ipynb/Gesture Typing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gesture Typing \n",
    "===\n",
    "\n",
    "Typing quickly and accurately on a smartphone screen is hard! One invention to make it easier is **gesture typing**, in which your finger can trace a **path** consisting of letter-to-letter **segments**. When you lift your finger the path (and the word) is complete. Below we see the path for the word \"hello.\" Note that the path is imprecise; it didn't quite hit the \"L\", but the word was recognized anyways, because \"Hello\" is a known word, whereas \"Hekko\", \"Hwerklo\", etc., are not.\n",
    "\n",
    "<img src=\"http://norvig.com/gesture.png\">\n",
    "\n",
    "Questions About Gesture Typing\n",
    "===\n",
    "\n",
    "My colleague Nicolas Schank examined (and answered) the question of what word has the longest path length. I mentioned this to [Shumin Zhai](http://www.shuminzhai.com/), the pioneer of gesture typing, and between the three of us we expanded the list of questions:\n",
    "\n",
    " 1. What words have the longest path length?\n",
    " 2. What words have the highest ratio of path length to word length? \n",
    " 3. What is the average segment length, over a typical typing work load?\n",
    " 3. Is there a better keyboard layout to minimize the average segment length over a work load?\n",
    " 4. How often are two words confused because they have similar paths?\n",
    " 5. Is there a better keyboard layout to minimize confusion? \n",
    " 6. Is there a better keyboard layout to maximize overall user satisfaction?\n",
    "\n",
    "Let's look at each of these questions, but first, let's get a rough idea for of the concepts we will need to model.\n",
    "\n",
    "Vocabulary\n",
    "===\n",
    "\n",
    "We will need to talk about the following concepts:\n",
    "\n",
    "* **Keyboard**: We'll need to know the **location** of each letter on the keyboard (we consider only letters, not the other symbols).\n",
    "* **Location**: A location is a **point** in two-dimensional space (we assume keyboards are flat).\n",
    "* **Path**: A path connects the letters in a word. In the picture above the path is curved, but a shortest path is formed by connecting straight line **segments**, so maybe we need only deal with straight lines.\n",
    "* **Segment**: A line segment is a straight line between two points.\n",
    "* **Length**: Paths and Segments have lengths; the distance traveled along them.\n",
    "* **Words**: We will need a list of allowable words (in order to find the one with the longest path).\n",
    "* **Work Load**: If we want to find the average path length over a typical work load, we'll have to represent a work load: not\n",
    "just a list of words, but a measure of how frequent each word (or each segment) is.\n",
    "* **Confusion**: We need some measure of *whether* (or perhaps *to what degree*) the paths for two words can be confused with each other.\n",
    "* **Satisfaction**: This is a very difficult concept to define.  A user will be more satisfied with a keyboard if it allows for fast, accurate typing, and if it gives the user a feeling of mastery, not frustration.\n",
    "\n",
    "**Note**: Before we get started writing any real code, I've taken all the `import`s I will need throughout this notebook and gathered them together here:\n",
    "\n",
    "\n",
    "             "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "from statistics import mean\n",
    "import matplotlib.pyplot as plt\n",
    "import urllib\n",
    "import itertools\n",
    "import random \n",
    "import re"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Representing Keyboards and Points\n",
    "===\n",
    "\n",
    "The representation for a keyboard needs to describe the location of each of the letters. Using the principle of *\"Do the simplest thing that could possibly work,\"* I represent a keyboard as a `dict` of `{letter: point}` pairs, where there will be 26 letters, A-Z,\n",
    "and each point will mark the x-y coordinates of the center of the corresponding key.  In a standard keyboard the letters are not all on a strict rectangular grid; the **A** key is half way between the **Q** and **W** in the horizontal direction. I would like to have a programmer-friendly way of defining keyboard layouts.  For example, a programmer should be able to write:\n",
    "\n",
    "    Keyboard(('Q W E R T Y U I O P',\n",
    "              ' A S D F G H J K L ',\n",
    "              '   Z X C V B N M   '))\n",
    "              \n",
    "and this would be equivalent to the `dict`:\n",
    "\n",
    "    {'Q': Point(0, 0),   'W': Point(1, 0), ...\n",
    "     'A': Point(0.5, 1), 'S': Point(1.5, 1), ...\n",
    "     'Z': Point(1.5, 2), 'X': Point(2.5, 2), ...}\n",
    "              \n",
    "Note that one key width is two characters in the input to `Keyboard`.  Here is the implementation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Keyboard(rows):\n",
    "    \"A keyboard is a {letter: location} map, e.g. {'Q':Point(0, 0), 'A': Point(0.5, 1)}.\"\n",
    "    return {ch: Point(x/2, y) \n",
    "            for (y, row) in enumerate(rows)\n",
    "            for (x, ch) in enumerate(row)\n",
    "            if ch != ' '}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What about `Point`? At first glance, Python does not appear to have a two-dimensional point as a builtin data type, but\n",
    "on second thought, it does: `complex`. A complex number is a point in the two-dimensional complex plane;\n",
    "we can use that to model the two-dimensional (x, y) plane. Because complex numbers are built in, manipulating them will be efficient. A bonus is that the distance between points `A` and `B` is simply  `abs(A-B)`; easier than the usual formula involving squares and a square root. Thus, the simplest possible thing I could do to represent points is\n",
    "\n",
    "<pre>\n",
    "Point = complex\n",
    "</pre>\n",
    "\n",
    "That would work fine. However, I would like to refer to the x coordinate of point `p` as `p.x` rather than `p.real`, and I would like points to display nicely, so I will do the second-simplest thing: make `Point` be a subclass of `complex` with `x` and `y` properties and a  `__repr__` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Point(complex):\n",
    "    \"A point in the (x, y) plane.\"\n",
    "    def __repr__(self): return 'Point({}, {})'.format(self.x, self.y)\n",
    "    x = property(lambda p: p.real)\n",
    "    y = property(lambda p: p.imag)\n",
    "    \n",
    "def distance(A, B):\n",
    "    \"The distance between two points.\"\n",
    "    return abs(A - B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternative representations for points include an `(x, y)` tuple or namedtuple, or a NumPy two-element array, or a class. \n",
    "\n",
    "Alternatives for `Keyboard` include a subclass of `dict`, or a class that contains a `dict`.  \n",
    "\n",
    "\n",
    "Now we can check that `Keyboard` works:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'A': Point(0.5, 1.0),\n",
       " 'B': Point(5.5, 2.0),\n",
       " 'C': Point(3.5, 2.0),\n",
       " 'D': Point(2.5, 1.0),\n",
       " 'E': Point(2.0, 0.0),\n",
       " 'F': Point(3.5, 1.0),\n",
       " 'G': Point(4.5, 1.0),\n",
       " 'H': Point(5.5, 1.0),\n",
       " 'I': Point(7.0, 0.0),\n",
       " 'J': Point(6.5, 1.0),\n",
       " 'K': Point(7.5, 1.0),\n",
       " 'L': Point(8.5, 1.0),\n",
       " 'M': Point(7.5, 2.0),\n",
       " 'N': Point(6.5, 2.0),\n",
       " 'O': Point(8.0, 0.0),\n",
       " 'P': Point(9.0, 0.0),\n",
       " 'Q': Point(0.0, 0.0),\n",
       " 'R': Point(3.0, 0.0),\n",
       " 'S': Point(1.5, 1.0),\n",
       " 'T': Point(4.0, 0.0),\n",
       " 'U': Point(6.0, 0.0),\n",
       " 'V': Point(4.5, 2.0),\n",
       " 'W': Point(1.0, 0.0),\n",
       " 'X': Point(2.5, 2.0),\n",
       " 'Y': Point(5.0, 0.0),\n",
       " 'Z': Point(1.5, 2.0)}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qwerty = Keyboard(('Q W E R T Y U I O P',\n",
    "                   ' A S D F G H J K L ',\n",
    "                   '   Z X C V B N M   '))\n",
    "qwerty"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing Path Length\n",
    "===\n",
    "\n",
    "Now let's figure out the path length for a word: the sum of the lengths of segments between letters.  So the path length for `'WORD'` would be the sum of the segment lengths for `'WO'` plus `'OR'` plus `'RD'`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.118033988749895"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W, O, R, D = qwerty['W'], qwerty['O'], qwerty['R'], qwerty['D'], \n",
    "distance(W, O) + distance(O, R) + distance(R, D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's make a function to compute this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def path_length(word, kbd=qwerty):\n",
    "    \"The total path length for a word on this keyboard: the sum of the segment lengths.\"\n",
    "    return sum(distance(kbd[word[i]], kbd[word[i+1]])\n",
    "               for i in range(len(word)-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "13.118033988749895"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path_length('WORD')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check with a simpler example that we know the answer to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path_length('TO')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That makes it clearer&mdash;the **O** is four keys to the right of the **T**, on the same row, so the distance between them is 4.\n",
    "\n",
    "Here's another one that you can verify on your own:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path_length('TYPEWRITER') == 1 + 4 + 7 + 1 + 2 + 4 + 3 + 2 + 1 == 25"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 1: Longest Path Length?\n",
    "===\n",
    "\n",
    "To know what the longest word is, we'll have to know what the allowable words are. The so-called TWL06 word list gives all the words that are legal in the game of Scrabble; that seems like a reasonable list (although it omits proper nouns). Here's how to load a copy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "! [ -e TWL06.txt ] || curl -O http://norvig.com/ngrams/TWL06.txt\n",
    "    \n",
    "WORDS = set(open('TWL06.txt').read().split())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "178691"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(WORDS)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a lot of words; which one has the longest path?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'PALEOMAGNETISMS'"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max(WORDS, key=path_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the ten longest paths? Including the lengths? We'll use a helper function, `print_top`, which prints the top *n* items in a seqence according to some key function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "72.2 PALEOMAGNETISMS\n",
      "70.0 ANTIQUARIANISMS\n",
      "69.9 ELECTROANALYSIS\n",
      "69.9 ANTIAPHRODISIAC\n",
      "69.3 PARLIAMENTARIAN\n",
      "68.9 BLEPHAROSPASMS\n",
      "68.6 BIDIALECTALISMS\n",
      "67.6 PALEOGEOGRAPHIC\n",
      "67.6 SPERMATOZOANS\n",
      "67.1 APOCALYPTICISMS\n"
     ]
    }
   ],
   "source": [
    "def print_top(n, sequence, key=None, formatter='{:.1f} {}'.format):\n",
    "    \"Find the top n elements of sequence as ranked by key function, and print them.\"\n",
    "    for x in sorted(sequence, key=key, reverse=True)[:n]:\n",
    "        print(formatter(key(x), x))\n",
    "              \n",
    "print_top(10, WORDS, path_length)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 2: Highest Path Length to Word Length Ratio?\n",
    "===\n",
    "\n",
    "Very long words tend to have long path lengths.  But what words have the highest *ratio*\n",
    "of path length to word length? (I decided to measure word length by number of letters; an alternative is number of segments.) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def path_length_ratio(word, kbd=qwerty): return path_length(word, kbd) / len(word)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6.9 PALAPA\n",
      "6.7 PAPAL\n",
      "6.4 PAPA\n",
      "6.4 JALAPS\n",
      "6.2 SLAPS\n",
      "6.2 KAMALA\n",
      "6.2 LAPELS\n",
      "6.2 PAPS\n",
      "6.2 HALALA\n",
      "6.1 SPALE\n"
     ]
    }
   ],
   "source": [
    "print_top(10, WORDS, path_length_ratio)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 3: Average Segment Length on Work Load?\n",
    "===\n",
    "\n",
    "What is the average segment length for a typical typing work load? To answer that, we need to know what a typical work load is. We will read a file of \"typical\" text, and count how many times each segment is used. A `Workload` is a `dict` of the form `{segment: proportion, ...},` e.g. `{'AB': 0.02}`, where each key is a two-letter string (or *bigram*) representing a segment, and each value is the proportion of time that segment appears in the workload.  Since the distance from `A` to `B` on a keyboard is the same as the distance from `B` to `A`, we can combine them together into one count;\n",
    "I'll arbitrarily choose count them both under the alphabetically first one. I make a `Counter` of all two-letter segments, and `normalize` it so that the counts sum to 1 (and are thus probabilities)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def Workload(text):\n",
    "    \"\"\"Create a Workload--a dict of the form {'AB': 1000, ...} \n",
    "    saying how often each letter pair occurs in text.\"\"\"\n",
    "    segments = (min(AB, AB[::-1]) for AB in bigrams(text))\n",
    "    return normalize(Counter(segments))\n",
    "\n",
    "def bigrams(text): return re.findall(r'(?=([A-Z][A-Z]))', text)\n",
    "\n",
    "def normalize(dictionary):\n",
    "    \"Normalize a {key: val} dict so that the sum of the vals is 1.0.\"\n",
    "    total = sum(dictionary.values())\n",
    "    for k in dictionary:\n",
    "        dictionary[k] /= total\n",
    "    return dictionary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Note:* Some `re` trickiness here: `([A-Z][A-Z])` means a group of two consecutive letters. But if I only looked for that, then in the text `'FOUR'` I would find `['FO', 'UR']`. So I use the `?=` operator, which says to check for a match, but don't consume the matched characters. So I can find `['FO', 'OU', 'UR']`, which is what I want.\n",
    "\n",
    "Let's see what a workload looks like for a tiny text:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Counter({'AL': 0.05,\n",
       "         'AO': 0.05,\n",
       "         'DO': 0.05,\n",
       "         'GO': 0.1,\n",
       "         'HO': 0.05,\n",
       "         'HS': 0.05,\n",
       "         'IS': 0.05,\n",
       "         'OO': 0.55,\n",
       "         'OT': 0.05})"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Workload('SHOT IS GOOD -- GOOOOOOOOOOOAL!')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I happened to have a file of about a megabyte of random text, `smaller.txt`; that should work fine as a typical work load:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "! [ -e smaller.txt ] || curl -O http://norvig.com/ngrams/smaller.txt\n",
    "    \n",
    "WORKLOAD = Workload(open('smaller.txt').read().upper())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's peek at the most common segments:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('HT', 0.04144819308354474),\n",
       " ('ER', 0.04050225926898767),\n",
       " ('EH', 0.03576926529702987),\n",
       " ('IN', 0.02699818128015268),\n",
       " ('AN', 0.02320447132440709),\n",
       " ('NO', 0.022344984888333263),\n",
       " ('EN', 0.021994208025641622),\n",
       " ('IT', 0.021467211506811055),\n",
       " ('ES', 0.020667573255841017),\n",
       " ('DE', 0.020619362217840744)]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WORKLOAD.most_common(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The most popular segments, at about 4% each are `HT/TH` and `ER/RE`. Now we can compute the workload average:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def workload_average(kbd, workload=WORKLOAD):\n",
    "    \"The average segment length over a workload of segments.\"\n",
    "    return sum(distance(kbd[A], kbd[B]) * workload[A+B]\n",
    "               for (A, B) in workload)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.2333097802127653"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "workload_average(qwerty)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, on average, your finger has to travel a little over 3 keys from one letter to the next over a typical workload."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aside: Visualizing a Keyboard\n",
    "---\n",
    "\n",
    "We'll need a way of visualizing what a keyboard looks like. I could just `print` letters, but I think it is more compelling to use IPython's `matplotlib` module. In the function `show_kbd` we'll draw a square around the center point of each key, and annotate the square with the key letter. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def show_kbd(kbd, name='keyboard'):\n",
    "    \"Plot the keyboard and show title/stats.\"\n",
    "    for L in kbd:\n",
    "        plot_square(kbd[L].x, -kbd[L].y, label=L)\n",
    "    plt.axis('equal'); plt.axis('off')\n",
    "    plt.title(title(kbd, name));\n",
    "    plt.show()\n",
    "\n",
    "def plot_square(x, y, label='', style='k:'):\n",
    "    \"Plot a square with center (x, y) and optional label.\"\n",
    "    H = 1/2\n",
    "    plt.plot([x-H, x+H, x+H, x-H, x-H], \n",
    "             [y-H, y-H, y+H, y+H, y-H], style)  \n",
    "    plt.annotate(label, (x-H/4, y-H/4)) # H/4 seems to place label well.\n",
    "    \n",
    "def title(kbd, name): \n",
    "    X, Y = span(kbd[L].x for L in kbd), span(kbd[L].y for L in kbd)\n",
    "    return ('{}: size: {:.1f}×{:.1f}; path length: {:.1f}'\n",
    "            .format(name, X, Y, workload_average(kbd)))\n",
    "            \n",
    "def span(numbers):\n",
    "    numbers = list(numbers)\n",
    "    return max(numbers) - min(numbers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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au7cI+GNgaRSWAdtQo9hr5rU9YxdA+rbBAuBkSTP5OdqZyXkRsQL4laSXwRPfBHoF8N1R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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b382588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(qwerty, 'qwerty')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 4: Keyboard with Minimal Workload Average?\n",
    "===\n",
    "\n",
    "Now for a much harder question: can we find a different keyboard layout that has a smaller average segment length over the workload? First, let's note that there are two ways to modify a keyboard:  \n",
    "\n",
    "* Keep the keys in the same locations but swap letters. (This is an operation you can do on a physical keyboard just by prying off the key caps and swapping them.) \n",
    "* Change the locations of keys.  (On a physical keyboard you'd need a saw and glue to do this, but it is easier on a virtual keyboard.)  \n",
    "\n",
    "Let's start by limiting ourselves to just swapping letters.  \n",
    "\n",
    "This is an **optimization** problem.  There are many permutations of letters; too many to try them all. To be precise, there are 26! (26 factorial) permutations, which is about 10<sup>26</sup> (fun fact: 25 and 26 are the only integers for which n! &approx; 10<sup>n</sup>).  If we can't try them all, we need some way to sample the configurations, trying to make progress towards a better one. Again, we'll try the simplest thing that could possibly work: \n",
    "\n",
    "  1. Pick two keys at random.\n",
    "  2. Swap them.\n",
    "  3. If that gives a better (lower) workload total, keep them that way.\n",
    "  4. If not, swap back.\n",
    "  5. Repeat this for a given number of times, say 1000."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def improved(kbd, swaps=1000, scorer=workload_average):\n",
    "    \"Minimize scorer(kbd) by swapping keys and keeping improvements.\"\n",
    "    kbd = kbd.copy()\n",
    "    score = scorer(kbd)\n",
    "    letters = list(kbd)\n",
    "    for _ in range(swaps):\n",
    "        A, B = random.sample(letters, 2)   # Step 1: pick two keys\n",
    "        swap(kbd, A, B)                    # Step 2: swap them\n",
    "        score2 = scorer(kbd)\n",
    "        if score2 < score:                 # Step 3: If better, keep them\n",
    "            score = score2                 # (and record the new best total)\n",
    "        else:\n",
    "            swap(kbd, B, A)                # Step 4: swap back if not better\n",
    "    return kbd\n",
    "\n",
    "def swap(kbd, A, B): kbd[A], kbd[B] = kbd[B], kbd[A]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Note 1:** This strategy is called **hillclimbing**, drawing on the metaphor of getting to a high peak by trying to take a step, and continuing if the step is uphill, and returning if it is not.  This technique often finds a local maximum&mdash;a solution that is better than all its neighbors, but not as good as another solution that is many steps away.\n",
    "\n",
    "**Note 2:** I make `scorer` be a parameter, in case we later decide we want to minimize something else other than `workload_average`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how well we can do:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Pjc6vBXYDXkO67F5aYTsLa+xbAOcDBxej9lcVmaMiAEmrAnsDF9dcv1tz+bUlXVc8rgeO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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x102cfc710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(improved(qwerty, 3000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's a pretty good improvement! We decreased the workload average  by about a third. (If you are reading this in an active IPython notebook, you can re-run the cell above and see a different result each time.) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Visualizing `improved`\n",
    "---\n",
    "\n",
    "Let's get a better feeling for what `improved` does.  We will keep track of the workload average after each swap, and plot that as a curve. We will repeat that 10 times (because each run has random variation).\n",
    "\n",
    "I'll add another parameter, `scores`, to `improved`, If it is not `None`, then it should be a list into which we can accumulate all the scores (after each time step). Then I'll add a new function, `plot_improvements`, that plots the scores that `improved` accumulates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def improved(kbd, swaps=1000, scorer=workload_average, scores=[]):\n",
    "    \"Minimize scorer(kbd) by swapping keys and keeping improvements.\"\n",
    "    kbd = kbd.copy()\n",
    "    score = scorer(kbd)\n",
    "    letters = list(kbd)\n",
    "    for _ in range(swaps):\n",
    "        A, B = random.sample(letters, 2)   # Step 1: pick two keys\n",
    "        swap(kbd, A, B)                    # Step 2: swap them\n",
    "        score2 = scorer(kbd)\n",
    "        if score2 < score:                 # Step 3: If better, keep them\n",
    "            score = score2                 # (and record the new best total)\n",
    "        else:\n",
    "            swap(kbd, B, A)                # Step 4: swap back if not better\n",
    "        scores.append(score)               # <<< NEW\n",
    "    return kbd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mZaZHzbExvfl6L3wFBTxQWEiwdCv/9aMI/j7LWJdzFCU969h4XQ5zb1mG+NLp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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c425898>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def plot_improvements(kbd, swaps, scorer=workload_average, repeats=10):\n",
    "    plt.ylabel('Workload average segment length') \n",
    "    plt.xlabel('Number of swaps');  \n",
    "    for _ in range(repeats):\n",
    "        scores = []\n",
    "        improved(kbd.copy(), swaps, scorer, scores)\n",
    "        plt.plot(scores)\n",
    "    plt.show()\n",
    "    \n",
    "plot_improvements(qwerty, 2500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This plot is interesting. I note the following:\n",
    "\n",
    "* Each curve follows a slightly different path (because the swaps are random).\n",
    "* The curves are grouped very tightly together; the variance is small.  Almost everywhere, the difference between the best and the worst is about 0.2 or less. By the end, almost all the curves are between 1.9 and 2.0.\n",
    "* We make rapid progress, decreasing from 3.2 to around 2.2 in about 200 swaps, and to around 2.0 in about 500 swaps.\n",
    "* After 1000 swaps, progress is slow, and after 2000, even slower."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I can't say we've found the best possible keyboard, but it looks like progress will be slow in improving over about 1.9. \n",
    "\n",
    "Keys in Different Physical Locations\n",
    "---\n",
    "\n",
    "Now let's allow keys  to be in different physical locations.  Rather than allowing complete freedom of movement, we'll start from a few different fixed key layouts and swap keys from there. I'll define three layouts and gather them into a `dict`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "keyboards = {\n",
    "    'qwerty': Keyboard(('Q W E R T Y U I O P',\n",
    "                        ' A S D F G H J K L ',  \n",
    "                        '   Z X C V B N M   ')),\n",
    "\n",
    "    '4-by-7': Keyboard((' A B C D E F ',\n",
    "                        'G H I J K L M',\n",
    "                        ' N O P Q R S ',\n",
    "                        'T U V W X Y Z')),\n",
    "\n",
    "    '5-by-6': Keyboard((' A B C D E ',\n",
    "                        'F G H I J K',\n",
    "                        'L M N O P Q',\n",
    "                        ' R S T U V',\n",
    "                        '  W X Y Z '))\n",
    "    }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a function to report on a collection of keyboards such as this (after improving them):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Y1FzMzfLVSR3AJoeLeYvleJrunvm09FIHGJTjMWUFF3OzTLlw2nT4H221WL+/\n2S8KmSx3gJ4zrX65L3WOYcuWFxdzM7MWcJvFzKwFXMzNzFrAxdzMrAVczM3MWsDF3MysBVzMzcxa\nwMXczKwFXMzNzFrAxdzMrAVczM3MWsDF3MysBVzMzcxawMXczKwFXMzNzFrAxdzMrAVczM3MWsDF\n3MysBVzMzcxawMXczKwFXMzNzFrAxdzMrAVczM3MWsDF3MysBVzMzcxawMXczKwFXMzNzFrAxdzM\nrAVczM3MWsDF3MysBVzMzcxawMXczKwFXMzNzFrAxdzMrAVczM3MWsDF3MysBVzMzcxawMXczKwF\nXMzNzFrg/wMexNo/VCenYwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c41ac18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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FTP0JVz0HeFIaQ8CmwA61+58EnNso3Nz9C3Dw0G0Hpdsbm/T4RcQfgGOBj6Sb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\nsw5wMTcz64D/D3/o6/JC78pnAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a4b2400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e09db70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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2EZHEqLCLiCRGhV1EJDEq7CIiiVFhFxFJzP8DWgp3AD+FGSUAAAAASUVORK5C\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c4f3cf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a41ea20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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GvfH52xM4CDgYOCXBuE1Vx/X/hXWfwyPj7TIDFfP0LiEWc+oifgWw0sy2MrMHAbsBy7tu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\nMRcR6QEVcxGRHlAxFxHpARVzEZEe+B8R50F3hN4/2AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c414550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def report(keyboards=keyboards, swaps=3000, scorer=workload_average):\n",
    "    \"Iterate through a dict of {name: kbd} pairs, showing kbd before and after improved(kbd).\"\n",
    "    for (name, kbd) in keyboards.items():\n",
    "        show_kbd(kbd, name)\n",
    "        show_kbd(improved(kbd, swaps=swaps, scorer=scorer), 'improved ' + name)\n",
    "        \n",
    "report()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(**Note:** The plots of different-shaped keyboards have different-sized squares around the keys. Some of the plots have a lot of whitespace around them. If anyone knows an easy way to tell `plot` to display them better, let me know.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "So the more compact keyboards with a smaller diameter (`4-by-7` and `5-by-6`) seem to perform slightly better than `qwerty`. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 5: How often are two words confused because of similar paths?\n",
    "===\n",
    "\n",
    "When can one word be confused with another?  When their paths are similar (which means that their corresponding letters are in similar locations). For example, on a Qwerty keyboard, the paths for \"HELLO\" and \"JELLO\" are similar, because **H** and **J** are adjacent, and the other letters are the same.\n",
    "\n",
    "<img src=\"http://norvig.com/gesture.png\">\n",
    "\n",
    "We'd like to know, for a given keyboard, how confusing is it? How many words have paths on the keyboard that can be confused for other words? We have our work cut out for us:\n",
    "\n",
    "1. Determine what letters could be confused for each other.\n",
    "2. Determine what words/paths can be confused.\n",
    "3. Invent some metric for the overall confusingness of a keyboard. \n",
    "4. Try to find less-confusing keyboards.\n",
    "\n",
    "Letter Confusions\n",
    "---\n",
    "\n",
    "So, as a first step, we will make a mapping from each key to the keys that it can be confused with. I'll say that any key within a distance of 1.5 units on the keyboard is a **neighboring** key, and thus a potential confusion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def neighboring_keys(kbd, radius=1.5):\n",
    "    \"Build a dict of {Letter:NeighboringLetters}, e.g. {'Q':'AQW', ...}.\"\n",
    "    return {A: cat(sorted(B for B in kbd if distance(kbd[A], kbd[B]) <= radius))\n",
    "            for A in kbd}\n",
    "\n",
    "cat = ''.join  ## Function to join letters (or strings) into one string"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'A': 'AQSWZ',\n",
       " 'B': 'BGHJNV',\n",
       " 'C': 'CDFGVX',\n",
       " 'D': 'CDEFRSXZ',\n",
       " 'E': 'DERSW',\n",
       " 'F': 'CDFGRTVX',\n",
       " 'G': 'BCFGHTVY',\n",
       " 'H': 'BGHJNUVY',\n",
       " 'I': 'IJKOU',\n",
       " 'J': 'BHIJKMNU',\n",
       " 'K': 'IJKLMNO',\n",
       " 'L': 'KLMOP',\n",
       " 'M': 'JKLMN',\n",
       " 'N': 'BHJKMN',\n",
       " 'O': 'IKLOP',\n",
       " 'P': 'LOP',\n",
       " 'Q': 'AQW',\n",
       " 'R': 'DEFRT',\n",
       " 'S': 'ADESWXZ',\n",
       " 'T': 'FGRTY',\n",
       " 'U': 'HIJUY',\n",
       " 'V': 'BCFGHV',\n",
       " 'W': 'AEQSW',\n",
       " 'X': 'CDFSXZ',\n",
       " 'Y': 'GHTUY',\n",
       " 'Z': 'ADSXZ'}"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qwerty_neighbors = neighboring_keys(qwerty) \n",
    "qwerty_neighbors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see, for example, that **Q**, off in the corner, has only **A**, **W**, and itself as neighbors, while **G**, in the middle of the keyboard, has 8 neighbors. \n",
    "\n",
    "Word Confusions\n",
    "---\n",
    "\n",
    "Consider each of the letters in the word \"HELLO,\" and all the possible choices for neighbors of each letter:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['BGHJNUVY', 'DERSW', 'KLMOP', 'KLMOP', 'IKLOP']"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "columns = [qwerty_neighbors[L] for L in 'HELLO']\n",
    "columns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are five columns of letters, and if we pick one letter from each column, we get a path that is formed by letters that are each confusions of letters in the original word, and so the whole path is a confusion for the original word.  So \"JELLO\" is a confusion for \"HELLO\", as would be \"BDKKI\" and \"YWPPP\", except those are not words.\n",
    "\n",
    "It turns out that there is a library function, `itertools.product`, that will take alternative choices, and generate all possible ways of assembling a sequence consisting of one selection (letter) from each alternative choice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "paths = {cat(picks) for picks in itertools.product(*columns)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How many paths are there?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5000"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(paths)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at a few of them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['NELKI', 'NWLKL', 'VEMKI', 'VEOOP', 'UWPMK', 'JSLML', 'GRMMO', 'UWKKP']"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "random.sample(paths, 8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And let's see all the paths that are also words:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'BROMO', 'BROOK', 'HELLO', 'JELLO'}"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "WORDS & paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only 4 out of 5000;  That's pretty good, but it means \"HELLO\" is still a confusing word. We can package this as the function  `confusions`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def confusions(word, neighbors=neighboring_keys(qwerty)):\n",
    "    \"All valid words whose paths could be confused with the path for the given word.\"\n",
    "    columns = [neighbors[L] for L in word]\n",
    "    return {cat(picks) for picks in itertools.product(*columns)} & WORDS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'BROMO', 'BROOK', 'HELLO', 'JELLO'}"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "confusions('HELLO')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'ALTOS', 'EIDOS', 'SIDLE', 'SKEPS', 'WIELD', 'WORKS', 'WORLD', 'WORMS'}"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "confusions('WORLD')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'FEARING',\n",
       " 'FRAYING',\n",
       " 'GEARING',\n",
       " 'GRATING',\n",
       " 'GRAYING',\n",
       " 'GREYING',\n",
       " 'REARING',\n",
       " 'REEFING',\n",
       " 'RESTING',\n",
       " 'TEARING',\n",
       " 'TESTING'}"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "confusions('TESTING')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far, so good.  But I'm worried about the efficiency of `confusions`.\n",
    "\n",
    "My first concern is that `WORDS` has 178,691 words, including many unusual words, so there might be too many confusions. I will read in a smaller word set, with only common words:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "! [ -e words.js ] || curl -O https://xkcd.com/simplewriter/words.js\n",
    "    \n",
    "COMMON = set(re.findall('[A-Z]+', open('words.js').read().upper()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3614, 'SPECIALS', 'UNDERSTANDINGS')"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(COMMON), max(COMMON, key=path_length), max(COMMON, key=len)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "More Efficient `confusions`\n",
    "---\n",
    "\n",
    "Another issue is that `confusions` is really slow:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 3.12 s, sys: 395 ms, total: 3.51 s\n",
      "Wall time: 3.52 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'SOMETHING'}"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time confusions('SOMETHING')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It took (on my computer) 3 seconds to compute this.  Why so long? Let's count:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7, 5, 5, 5, 5, 8, 5, 6, 8]"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[len(neighboring_keys(qwerty)[L]) for L in 'SOMETHING']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are 7 &times; 5 &times; 5 &times; 5 &times; 5 &times; 8 &times; 5 &times; 6 &times; 8 = 8,400,000 paths for `confusions` to consider. `'PALEOMAGNETISMS'` would have about 10,000 times more paths. Looking at every path is slow, but for most paths, we're wasting our time.  For example, one choice for the first two neighboring letters of 'SO' is 'XP', but 'XP' does not start any word in the dictionary.  Nevertheless, `itertools.product` will generate 240,000 combinations that start with 'XP', and will then rule them out one at a time. It would be better to stop as soon as we see 'XP', and never consider continuations of this path.\n",
    "\n",
    "So that gives us the key idea for a more efficient version of `confusions`: *only follow paths that form a prefix of some word.*  I'll make a set of all the prefixes of the `COMMON` words:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def prefixes(words):\n",
    "    \"Return a set of prefixes (1 to N characters) of this collection of words.\"\n",
    "    return {word[:i] for word in words for i in range(1, len(word)+1)}\n",
    "\n",
    "PREFIXES = prefixes(COMMON)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'T', 'TH', 'THE', 'THES', 'THESE', 'THEY', 'THO', 'THOS', 'THOSE'}"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prefixes(['THESE', 'THEY', 'THOSE'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can describe the more efficient version of the `confusions` algorithm:\n",
    "\n",
    "1. Maintain a queue of partial paths, where a partial path is a string representing choices for neighboring letters for a prefix of the word. For example, 'HE' and 'JE' are both partial paths for the word 'HELLO'.  Initialize the queue to have just one partial path, the empty string.\n",
    "2. Remove a partial path from the queue. Find all possible ways to extend the path by adding a neighboring letter, but only if doing so creates a path that is a prefix of some word in the dictionary.  For example, given the word 'HELLO', and the partial path 'JE', consider all the neighbors of 'L' (namely, 'K', 'M', 'L', 'O', or 'P'), but only 'JEM', 'JEL', and 'JEO' are prefixes of words, so add just those to the queue.\n",
    "3. When a partial path reaches the same length as the word ('HELLO' in this example), then don't extend it any farther; instead check to see if the path is a word.  If it is, yield it as a result.\n",
    "\n",
    "A word is `confusable` if it has a confusion word (other than itself)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def confusions(word, neighbors=neighboring_keys(qwerty)):\n",
    "    \"All valid words whose paths could be confused with the path for this word.\"\n",
    "    Q = ['']        # A queue of partial paths \n",
    "    while Q:\n",
    "        path = Q.pop()\n",
    "        if len(path) < len(word):\n",
    "            for L in neighbors[word[len(path)]]:\n",
    "                if path + L in PREFIXES:\n",
    "                    Q.append(path + L)\n",
    "        elif path in COMMON:\n",
    "            yield path\n",
    "\n",
    "def confusable(word, neighbors=neighboring_keys(qwerty)) -> bool: \n",
    "    \"Is this word confusable with another, given this keyboard's neighboring keys?\"\n",
    "    return any(c != word for c in confusions(word, neighbors))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check the speed (remember the old version took over 3 seconds):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 119 µs, sys: 0 ns, total: 119 µs\n",
      "Wall time: 122 µs\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'SOMETHING'}"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time set(confusions('SOMETHING'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We went from about 3 seconds to about 100 microseconds: that's 30,000 times faster!  We can look at some more confusions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'A': 'S A',\n",
       " 'ACCEPT': 'ACCEPT',\n",
       " 'AIRS': 'WORE WORD WIFE WIDE SURE SITS SIRS SIDE AIRS',\n",
       " 'BEATER': 'NEARER NEARED HEATER HEATED GRAYER GRAYED BESTED BEATER',\n",
       " 'BEGAN': 'BEGAN',\n",
       " 'BLOWER': 'NOISED BLOWER',\n",
       " 'DISCOVERS': 'DISCOVERS',\n",
       " 'DRINK': 'DRUNK DRINK',\n",
       " 'DROVES': 'DROVES DRIVES',\n",
       " 'ENTER': 'ENTER',\n",
       " 'EXCEPT': 'EXCEPT',\n",
       " 'GATHER': 'GATHER FATHER BAGGER BAGGED',\n",
       " 'GUESSING': 'GUESSING',\n",
       " 'LEADINGS': 'LEADINGS',\n",
       " 'LOOKER': 'POOLED MOONED LOOKER LOOKED KILLER KILLED',\n",
       " 'MAPS': 'MAPS',\n",
       " 'NEEDED': 'NEEDED BEDDED',\n",
       " 'NIGHTS': 'NIGHTS MIGHTS',\n",
       " 'RETURNS': 'RETURNS',\n",
       " 'ROOMED': 'ROOMED ROLLED FILLED',\n",
       " 'SEXING': 'SEXING ADDING',\n",
       " 'SHORTEST': 'SHORTEST',\n",
       " 'SHOVING': 'SHOVING',\n",
       " 'SKINNED': 'SKINNED',\n",
       " 'SOUND': 'WOUND SPINS SOUND SKINS',\n",
       " 'STOPS': 'STOPS STOOD STOLE DROPS',\n",
       " 'THANKS': 'THANKS',\n",
       " 'TRIPE': 'TRIPS TRIPE',\n",
       " 'TV': 'TV',\n",
       " 'VIEWED': 'VIEWED HORSES'}"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{w: ' '.join(confusions(w))\n",
    " for w in random.sample(COMMON, 30)}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Visualizing Paths on a Keyboard\n",
    "----\n",
    "\n",
    "It would be nice to see potentially confusable word paths on a keyboard.\n",
    "I'll add a call to `plot_paths` in `show_kbd`:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def show_kbd(kbd, name='keyboard', words=()):\n",
    "    \"Plot the keyboard and show title/stats.\"\n",
    "    for L in kbd:\n",
    "        plot_square(kbd[L].x, -kbd[L].y, label=L)\n",
    "    plot_paths(kbd, words)\n",
    "    plt.axis('equal'); plt.axis('off')\n",
    "    plt.title(title(kbd, name));\n",
    "    plt.show()\n",
    "    \n",
    "def plot_paths(kbd, words):\n",
    "    \"Plot paths for each word.\"\n",
    "    for (i, word) in enumerate(words):\n",
    "        Xs = [kbd[L].x for L in word]\n",
    "        Ys = [-kbd[L].y for L in word]\n",
    "        plt.plot(Xs, Ys, '-o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how it works on three similar paths:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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276p101eNBVpPvxPMd8X0sV7cXroVuKQU9Qol9Hr50Ii9QMrw58cyHl2jezTR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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c31da58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(qwerty, words=['VALUE', 'VALOR', 'CAKE'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " OK, we're on the right track, but I see three problems, which I think I can solve: \n",
    "\n",
    "- The letters are obscured by the circles.  Solution: offset paths away from the center.\n",
    "- When the paths are the same, they overwrite each other.  Solution: offset each path towards a different corner.\n",
    "- There is no indication what direction the path is going in. Solution: put a black square on the start position.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_paths(kbd, words):\n",
    "    \"Plot paths for each word, each with a different offset (and color).\"\n",
    "    Q = 1/5  # Q originally meant a quarter of a key width; but 1/5 looks nicer.\n",
    "    offsets = [Point(-Q, -Q), Point(-Q, +Q),  Point(Q, +Q), Point(Q, -Q)]\n",
    "    for (i, word) in enumerate(words):\n",
    "        Xs = [kbd[L].x + offsets[i % 4].x for L in word]\n",
    "        Ys = [-kbd[L].y + offsets[i % 4].y for L in word]\n",
    "        plt.plot(Xs, Ys, '-o')\n",
    "        plt.plot(Xs[:1], Ys[:1], 'ks')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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MAM5w9/fLHads2vsbUp39AUwGTo/+N+BWYFyK+RsSTvcp4N5YnH8BT8Wen0RISGWJF027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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dfeb470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(qwerty, words=['VALUE', 'VALOR', 'CAKE'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That looks busier, but it is  easier to follow the paths. Another example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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J/tayFp5Lva0pf42IWO/8czCHOvXOS8ilaFuL8epp4lWtq/fWQTX2zLHetg8b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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dff4470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(qwerty, words={'HOUSED', 'HOUSES', 'NOISES'})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Average Confusions of a Keyboard\n",
    "===\n",
    "\n",
    "The question is: how confusing is a keyboard? One metric for confusingness is the percentage of words that are confused with other words:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "def confusingness(kbd):\n",
    "    \"The proportion of words in COMMON that are confused with other words on this keyboard.\"\n",
    "    neighbors = neighboring_keys(kbd)\n",
    "    return mean([confusable(w, neighbors) for w in COMMON])\n",
    "\n",
    "    \n",
    "def title(kbd, name): \n",
    "    X, Y = span(kbd[L].x for L in kbd), span(kbd[L].y for L in kbd)\n",
    "    return ('{}: size: {:.1f}×{:.1f}; path length: {:.1f}; confusingness = {:.0%}'\n",
    "            .format(name, X, Y, workload_average(kbd), confusingness(kbd)))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'4-by-7': 0.4571112340896514,\n",
       " '5-by-6': 0.5282235749861649,\n",
       " 'qwerty': 0.5398450470392916}"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{name: confusingness(kbd) for (name, kbd) in keyboards.items()}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "About half the common words are confusable, with a little bit of variation between keyboards."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 6: Is there a Keyboard that Minimizes Confusion?\n",
    "===\n",
    "\n",
    "Consider this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 134 ms, sys: 3.99 ms, total: 138 ms\n",
      "Wall time: 143 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.5398450470392916"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time confusingness(qwerty)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 188 µs, sys: 3 µs, total: 191 µs\n",
      "Wall time: 202 µs\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "3.2333097802127653"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time workload_average(qwerty)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing `confusingness` takes about 500 times longer than computing `workload_average`, so if we want to use `confusingness` as a scoring function, we will have to settle for fewer swaps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9WaMVqXC+kToTR8Rvge8AHwE+Bnw9Im7pVV6Ta4BNas5TN0+SXg58Gdi75mcr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\nAmJmZkVcQMzMrIgLiJmZFXEBMTOzIv8fhAQEAd67WmcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c378f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 10.6 s, sys: 26.7 ms, total: 10.6 s\n",
      "Wall time: 10.6 s\n"
     ]
    }
   ],
   "source": [
    "%time show_kbd(improved(qwerty, swaps=100, scorer=confusingness))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This did indeed reduce confusingness (which was 54%); not bad for only 100 swaps."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Question 7: Is there a Keyboard that Maximizes User Satisfaction?\n",
    "===\n",
    "\n",
    "What is user satisfaction? I don't know, but for now I'll approximate satisfaction (or rather, *dissatisfaction*, since lower scores are better) with a combined score that is the product of workload average and confusingness. Then I (rather arbitrarily) scale the result by 5, just because I think a number like \"2.1\" looks better than \"0.42\".\n",
    "\n",
    "First we'll define the combined scorer function and incorporate it into `title`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def dissatisfaction(kbd, scale=5):\n",
    "    \"The product of workload average and confusingness, scaled.\"\n",
    "    return scale * workload_average(kbd) * confusingness(kbd)\n",
    "\n",
    "def title(kbd, name): \n",
    "    X, Y = span(kbd[L].x for L in kbd), span(kbd[L].y for L in kbd)\n",
    "    return ('{}: size: {:.1f}×{:.1f}; path length: {:.1f}; confusingness: {:.0%}; overall: {:.1f}'\n",
    "            .format(name, X, Y, workload_average(kbd), confusingness(kbd), dissatisfaction(kbd)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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GZvdx+rb2dP6u1ePp2/oQo9WnfemcftjxpnS829rT/aqLE93eTLc/d+1aTwxy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\nZmYFnEjNzMwKOJGamZkV+F+j7QLr+TOA7AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b379518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(qwerty)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's try to minimize confusion. This should take around 2 minutes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Zk9wDNLMZnPxsEjgBjoES6xCF1mu6bccwKnxOLUyJ+8kWzgnQzAbptB2A2bC5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\ndiMRvXUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10cac4e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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jrGqhvQbK2A54L/Dy2gHR2Py5GihjeWH1YjoZeLKko0lt68ONBFXPFsCiiLgX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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e011710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a3c3a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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napRtPeBU4LCIuKtGjL7iRcRPSM3yh+b4KwIfAT4VEXf3uJtWjbKN8xiAdn1p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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c4715f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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3St5MGC4/Ari78to9CL35W4APDWjvpEp721SPnwG5/6/vWPta5fnTgbf1rXMJS9wRC7wM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\nIiIyl1QARURkLqkAiojIXFIBFBGRufT/6fQ4TkUprgcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c2eea90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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8/vclGdqcI+M438nAHsyo6GfK+TigMbOLzOy7hIe7XpGh3Vtlyjnzp3YzfMXs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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a4dd2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 2min 24s, sys: 188 ms, total: 2min 25s\n",
      "Wall time: 2min 25s\n"
     ]
    }
   ],
   "source": [
    "%time report(swaps=500, scorer=dissatisfaction)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Success!  We were in fact able to make progress on the combined metric.  If we could run 50,000 swaps instead of just 500, we could probably do even better. To do so would require either (1) more computers, (2) more patience, or (3) more efficient algorithms.  I'll leave it up to you to make more progress.\n",
    "\n",
    "**Note**: Each time this notebook is run, different random results can occur.  I'll record a keyboard found by one\n",
    "of the good runs (only 11% confusingness) here, just in case another run is not as good:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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eny8Splx/QXy6uvH+QcC+Q69tAhzXWP7jxv69Mx6HR2/INuO+vY/G09vreR78GSEJLSXU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\ntBY92c7W9wcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b7b6080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(Keyboard((' U D J K N W ',\n",
    "                   'T V H E B Q R',\n",
    "                   ' Z I M X A C ',\n",
    "                   'S P G O F Y L')))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bonus Question: What about the Keybee Keyboard?\n",
    "\n",
    "In December 2017 the [Keybee keyboard](http://keybee.it/) was announced, claiming to cut the path length in half. The keyboard uses hexagonal keys:\n",
    "\n",
    "![Keybee keyboard](http://keybee.it/img/phones/theme_002.png)\n",
    "\n",
    "Let's see if we can replicate it. I'll give `keyboard` optional arguments to scale the (x, y) positions to get the proper hex grid layout (even though I will still draw each key with a square around it rather than a hexagon; I'll leave it to you to fix that if you want):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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3yg8FLmkcryKFAu7V0KybtJPZbZp2XtB4/1TgJ6QBfgtpV7nHgDa/RwppLyM52YMbZWvl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\ncIxBEARB0CAcYxAEQRA0+F+RfqEz4BJC3gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a2a9940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def Keyboard(rows, xscale=1/2, yscale=1):\n",
    "    \"A keyboard is a {letter: location} map, e.g. {'Q':Point(0, 0), 'A': Point(0.5, 1)}.\"\n",
    "    return {ch: Point(xscale * x, yscale * y) \n",
    "            for (y, row) in enumerate(rows)\n",
    "            for (x, ch) in enumerate(row)\n",
    "            if ch != ' '}\n",
    "\n",
    "def HexKeyboard(rows): return Keyboard(rows, xscale=(3 ** 0.5)/2, yscale=1/2)\n",
    "\n",
    "keybee = HexKeyboard((\n",
    "    ' Q W C ',\n",
    "    'J U I K',\n",
    "    ' F H N ',\n",
    "    'Z O T G',\n",
    "    ' R . Y ',\n",
    "    'B E S V',\n",
    "    ' P A M ',\n",
    "    \"X L D '\"))\n",
    "\n",
    "show_kbd(keybee, 'Keybee')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's not pretty, but the keys are in the right places. The path length of 1.9 is better than the 3.2 for the QWERTY layout, but not twice as good, and not as good as some of the layouts we were able to achieve with square keys, and the advantage may be due to the fact that the keys overlap by 13%, not because the layout is especially clever. Can we improve the layout?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ghFYfvXMI801hOv2TGjtiz6v6+c4Qt/t5YHlE7BQRjwQOAbYZohxTGW+lJkDS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\n4FHA0ZK2XGBJZkC6NMkYJ5J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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c2d54a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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fSjPbKufiR5AKsj+SLnaDcAqwr6Vf+qwN/D/SXeqy3OcOOQ/OBT4O3Ozu3x33wVO+ex7p\njhzSCM9uZrYJqRC4Ob/uEjN7bx+Puq4pvZjDxOf7lDCznfN3YZZ+pXEicEknln3e8yzSqMjXBu1n\nkGJg2LuC7tefTprvuY00d3RYv9e6+/WkaurTpBXie5HmhFdUXvZl0hzWl7r6OSi3/yvSXNVXWVWd\nn0waKv8FaR7qvwf5DO6+lDSPtKeZvS8Pwe0N7ECa/7krtz238hkuI50wi929exjvctIiyXtIK1Nf\n5u73Tebv7l8nVeRnWhp+uopUqAAP//Ggd/b5LI8nHRB/IM2HX0K64D78OTP7kU7eZbbqd6q7uPst\npLmvo0nfyU2kC/Aaue93mdm3+vS9NvDBHKe7gTcD+7j7r/N7N8/9dO5K30OK5fkVh2rbvwMOd/f/\n6Dzh7l8mLeK5q/KZ3kSq7O8FPgIc5u7f6tPn7sBVZraMtAD0a6Q7+g6bk9ZcTMS5wM9IK73PIy3w\ngVT8Po20UPQ8Vj/ujgP+n6VhyMMr/lUmOv+qd1o/Iy38/LSlYeDrSfOak7bj7hcA/0Y6Nq4nXXgg\nre+BdEHdNnuePYBX57fOu0zwkj+S1o84aV7/gcp7P2tmJ1VeeyhpPvw20jTTF939v/K+zUkL6/rd\n0Z9AuiBcaGZ/IK0OXzcPVe9NOpbvyf/dq3I+Tpb3uve/jXRXfR9puPmcyr4nA/+Tj7EfAp/xVT9f\nHagfd3+Q9FPMQ3IfB5COqT/3f+u4tt9JWux8ec4hF5IW1/Xz+x5pUdxxpHP3NtK05bsAzOzZZra0\nb8ful5ByxvmsKnoPqLzkKFLcbyLNz+/bo5lPA2+t3DweTbp+XE0aBeuc833P0RqvKb2Y7Hzv+11b\n+rsK+/fZvQXpBmopKff/iUos++Tgg4D/dvflE/iOd1gV5/KY2SWkXw9Mdmc1UpjZRcCX2va5R418\nR/Nz4Knu3vNOzsxWklaq95qXDkkeIbmatHhq0DvBWcHSH1+5y91Pnm2XmcbMLictujt1tl1mi3wH\nfpa7P3u2XaKz1mwLjBpm9nTSz7xeMtsuYnrku7FtZ9tjJrC0aPN80srl40k/F210IQDgE/zxlVEj\nD1FfR7qjPpA0h95zyq0tePoDRCoEClD3moHZXHg045jZf5GG3w4bZnhGhGZUjvE3kKZabmDV76dF\ns3gKq/7I09tJ04x3TvwWIQaj1mkCIYQQQjSfxvy/CYQQQggxO6gYEEIIIVqOigEhhBCi5agYEEII\nIVqOigEhhBCi5agYEDOOmY2Z2VjU7SYx27FQLJuzLcR00E8LxYxhZmPuPjbbHqOAYlkOxVIIjQwI\nMTC6CyuHYlkOxVKUQMWAmDF091UOxbIciqUQmiYQQgghWo9GBsSMoeHMciiW5VAshVAxIMTA6KJR\nDsWyHIqlKIGKATFjaG62HIplORRLIbRmQAghhGg9GhkQM4aGM8uhWJZDsRRCxYAQA6OLRjkUy3Io\nlqIEKgbEjKG52XIoluVQLIXQmgEhhBCi9WhkQMwYGs4sh2JZDsVSCBUDQgyMLhrlUCzLoViKEqgY\nEDOG5mbLoViWQ7EUQmsGhBBCiNajkQExY2g4sxyKZTkUSyFUDAgxMLpolEOxLIdiKUqgYkDMGJqb\nLYdiWQ7FUgitGRBCCCFaj0YGxIyh4cxyKJblUCyFUDEggmJml5rZnpXtl5vZ+TX3OVZn+7OFmT1k\nZovN7Mr833kz0OdY3X3MFma2bIb7G5vJ/sRostZsC4j2UHhu9o3AV83sYuARwIeA5xdsv9EUjuVy\nd19QsL1Q1LBmQHOvIhxaMzBidO4SOgku2vaQn/U44AFgPWCpu39o2DYmaX+cW7TtIT7nMnefM8x7\nhmW2YzHDx+VSd5877PsGaHecU7Rt0WxUDIwIZjbW9JOutKOZPQpYDPwZ2MndHyzUbqtiaWYrgKsA\nA25095eVaDe33apY5vZqKQYiE+E4aDtaMyAawVTmPd39AeAs4PRShcBERJmbnYLnA+6+wN13LFkI\nTMQIx3LGieAomo+KgREhQtVdk+PK/K8YLY5lcSJ4RnCE2Bf9KDFuM5omEKExs2OAZe5+wmy7RGUm\n1gy0iTrXDOiiKupCIwMjQoS7hgiOEMOzsGNtdwRti6WZrUlaw1KcyIVAhOOg7agYEI1gqsnC3d83\nU6MCURLasJ6zsdhtVGMJbAf8pgaVvkSJpWg2KgZGhAh3DREcIYZnBEeI4VnwVxlvAL4EvLtEez3a\nH6uj3ZkgwnHQdrRmQAghAqA1A6JONDIwIkS4a4jgCDE8IzhCDM8IjhD77jpKjNuMigHRCCIkiwiO\nEMMzgiPE8IzgKJqPioERIcJdQwRHiOEZwRFieEZwhNgX/SgxbjNaMyCEEAHQmgFRJxoZGBEi3DVE\ncIQYnhEcIYZnBEeIfXcdJcZtRsWAaAQRkkUER4jhGcERYnhGcBTNR8XAiBDhriGCI8TwjOAIMTwj\nOELsi36UGLcZrRkQQogAaM2AqBONDIwIEe4aIjhCDM8IjhDDM4IjxL67jhLjNqNiQDSCCMkigiPE\n8IzgCDE8IziK5qNiYESIcNcQwRFieEZwhBieERwh9kU/SozbjNYMCCFEALRmQNSJRgZGhAh3DREc\nIYZnBEeI4RnBEWLfXUeJcZtRMSBWw8w2NbOvm9n1ZvZrM/s3M1u75j7H6my/BFNxNLNlNahM1ufY\nTPc5LFOM5SZmdoaZ3WBmPzGzb5rZljXoVfscq7P9EkRwFM1HxcCIUPiu4WzgbHffCngy8CjgowXa\nXdhJXGY2ZmaLKolsYXW7e/9Mbrv7WK/9wMIpfOa65uHaGMtzgIvd/cnu/nTgXcAmU2inmxCxrPwr\nEcsZJfKoRlvQmgExDjPbDXivuy+sPDcHuAnYzN0fmEbbrZvzNLOl7j63hnZbFUsz2xU4pnpcFmw7\nRCyjeIqYaGRgRCg4VLgt8LPqE+6+DPgtMK0h2SiJLMKwawtjuR1dx2UposQyimcvIpxTbUfFgBgU\nq7XxAMkigiPE8IzgCDE8IziK5qNiYEQoeNfwK2Cn6hNmNpc0N3vddBqOkrQi3IG1MJa/pOu4LEWU\nWEbx7EWEc6rtqBgQ43D3i4B1zexAADNbE/gY8Cl3/3ON/Y7V1XYppuhY64hKL0Yxlu5+MfAIMzuk\n85yZbW9mu5R26+p3rM72SxDBUTQfFQMjQuG7hn2Bl5vZ9cA9wEPuftx0G42StArHcl0zu9nMluT/\nvq1Eoy2N5b7AHpZ+7no1cCxwx3QbjRLLkp5m9i0ze3yp9gbob2ym+hJTQ8WAWA13v9Xd98k/LXwh\nsKeZ7VBnnxGSxVQc3X0td5/n7pvn/36yBrVxjHAs73D3/dx9S3ff3t1f7O6/qUHvYUY4lnu5+7QL\nKTE6qBgYEeq6u3H3y939r93959NtK0JihRh3ioplOaLEMopnLyIcB21HxYBoBBGSRQRHiOEZwRFi\neEZwFM1HxcCIEOGuIUrSUizLoViWI4pnLyIcB21HxYBoBBGSRQRHiOEZwRFieEZwFM1HxcCIEOGu\nIUrSUizLoViWI4pnLyIcB21HxYBoBBGSRQRHiOEZwRFieEZwFM1HxcCIEOGuIUrSUizLoViWI4pn\nLyIcB21HxYBoBBGSRQRHiOEZwRFieEZwFM1HxcCIEOGuIUrSUizLoViWI4pnLyIcB21HxYBoBBGS\nRQRHiOEZwRFieEZwFM1HxcCIEOGuIUrSUizLoViWI4pnLyIcB21HxYBoBBGSRQRHiOEZwRFieEZw\nFM1HxcCIEOGuIUrSUizLoViWI4pnLyIcB21HxYBoBBGSRQRHiOEZwRFieEZwFM1HxcCIEOGuIUrS\nUizLoViWI4pnLyIcB21HxYBoBMMmCzObb2ZXdz13jJkdXlRsfPtjdbVdkql4mtm7zewaM/uFmS02\ns6fXoFbtb6zO9ksxheNyWeXxi8zsWjPbvLjY+D7H6mxftAMVA6PDwk5SMLMxM1vUwO1F/fYDC6fw\nmX0K7xmEVsXSzJ4BvAjYwd3/Fvh7YMlQEetPq2JJPibNbHfgk8Ce7l4klv38O44Niudq25FHNdqC\nudeVT8VMEuGEK+loZvOB89z9qZXnjgGWufsJ02y7bbHcF/hnd9+nRHtdbbctlstIhdUpwAvd/YYS\n7QpRNyoGREjqLAbahpmtB/wAWBe4CDjL3b8/u1YxMbO/AEuBhe5+TeG2G19Y9SOye1vQNIFoBJ2h\nxSHoV8XWVt1OwXFWGNbT3ZcDC4DXA3cDZ5rZQTWoPcyoxhJ4ELgMOKS8TW+ixFI0GxUDI0KEhFDY\n8XfAxl3PbQzcM92GWxhLPPH9fPd2KPCyEu22MJYPAa8AdjazdxVsN/SvCSK7twUVA6IRDJss8t3s\nbWa2K4CZbQy8gDTcXQtREtqwnma2lZltWXlqB+CmolJdjGosSVOvfwL2Ag4ws9eUtxpPlFiKZqM1\nAyIsZrY1cBKwEWl64CPufubsWsXDzBYAnwI2AFYAvwZe7+73zqpYQMxsqbvPzY83A74HHObu3yzQ\ndth598jubUEjA6IRTGWo1t2vdffd3H1Hd19QdyEQYcgbprRmYLG77+Lu27n7Du7+j3UXAiMcy7mV\nx7e4+5NKFAITESWWotmoGBgRIiSECI4QwzOCI8TwjOAIsacDIru3BRUDohFESBYRHCGGZwRHiOEZ\nwVE0H60ZEEKIAESed4/s3hY0MiAaQYSh2giOEMMzgiPE8IzgKJqPioERIUJCiOAIMTwjOEIMzwiO\nEHs6ILJ7W1AxIBpBhGQRwRFieEZwhBieERxF89GaASGECEDkeffI7m1BIwOiEUQYqo3gCDE8IzhC\nDM8IjqL5qBgYESIkhAiOEMMzgiPE8IzgCLGnAyK7twUVA6IRREgWERwhhmcER4jhGcFRNB+tGRBC\niABEnneP7N4WNDIgGkGEodoIjhDDM4IjxPCM4Ciaj4qBESFCQojgCDE8IzhCDM8IjhB7OiCye1tQ\nMSAaQYRkEcERYnhGcIQYnhEcRfPRmgEhhAhA5Hn3yO5tQSMDYjXM7CEzW2xmPzezn5rZM2agz7G6\n+5guU3E0s03N7Otmdr2Z3WBmnzCztWrQq/Y5Vmf7JRjW0cw2NrMr83F5u5ndUtmuLZ4jGsvNzOxG\nM9swb2+Ut+fVIihCoGJgRCictJa7+wJ33wE4GjiuRKNmtqjjaWZj1W1g4UT7Z3K78m/cfmDhFD72\n2cDZ7r4VsBUwBzh2Cu2Mo22xdPd73X1Hd18AfBY4obPt7iuGDN84osTS3UvF8hbgJOD4/NRxwL+7\n+83DtDNkn2N1tS3KoGmCEaGTLAq1tczd5+THLwf2d/d/KNBuiKHCUp5mthvwXndfWHluDvBbYDN3\n/9NsO9ZNHZ5mdgywzN1PKNReiFiWJI+m/BQ4BTgE2MHdH5pdKzGbqBgQq2FmK4CrgHWBxwO7ufuV\ns2sVDzM7FHiiux/R9fzPgIPd/ZrZMYtN6WIgCqWLFjN7PnAB8PfufnGpdvv01bqCKxqaJhC9eCAP\nv24DvBA4ve4OC09z1EJBRyvUTu/G2xXLWongOQ3HFwG3AduXsxFRUTEwItSVtNz9cuAxZvaY6bYV\nIbFCUc9fATt1tT0X2Bz49XQabmEsayOCI5SddzezHYDdgWcAh5vZJqXa7oVGBZqPigHRi4fvXM1s\na9Jx8rs6O4yQLIZ1dPeLgHXN7EAAM1sT+BhwynTWCwzQ71hdbZcigiPE8Jyi40nAYXkx4UeAjxeV\nEuHQmgGxGmb2IHA1q4qCd7n7BbOoFBYz25S0+n1rUjzPB4509wdnVSwwWjMw7XZeR1oHtH/eXgP4\nMfB2d790uu336VNrBhqORgbEarj72nnNwI75X+2FQISh2qk4uvut7v4Sd9/K3Z/s7ofVXQiMaiw7\nuPv7ZqoQGMVYuvvJnUIgb690953qKgREDFQMjAijmLRmiwieERwhhmcER4gxZdGPyO5tQcWAaAQR\nkkUER4jhGcERYnhGcBTNR2sGhBAiAJHn3SO7twWNDIhGEGGoNoIjxPCM4AgxPCM4iuajYmBEiJAQ\nIjhCDM8IjhDDM4IjxJ4OiOzeFlQMiEYQIVlEcIQYnhEcIYZnBEfRfLRmQAghAhB53j2ye1vQyIBo\nBBGGaiPMoyW+AAAOvUlEQVQ4QgzPCI4QwzOCo2g+KgZGhAgJIYIjxPCM4AgxPCM4QuzpgMjubUHF\ngGgEEZJFBEeI4RnBEWJ4RnAUzUdrBoQQIgCR590ju7cFjQyIRhBhqDaCI8TwjOAIMTwjOIrmo2Jg\nRIiQECI4QgzPCI4QwzOCI8SeDojs3hZUDIhGECFZRHCEGJ4RHCGGZwRH0Xy0ZkAIIQIQed49sntb\n0MiAaATDDtWa2UNmttjMrjazc81sbk1q1T7H6u6jBFPxNLOXmtlKM9uqBqVe/Y3NRD/TZQrH5bKu\n7YPN7FNFpVbvc6zO9kU7UDEwIpjZok5SMLOxhm4v6rcfWDjkR17u7gvcfXvgPuDNQ76/Lw2J1WTb\ni/rtZ/hYArwSuBTYfwrv7UtDYjXZ9qJ++xk+lr2GWosNv/byJzs2KJ6rbWtUoPlommBEiHDClXQ0\ns6XuPjc/fgOwvbu/pVDbbYvlesC1wK7AN9196xLt5rbbFsuHj8u8fTDwNHd/a4n2hagLFQMiJGa2\nzN3nmNmawBnAf7r7hbPtFREzOwDY1d1fZ2Y/AA519ytn2ysiZrYCuKqzCWwEfKNEMRChsOpHZPe2\noGkC0Qg6Q4tDsK6ZLQZuBx4HfLe4VBdTcJwVpuC5P3BmfnwWcEBRoR6McCwfyNNXC9x9R+CYGrTG\nESWWotmoGBgRIiSEwo4PuPsCYB7pDqzIFAG0K5ZmthGwG/CfZnYjcCTw8hJt5/bHSrVVFxEcIfZP\nCCO7twUVA6IRTCFZWH7fn4DDgCPMrNbjOUpCG9Lz5cBp7v7X7r6Fu88Hfmtmz67HLjGisYR8XM4k\nUWIpmo2KgREhQkIo7PjwYhd3/znwCwqthG9ZLPcDzul67mwUyyk3V7CtcUQZwehFZPe2oGJANIJh\nk0V1xXbe3sfdv1RUqosoCW0YT3ffvXvhpbt/yt2L/VSzF6MYS+h5XJ5a9y8JosRSNBsVAyNChIQQ\nwRFieEZwhBieERwhxihLPyK7twUVA6IRREgWERwhhmcER4jhGcFRNB/9nQEhhAhA5N/qR3ZvCxoZ\nEI0gwlBtBEeI4RnBEWJ4RnAUzUfFwIgQISFEcIQYnhEcIYZnBEeIPR0Q2b0tqBgQjSBCsojgCDE8\nIzhCDM8IjqL5aM2AEEIEIPK8e2T3tqCRAdEIIgzVRnCEGJ4RHCGGZwRH0XxUDIwIERJCBEeI4RnB\nEWJ4RnCE2NMBkd3bgooB0QgiJIsIjhDDM4IjxPCM4Ciaj9YMCCFEACLPu0d2bwsaGRCNIMJQbQRH\niOEZwRFieEZwFM1HxcCIECEhRHCEGJ4RHCGGZwRHiD0dENm9LagYEI0gQrKI4AgxPCM4QgzPCI6i\n+WjNgBBCBCDyvHtk97agkQGxGmb2kJktNrOrzewsM3vkDPQ5Vncf02UqjpVYXpn/e1QNat19jtXd\nx3SZYiyX1aAyWZ9jM93nsAzraGYvrRyPnWPzITN7QU2KIgAqBkaEwklrubsvcPftgQeBN5Zo1MwW\ndTzNbKy6DSycaP9Mblf+jdsPLJzCx+7Ecsf8349MoY3VaGksaxnGjBJLdy8SS3f/euV4XACcBHzf\n3b8zZOiG6XOsrrZFGTRNMCJ0kkWhtpa6+9z8+A3A9u7+lgLthhgqLBzLZe4+p0RbXe22MZYPH5cl\niRLLOjCzrYCLgGe4+62z7SNmDxUDYjU6FzAzWwv4GvBtd//cbHtFxMxWAFcBRrqz/bC7f3V2rWJS\nVzEQhdJFSz6/fwQc7+5fK9Vun75aW3BFQdMEohfrmtli4MfATcDn6+6w8DRHLUzR8YGuaYLaC4ER\njuWME8FzGo4fBK6puxAQMVAxMCIUTlqdC9gCdz/M3VeUaDRCYoUYnhEcIYZnBEcoO+9uZguBfYE3\nl2pzIjQq0HzWmm0B0UhspjuMkCym6KhY9kCxLMewjma2EfAFYH93f6AWKREOrRkQq9H2udmSmNmD\nwNWsWjNwgbsfPbtWMWn7cVlq3t3M3gm8G7ih8xQ1r2fRmoHmo2kCsRqzkXAjDNVOxdHd1+5aM1B7\nITDCsdRx2YNhHd39OHefU5kKnLH1LKK5qBgYEUYxac0WETwjOEIMzwiOEGPKoh+R3duCigHRCCIk\niwiOEMMzgiPE8IzgKJqP1gwIIUQAIs+7R3ZvCxoZEI0gwlBtBEeI4RnBEWJ4RnAUzUfFwIgQISFE\ncIQYnhEcIYZnBEeIPR0Q2b0tqBgQjSBCsojgCDE8IzhCDM8IjqL5aM2AEEIEIPK8e2T3tqCRAdEI\nIgzVRnCEGJ4RHCGGZwRH0XxUDIwIERJCBEeI4RnBEWJ4RnCE2NMBkd3bgooB0QgiJIsIjhDDM4Ij\nxPCM4Ciaj9YMCCFEACLPu0d2bwsaGRCNIMJQbQRHiOEZwRFieEZwFM1HxcCIECEhRHCEGJ4RHCGG\nZwRHiD0dENm9LagYEI0gQrKI4AgxPCM4QgzPCI6i+WjNgBBCBCDyvHtk97agkQHRCIYdqjWzlWZ2\nWmV7TTO728y+UVxuVR9jdbVdkinG8qOV7SPM7L3Fxcb3OVZn+6UYxtPMTjCzt1a2LzCz/6hsf8zM\n3lZYMUwsRbNRMTAimNmiTlIws7GGbi/qtx9YOORHXg5sZ2br5O09gCVDttGThsRqsu1F/fYzfCz/\nDPyDmW085PsmpSGxmmx7Ub/9DBfLHwLPyu814DHAtpX9zwIuG6K91ejl33FsUDxX29aoQPPRNMGI\nEOGEK+loZsuAE4HF7n62mZ0KXAM8x91fMs222xjLDwJz3P09ZnYEsJ67v79A262JpZk9AbjC3eeZ\n2XbAkcDjgf2APwJ3AI9z9xXT7UuI0mhkYERoesKF4o4OnAnsn0cHngpcUaThdsbyM8A/mdmcgu22\nKpbufjvwoJltxqpRgCuAZwI7AVdPpxDo3HFHJLJ7W1AxIBrBVJKFu18DPBHYH/gWYGWtxhMloU0x\nlvcDpwKHFRfqwQjH8jJgF1Ix8CPg8sr2D4vKZaLEUjQbFQMjQoSEUJPjN4CPAmeUarDFsTwReC3w\nqFINtjCWl5Eu/NuRpq0uJ40MPJNprheIMMrSj8jubUHFgGgEU0gWnVGALwDvc/dfljVanSgJbaqx\ndPf7gK8Ah5R26maEY3kZsDdwryfuAzakQDHQjyixFM1GxcCIECEh1DDPjbvf6u6fLthua2OZ+Tjw\n6K7npt5w+2J5NSl+P+p67vfufu90Go4wytKPyO5tQcWAaATDJgt3n9vjue9N95cEExEloU0nlu5+\nl7uv7+4fKC5WYYRjudLdN3T3YyrPvdrd/6a4XCZKLEWzUTEwIkRICBEcIYZnBEeI4RnBEWKMsvQj\nsntbUDEgGkGEZBHBEWJ4RnCEGJ4RHEXz0R8dEkKIAET4A079iOzeFjQyIBpBhKHaCI4QwzOCI8Tw\njOAomo+KgREhQkKI4AgxPCM4QgzPCI4QezogsntbUDEgGkGEZBHBEWJ4RnCEGJ4RHEXz0ZoBIYQI\nQOR598jubUEjA6IRRBiqjeAIMTwjOEIMzwiOovmoGBgRIiSECI4QwzOCI8TwjOAIsacDIru3BRUD\nohFESBYRHCGGZwRHiOEZwVE0H60ZEEKIAESed4/s3hY0MiAaQYSh2giOEMMzgiPE8IzgKJqPioER\nIUJCiOAIMTwjOEIMzwiOEHs6ILJ7W1AxIBpBhGQRwRFieEZwhBieERxF89GaATEOM7sY+LC7f7fy\n3GHAVu7+5tkzi4mZPQT8AngE8CBwOvAJ14k3NGb2OOATwN8B9wF/AT7i7ufOqtgMEXnePbJ7W9DI\ngOjmy8D+Xc+9Mj9fGxGGaqfouNzdF7j7dsAewAuBYyZ5z7QY4Vh+HVjk7lu6+9NJx+VmRcW6GMVY\nmtlva1IRgVlrtgVEGQpW3v8NfNDM1nL3FWY2H3iCu/9wug13klbHs6nbnee6908Xd7/HzF4P/ASY\nVptNidVMxdLMdgP+7O4nV9pdAnxmKu11tT3OLdr2FJjxUSmNCjQfTROMCCWH4czsG8DJ7n6emb0D\neLS7H1Wi7QgUjuVSd5/b9dy9wFPc/e4SfTSZUrE0s0OBJ7r7EdO3ajdmdoW7/91se4hmoWmCEaFw\n5X0maQiW/N8zCrbdeGbgLsZqbr8x1BVLM/u0mf3czK6oo/0mUmrKYjYKgQjTLW1HxYDoxbnA7ma2\nI7Cuu19Zd4dtSRZmtgWwos5RgRGN5S+Bp3U23P0twO7AY+vsdERjKcRqqBgYEUomLXdfDiwCvkDL\nRgWg+AXg4VEAM3ss8FngUwXbbzQF72YvBtYxszdUnl6vRNtRiDzvHtm9LWgBoejHGcDZwH4z0dkI\nJ4tHmtliVv208DR3/0SdHY5wLF8KfNLMjgLuBpYDta5lGeFYCjEOLSAUQogARP6tfmT3tqBpAtEI\nNDdbDsWyHIqlaAsqBkYEJa1yKJblUCzLEfnOOrJ7W1AxIBqBkkU5FMtyKJaiLWjNgBBCBCDyvHtk\n97agkQHRCDScXA7FshyKpWgLKgZGBCWtciiW5VAsyxH5zjqye1tQMSAagZJFORTLciiWoi1ozYAQ\nQgQg8rx7ZPe2oJEB0Qg0nFwOxbIciqVoCyoGRgQlrXIoluVQLMsR+c46sntbUDEgGoGSRTkUy3Io\nlqItaM2AEEIEIPK8e2T3tqCRAdEINJxcDsWyHIqlaAsqBkYEJa1yKJblUCzLEfnOOrJ7W1AxIBqB\nkkU5FMtyKJaiLWjNgBBCBCDyvHtk97agkQHRCDScXA7FshyKpWgLKgZGBDMbqyauaNtNYrZjoVg2\nZ7tpNCk2w2xrVKD5aJpACCGEaDkaGRBCCCFajooBIYQQouWoGBBCCCFajooBIYQQouWoGBBCCCFa\njooBIYQQouWoGBBCCCFajooBIYQQouWoGBBCCCFajooBIYQQouWoGBBCCCFajooBIYQQouWoGBBC\nCCFajooBIYQQouWoGBBCCCFajooBIYQQouWoGBBCCCFajooBIYQQouWoGBBCCCFajooBIYQQouWo\nGBBCCCFajooBIYQQouWoGBBCCCFazv8HqGK7wuWp6wgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d1ece48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "report({'keybee': keybee})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This does indeed cut the path length to\n",
    "half of QWERTY, so maybe there is something to be said for hex grids. \n",
    "Note: the space  key was moved to the edge, because it doesn't contribute to the workload.\n",
    "The Keybee FAQ says \"the space key is the most important key on a keyboard,\" so clearly they are aiming\n",
    "at two-thumb typing, not gesture typing; perhaps that's why it was so easy to improve their layout for our purposes.\n",
    "\n",
    "Here is Keybee with some word paths:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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I3p/h3ojvYMnAgsVp2gr0ceA666fi3jaPiXSsFR3bAJcHPgNGlRnebGKDTIrYbWH0twhY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\noEJjBACgQmMEAKBCYwQAoEJjBACg8i8i5ZulEQmc/wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x104639a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(keybee, 'keybee', words=['FOUR', 'PLEASANT', 'THINGS'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Summary\n",
    "=====\n",
    "  \n",
    "  So where are we? Let's revisit our initial questions and see what answers we have:\n",
    "  \n",
    " 1. What words have the longest path length? <br>**Answered**: \"PALEOMAGNETISMS\" etc.\n",
    " 2. What words have the highest ratio of path length to word length? <br>**Answered**: \"PALAPA\" etc.\n",
    " 3. What is the average segment length, over a typical typing work load? <br>**Answered**: 3.23 keys, for Qwerty keyboard, on our sample workload.\n",
    " 4. Is there a better keyboard layout to minimize the average segment length over a work load?  <br>**Answered**: Yes, many layouts at around 1.9 on Qwerty; or 1.7 or 1.8 on more square keyboards.\n",
    " 5. How often are two words confused because they have similar paths? <br>**Answered**: On Qwerty, 26% of the words in a small dictionary, and 55% of the words in running text have at least one possible confusion. Other layouts are worse.\n",
    " 6. Is there a better keyboard layout to minimize confusion? <br>**Partially Answered**: We found a keyboard with less confusingness than Qwerty.  The computation of confusingness takes too long to do very much hillclimbing search.  \n",
    " 7. Is there a better keyboard layout to maximize overall user satisfaction? <br>**Partially Answered**: We defined a combined metric, and found\n",
    "a keyboard with a better score. There are no doubt better metrics, and better keyboards to be found.\n",
    " \n",
    "\n",
    "\n",
    "\n",
    "Going Beyond\n",
    "===\n",
    "\n",
    "Now it is your turn to answer the open questions, or make up some questions of your own.  Good luck! Here are a few ideas to get you started:\n",
    "\n",
    "* Hillclimbing just keeps the one best keyboard it has found so far.  Other optimization techniques such as\n",
    "[beam search](http://en.wikipedia.org/wiki/Beam_search) or [genetic algorithms](http://en.wikipedia.org/wiki/Genetic_algorithm)  or [ant colony optimization](http://en.wikipedia.org/wiki/Ant_colony_optimization_algorithms) maintain several candidates at a time. Is that a good idea?\n",
    "\n",
    "* The code in this notebook emphasizes clarity, not efficiency.  Can you modify the code (or perhaps port it to another language) and make it twice as efficient? 10 times? 100 times?\n",
    "\n",
    "* What other factors do you think are important to user satisfaction with a keyboard.  Can you measure them?\n",
    "\n",
    "* Consider the 5 paths below. They all start at 'P', move in a straight line to 'T', and then go to 'S', but they all make different stops along the top row. In other words, the 5 paths all trace exacty the same lines, so they are very confusing, but our definition of  `confusions` makes most of them different. Can you think\n",
    "of a better way to handle confusions for paths like this?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
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6M22aM2HCGwwYMKar913+7Tx4DBz/Biz38Oa/5R6GB7dZ5p62fkn2XScdsyV7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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c2584a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "show_kbd(qwerty, words=['POUTS', 'POIUYTS', 'PUTTS', 'PUTS', 'POTS', 'PITS'])"
   ]
  }
 ],
 "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": "3.5.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}