.1, 0.1, 0.2, 0.2]):\n",
    "    distance += propagate(runners, t)\n",
    "    total_time += t\n",
    "    subplot(2, 2, ind + 1)\n",
    "    title('time=%.2f' % total_time)\n",
    "    bar(range(runners.size), distance)\n",
    "    ylim(0, 12)\n",
    "tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What does this look like if we let runners start from random points?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "pqy0IH2adwAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a9f64a8>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 281,
       "width": 424
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "distance = (rand(runners.size) - 0.5) * 6\n",
    "total_time = 0\n",
    "for ind, t in enumerate([0.1, 0.1, 0.2, 0.2]):\n",
    "    distance += propagate(runners, t)\n",
    "    total_time += t\n",
    "    subplot(2, 2, ind + 1)\n",
    "    title('time=%.2f' % total_time)\n",
    "    bar(range(runners.size), distance)\n",
    "    ylim(-3, 14)\n",
    "tight_layout()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}