daviddoji/pytudes
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Add files via upload

Browse Source
This commit is contained in:
Peter Norvig 2017-03-24 19:12:25 -07:00 committed by GitHub
parent c08ec2a796
commit 5ba68cb755
1 changed files with 504 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

504
Cheryl.ipynb Normal file
View File

@ -0,0 +1,504 @@
{
"metadata": {
"name": "",
"signature": "sha256:b9e2e6c3f7e82432833f1416dd0552c13753f683e1888682faf2910ef7d9f8ff"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"When is Cheryl's Birthday?\n",
"===\n",
"\n",
"<div align=right>Peter Norvig, April 2015</div>\n",
"\n",
"This logic puzzle has been [making the rounds](https://www.google.com/webhp?#q=cheryl%27s+birthday):<i>\n",
"\n",
"\n",
"1. Albert and Bernard just became friends with Cheryl, and they want to know when her birtxhday is. Cheryl gave them a list of 10 possible dates:\n",
"<pre>\n",
" May 15 May 16 May 19\n",
" June 17 June 18\n",
" July 14 July 16\n",
" August 14 August 15 August 17\n",
"</pre>\n",
" \n",
"2. Cheryl then tells Albert and Bernard separately the month and the day of the birthday respectively.\n",
" \n",
"3. **Albert**: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.\n",
" \n",
"4. **Bernard**: At first I don't know when Cheryl's birthday is, but I know now.\n",
" \n",
"5. **Albert**: Then I also know when Cheryl's birthday is.\n",
" \n",
"6. So when is Cheryl's birthday?</i>\n",
"\n",
"Problem-Solving Tools\n",
"---\n",
"\n",
"Cheryl's puzzle was designed to be solved with a pencil, the greatest problem-solving tool in the history of mathematics (although some prefer a pen, chalk, marker, or a [stick for drawing in the sand](http://www.hellenicaworld.com/Greece/Science/en/Archimedes.html)). But I will show how to solve it with another tool: **computer code**. I choose this tool for four reasons: \n",
"- It is a more direct way to find the solution. All I have to do is faithfully describe the problem with code that says: *\"for each of the 10 possible dates, tell Albert the month and Bernard the day and check if statements 3 through 5 are true.\"* The intended [pencil and paper solution](https://scontent-iad.xx.fbcdn.net/hphotos-xpa1/v/t1.0-9/s720x720/11111802_983395601695416_3208022346737572922_n.jpg?oh=15fcb7edc4689dd9c71385b613446465&oe=55DD4488) requires not just understanding the *problem*, but also creatively discovering the steps of the *solution*&mdash;a harder task.\n",
"- With tested, debugged code, you're less likely to make a mistake that leads you to a [wrong answer](http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/15/why-the-cheryl-birthday-problem-turned-into-the-maths-version-of-thatdress).\n",
"- You'll learn how to solve problems that are similar, but can't be solved with pencil and paper because they have millions of possibilities rather than just 10.\n",
"- Solving puzzles is fun; programming is fun; solving puzzles with programs is double fun.\n",
"\n",
"We will translate each of the 6 statements in the puzzle into Python code:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. Cheryl gave them a list of 10 possible dates:\n",
"---\n"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"DATES = ['May 15', 'May 16', 'May 19',\n",
" 'June 17', 'June 18',\n",
" 'July 14', 'July 16',\n",
" 'August 14', 'August 15', 'August 17']"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 1
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We'll also define accessor functions for the month and day of a date:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def Month(date): return date.split()[0]\n",
"\n",
"def Day(date): return date.split()[1]"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 2
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Month('May 15')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 3,
"text": [
"'May'"
]
}
],
"prompt_number": 3
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"Day('May 15')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 4,
"text": [
"'15'"
]
}
],
"prompt_number": 4
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"2. Cheryl then tells Albert and Bernard separately the month and the day of the birthday respectively.\n",
"---\n",
"\n",
"We can define the idea of **telling**, and while we're at it, the idea of **knowing** a birthdate:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def tell(part, possible_dates=DATES):\n",
" \"Cheryl tells a part of her birthdate to someone; return a new list of possible dates that match the part.\"\n",
" return [date for date in possible_dates if part in date]\n",
"\n",
"def know(possible_dates):\n",
" \"A person knows the birthdate if they have exactly one possible date.\"\n",
" return len(possible_dates) == 1"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 5
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note that we use a *list of dates* to represent someone's knowledge of the possible birthdates, and that someone *knows* the birthdate when they get down to only one possibility. For example: If Cheryl tells Albert that her birthday is in May, he would have a list of three possible birthdates:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"tell('May')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 6,
"text": [
"['May 15', 'May 16', 'May 19']"
]
}
],
"prompt_number": 6
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And if she tells Bernard that her birthday is on the 15th, he would end up with two possibilities:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"tell('15')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 7,
"text": [
"['May 15', 'August 15']"
]
}
],
"prompt_number": 7
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With two possibilities, Bernard does not know the birthdate:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"know(tell('15'))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 8,
"text": [
"False"
]
}
],
"prompt_number": 8
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Overall Strategy\n",
"---\n",
"\n",
"When Cheryl tells Albert `'May'` then *he* knows there are three possibilities, but *we* (the puzzle solvers) don't, because we don't know what Cheryl said. So what can we do? We will consider *all* of the possible dates, one at a time. For example, first consider `'May 15'`. Cheryl tells Albert `'May'` and Bernard `'15'`, giving them the lists of possible birthdates shown above. We can then check whether statements 3 through 5 are true in this scenario. If they are, then `'May 15'` is a solution to the puzzle. Repeat the process for each of the possible dates. If all goes well, there should be exactly one solution. \n",
"\n",
"Here is the main function, `cheryls_birthday`:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def cheryls_birthday(possible_dates=DATES):\n",
" \"Return a list of the possible dates for which statements 3 to 5 are true.\"\n",
" return filter(statements3to5, possible_dates)\n",
"\n",
"def statements3to5(date): return statement3(date) and statement4(date) and statement5(date)\n",
"\n",
"## TO DO: define statement3, statement4, statement5"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 9
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" (*Python note:* `filter(predicate, items)` returns a list of all items for which `predicate(item)` is true.)\n",
" \n",
" 3. Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function `statement3` takes as input a possible birthdate and returns true if Albert's statement is true for that birthdate. How do we go from Albert's English statement to a Python function? Let's paraphrase in a form that is closer to Python code:\n",
"\n",
"> **Albert**: After Cheryl told me the month of her birthdate, I didn't know her birthday. I don't know which day Cheryl told Bernard, but I know that for all of the possible dates, if Bernard is told that day, he wouldn't know the birthdate.\n",
"\n",
"That I can translate directly into code:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def statement3(date):\n",
" \"Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too.\"\n",
" possible_dates = tell(Month(date))\n",
" return (not know(possible_dates) \n",
" and all(not know(tell(Day(d)))\n",
" for d in possible_dates))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 10
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can try the function on a date:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"statement3('May 15')"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 11,
"text": [
"False"
]
}
],
"prompt_number": 11
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In fact, we can see all the dates that satisfy statement 3:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"filter(statement3, DATES)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 12,
"text": [
"['July 14', 'July 16', 'August 14', 'August 15', 'August 17']"
]
}
],
"prompt_number": 12
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"4. Bernard: At first I don't know when Cheryl's birthday is, but I know now.\n",
"---\n",
"\n",
"Again, a paraphrase:\n",
"\n",
"> **Bernard:** At first Cheryl told me the day, and I didn't know. Then I considered just the dates for which Albert's statement 3 is true, and now I know."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def statement4(date):\n",
" \"Bernard: At first I don't know when Cheryl's birthday is, but I know now.\"\n",
" at_first = tell(Day(date))\n",
" return (not know(at_first)\n",
" and know(filter(statement3, at_first)))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 13
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see which dates satisfy both statement 3 and statement 4:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"filter(statement4, filter(statement3, DATES))"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 14,
"text": [
"['July 16', 'August 15', 'August 17']"
]
}
],
"prompt_number": 14
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wait a minute&mdash;I thought that Bernard **knew**?! Why are there three possible dates remaining? Bernard does indeed know the birthdate,\n",
"because he knows something we don't know: the day. We won't know the birthdate until after statement 5.\n",
"\n",
"5. Albert: Then I also know when Cheryl's birthday is.\n",
"---\n",
"\n",
"Albert is saying that after hearing the month and Bernard's statement 4, he now knows Cheryl's birthday:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def statement5(date):\n",
" \"Albert: Then I also know when Cheryl's birthday is.\"\n",
" return know(filter(statement4, tell(Month(date))))"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 15
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"6. So when is Cheryl's birthday?\n",
"---\n",
"\n",
"Let's see:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"cheryls_birthday()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 16,
"text": [
"['July 16']"
]
}
],
"prompt_number": 16
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Success!** We have deduced that Cheryl's birthday is **July 16**. It is now `True` that we know Cheryl's birthday:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"know(cheryls_birthday())"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 17,
"text": [
"True"
]
}
],
"prompt_number": 17
}
],
"metadata": {}
}
]
}