6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
February 15, 2005
Lecture Notes
Induction III
1 Two Puzzles
Here are two challenging puzzles.
1.1 The 9Number Puzzle
The numbers 1, 2, . . . , 9 are arranged in a 3 3 grid as sh
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
May 10, 2005
Lecture Notes
Random Walks 1 Random Walks
A drunkard stumbles out of a bar. Each second, he either staggers one step to the left or staggers one step to the right,
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
May 12, 2005 Lecture Notes
Special Topics 1 Streaks
Was the table of H s and T s below generated by ipping a fair coin 100 times, or by someone tapping the H and T keys in a what
CS237. Practice Problems Set 1: Sets, Basic Counting.
Not Graded. Please complete by Thursday Jan 28, 2011. January 27, 2011
Reading.
Schaums Chapter 1, 2.1. L&L Counting I.
Exercise 1. Let A, B U. Prove the following. (Hint, use proofs by contradiction.)
CS237. Practice Problems Set 2: More Counting.
Assigned problems due Thurs Feb 3, 11:59PM. February 6, 2011
Reading.
Schaums Chapter 2.2- 2.6. L&L notes on counting. Any of problems 2.8 - 2.31 are good for extra practice.
Optional Extra Practice.
1
Practi
CS237. Practice Problems Set 3: Basic Probability.
Not Graded. Please complete by Thurs Feb 10. February 14, 2011
Reading. Schaums Chapter 3. L&L probability- Chapter 18. Optional Extra Practice. Any of the problems in Schaumss Chapter 3. Exercise 1. You
CS237. Practice Problems Set 4: Basic Probability.
Graded Problems due Thurs Feb 17 11:59PM. February 20, 2011
Reading.
Schaums Chapter 4. L&L Chapter 19-20. Any of the problems in Schaumss Chapter 4. Mitzenmacher-Upfal
Optional Extra Practice. problems 1
CS237. Practice Problems Set 5: Randomized Min Cut Algorithm. Random Variables
Ungraded. Please complete by Thurs Feb 24. March 8, 2011
Reading. M&U Chapter 1.4, L&L Notes on Random Variables. (See also Schaums Chapter 5.2,5.3,5.10,5.11, but the other boo
CS237. Problem Set 6: Random Variables & Binomial Distribution.
Graded Problems due Thurs March 3 11:59PM. February 25, 2011
A short problem set for a short week. Reading. Schaums Chapter 5.2,5.3,5.10,5.11., and 6.2 L&L Notes on Random Variables. In class
CS237. Problem Set 7: Expectation
Ungraded. Please complete by Thursday March 10 March 8, 2011
Reading. L&L Expectation I and II. M&U 2.1-2.4. Exercise 1. We independently roll two dice. Let X1 be the number that comes up on the rst die, X2 be the number
CS237. Problem Set 8: Variance, Geometric Distribution and Coupon Collector Problem
Graded Problems due Thurs March 31 11:59PM. April 9, 2011
Reading. Schaums Chapter 5.4, 5.5. M&U 2.4 and 3.2 and Lemma 3.8 on page 51.
1
Practice Problems.
VAR[X1 + X2 ] =
CS237. Problem Set 9: Geometric Distribution, Normal Distribution and Central Limit Theorem
Ungraded. Please complete by Thursday April 7. April 7, 2011
Reading. Schaums Chapter 6.1 - 6.6. Extra Practice. Schaums Problems 6.21 - 6.36. Exercise 1. A page c
Graph Theory Problems and Solutions
tomrdavis@earthlink.net http:/www.geometer.org/mathcircles November 11, 2005
Tom Davis
1 Problems
1. Prove that the sum of the degrees of the vertices of any nite graph is even. 2. Show that every simple graph has two v
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
February 1, 2005
Lecture Notes
Logic
Its really sort of amazing that people manage to communicate in the English language. Here are some typical sentences: 1. You may have cake
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
February 3, 2005
Lecture Notes
Proofs
Why do you believe that 3 + 3 = 6? Is it because your secondgrade teacher, Miss Dalrymple, told you so? She might have been lying, you know
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
February 8, 2005
Lecture Notes
Induction I 1 Induction
A professor brings to class a bottomless bag of assorted miniature candy bars. She offers to share in accordance with two
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
May 5, 2005
Lecture Notes
Expected Value II 1 The NumberPicking Game
Here is a game that you and I could play that reveals a strange property of expectation. First, you think of
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
May 3, 2005
Lecture Notes
Expected Value I
The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Rou
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
February 17, 2005
Lecture Notes
Number Theory I
Number theory is the study of the integers. Number theory is right at the core of math ematics; even Ug the Caveman surely had so
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
February 24, 2005
Lecture Notes
Number Theory II
Image of Alan Turing removed for copyright reasons.
The man pictured above is Alan Turing, the most important gure in the histor
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 1, 2005 Lecture Notes
Graph Theory 1 Introduction
Informally, a graph is a bunch of dots connected by lines. Here is an example of a graph:
B A F D G I C E H
Sadly, this de
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 3, 2005
Lecture Notes
Graph Theory II 1 Coloring Graphs
Each term, the MIT Schedules Office must assign a time slot for each final exam. This is not easy, because some stu
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 10, 2005
Lecture Notes
Sums and Approximations
When you analyze the running time of an algorithm, the probability some procedure succeeds, or the behavior of a loadbalanci
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 15, 2005 Lecture Notes
Sums, Approximations, and Asymptotics II 1 Block Stacking
How far can a stack of identical blocks overhang the end of a table without toppling over?
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 17, 2005
Lecture Notes
Recurrences
Recursion- breaking an object down into smaller objects of the same type- is a ma jor theme in mathematics and computer science. For exa
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 25, 2005
Lecture Notes
Counting I
20480135385502964448038 489445991866915676240992 1082662032430379651370981 1178480894769706178994993 1253127351683239693851327 1301505129
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
March 31, 2005
Lecture Notes
Counting II
We realize everyone has been working pretty hard this term1 , and were considering awarding some prizes for truly exceptional coursework
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
April 5, 2005 Lecture Notes
Counting III
Today well briey review some facts you dervied in recitation on Friday and then turn to some applications of counting.
1 The Bookkeeper R
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
April 7, 2005
Lecture Notes
Generating Functions
Generating functions are one of the most surprising, useful, and clever inventions in discrete math. Roughly speaking, generatin
6.042/18.062J Mathematics for Computer Science Srini Devadas and Eric Lehman
April 14, 2005
Lecture Notes
Introduction to Probability
Probability is the last topic in this course and perhaps the most important. Many algorithms rely on randomization. Inve