About this course: Mathematical thinking is crucial in all areas of computer science: algorithms, bioinformatics, computer graphics, data science, machine learning, etc. In this course, we will learn the most important tools used in discrete mathematics: induction, recursion, logic, invariants, examples, optimality. We will use these tools to answer typical programming questions like: How can we be certain a solution exists? Am I sure my program computes the optimal answer? Do each of these objects meet the given requirements? In the course, we use a try-this-before-we-explain-everything approach: you will be solving many interactive (and mobile friendly) puzzles that were carefully designed to allow you to invent many of the important ideas and concepts yourself. Prerequisites: 1. We assume only basic math (e.g., we expect you to know what is a square or how to add fractions), common sense and curiosity. 2. Basic programming knowledge is necessary as some quizzes require programming in Python.
Who is this class for: Our intended audience are all people that work or plan to work in IT, starting from motivated high school students. Prerequisites: 1. We assume only basic math (e.g., we expect you to know what is a square or how to add fractions), common sense and curiosity. 2. Basic programming knowledge is necessary as some quizzes require programming in Python.
Created by: University of California San Diego, National Research University Higher School of Economics
Taught by: Alexander S. Kulikov, Visiting Professor
Department of Computer Science and Engineering
Taught by: Michael Levin, Lecturer
Computer Science
Taught by: Vladimir Podolskii, Associate Professor
Computer Science Department
| Basic Info | Course 1 of 5 in the Introduction to Discrete Mathematics for Computer Science Specialization |
| Level | Beginner |
| Commitment | 6 weeks, 2–5 hours/week |
| Language | English |
| How To Pass | Pass all graded assignments to complete the course. |
| User Ratings |
Syllabus
WEEK 1
Making Convincing Arguments
Why some arguments are convincing and some are not? What makes an argument convincing? How to establish your argument in such a way that there is no possible room for doubt left? How mathematical thinking can help with this? In this week we will start digging into these questions. We will see how a small remark or a simple observation can turn a seemingly non-trivial question into an obvious one. Through various examples we will observe a parallel between constructing a rigorous argument and mathematical reasoning.
10 videos, 4 readings
- Video: Proofs?
- LTI 项目: Puzzle: Tile a Chessboard
- Video: Proof by Example
- Video: Impossibility Proof
- Video: Impossibility Proof, II and Conclusion
- 阅读: Slides
- 阅读: Python
- Video: One Example is Enough
- Video: Splitting an Octagon
- Video: Making Fun in Real Life: Tensegrities
- Video: Know Your Rights
- Video: Nobody Can Win All The Time: Nonexisting Examples
- 阅读: Slides
- 阅读: Acknowledgements
Graded: Tiles, dominos, black and white, even and odd
Graded: Puzzle: Two Congruent Parts
Graded: Puzzle: Pluses, Minuses and Splitting
WEEK 2
How to Find an Example?
How can we be certain that an object with certain requirements exist? One way to show this, is to go through all objects and check whether at least one of them meets the requirements. However, in many cases, the search space is enormous. A computer may help, but some reasoning that narrows the search space is important both for computer search and for "bare hands" work. In this module, we will learn various techniques for showing that an object exists and that an object is optimal among all other objects. As usual, we'll practice solving many interactive puzzles. We'll show also some computer programs that help us to construct an example.
16 videos, 5 readings, 1 practice quiz
- Video: Narrowing the Search
- Video: Multiplicative Magic Squares
- Video: More Puzzles
- Video: Integer Linear Combinations
- Video: Paths In a Graph
- 阅读: Slides
- LTI 项目: Puzzle: N Queens (Optional)
- Video: N Queens: Brute Force Search (Optional)
- 阅读: N Queens: Brute Force Solution Code (Optional)
- Video: N Queens: Backtracking: Example (Optional)
- Video: N Queens: Backtracking: Code (Optional)
- 阅读: N Queens: Backtracking Solution Code (Optional)
- LTI 项目: Puzzle: 16 Diagonals (Optional)
- Video: 16 Diagonals (Optional)
- 练习测验: Number of Solutions for the 8 Queens Puzzle (Optional)
- 阅读: Slides (Optional)
- Video: Warm-up
- Video: Subset without x and 100-x
- Video: Rooks on a Chessboard
- Video: Knights on a Chessboard
- Video: Bishops on a Chessboard
- Video: Subset without x and 2x
- 阅读: Slides
Graded: Puzzle: Magic Square 3 times 3
Graded: Puzzle: Different People Have Different Coins
Graded: Puzzle: Free accomodation
Graded: Is there...
Graded: Maximum Number of Two-digit Integers
Graded: Puzzle: Maximum Number of Rooks on a Chessboard
Graded: Puzzle: Maximum Number of Knights on a Chessboard
Graded: Puzzle: Maximum Number of Bishops on a Chessboard
Graded: Puzzle: Subset without x and 2x
WEEK 3
Recursion and Induction
We'll discover two powerful methods of defining objects, proving concepts, and implementing programs — recursion and induction. These two methods are heavily used, in particular, in algorithms — for analysing correctness and running time of algorithms as well as for implementing efficient solutions. You will see that induction is as simple as falling dominos, but allows to make convincing arguments for arbitrarily large and complex problems by decomposing them and moving step by step. You will learn how famous Gauss unexpectedly solved his teacher's problem intended to keep him busy the whole lesson in just two minutes, and in the end you will be able to prove his formula using induction. You will be able to generalize scary arithmetic exercises and then solve them easily using induction.
13 videos, 2 readings
- Video: Coin Problem
- Video: Hanoi Towers
- 阅读: Slides
- Video: Introduction, Lines and Triangles Problem
- Video: Lines and Triangles: Proof by Induction
- Video: Connecting Points
- Video: Odd Points: Proof by Induction
- Video: Sums of Numbers
- Video: Bernoulli's Inequality
- 笔记本: Bernoulli's Inequality
- Video: Coins Problem
- Video: Cutting a Triangle
- Video: Flawed Induction Proofs
- Video: Alternating Sum
- 阅读: Slides
Graded: Largest Amount that Cannot Be Paid with 5- and 7-Coins
Graded: Pay Any Large Amount with 5- and 7-Coins
Graded: Puzzle: Hanoi Towers
Graded: Number of Moves to Solve the Hanoi Towers Puzzle
Graded: Puzzle: Two Cells of Opposite Colors
Graded: Puzzle: Connect Points
Graded: Induction
WEEK 4
Logic
We have already invoked mathematical logic when we discussed how to make convincing arguments by giving examples. This week we will turn mathematical logic full on. We will discuss its basic operations and rules. We will see how logic can play a crucial and indispensable role in creating convincing arguments. We will discuss how to construct a negation to the statement, and you will see how to win an argument by showing your opponent is wrong with just one example called counterexample!. We will see tricky and seemingly counterintuitive, but yet (an unintentional pun) logical aspects of mathematical logic. We will see one of the oldest approaches to making convincing arguments: Reductio ad Absurdum.
10 videos, 2 readings
- Video: Counterexamples
- Video: Basic Logic Constructs
- Video: If-Then Generalization, Quantification
- 阅读: Slides
- Video: Reductio ad Absurdum
- Video: Balls in Boxes
- Video: Numbers in Tables
- Video: Pigeonhole Principle
- Video: An (-1,0,1) Antimagic Square
- Video: Handshakes
- 阅读: Slides
Graded: Puzzle: Always Prime?
Graded: Examples, Counterexamples and Logic
Graded: Puzzle: Balls in Boxes
Graded: Puzzle: Numbers in Boxes
Graded: Puzzle: Numbers on the Chessboard
Graded: Numbers in Boxes
Graded: How to Pick Socks
Graded: Pigeonhole Principle
Graded: Puzzle: An (-1,0,1) Antimagic Square
WEEK 5
Invariants
"There are things that never change". Apart from being just a philosophical statement, this phrase turns out to be an important idea that can actually help. In this module we will see how it can help in problem solving. Things that do not change are called invariants in mathematics. They form an important tool of proving with numerous applications, including estimating running time of programs and algorithms. We will get some intuition of what they are, see how they can look like, and get some practice in using them.
11 videos, 4 readings, 3 practice quizzes
- Video: `Homework Assignment' Problem
- 阅读: Slides
- 练习测验: Coffee with Milk
- Video: Invariants
- 练习测验: More Coffee
- Video: More Coffee
- Video: Debugging Problem
- 阅读: Slides
- 练习测验: Football Fans
- Video: Termination
- Video: Arthur’s Books
- 阅读: Slides
- Video: Even and Odd Numbers
- Video: Summing up Digits
- Video: Switching Signs
- Video: Advanced Signs Switching
- 阅读: Slides
Graded: Puzzle: Sums of Rows and Columns
Graded: 'Homework Assignment' Problem
Graded: 'Homework Assignment' Problem 2
Graded: Chess Tournaments
Graded: Debugging Problem
Graded: Merging Bank Accounts
Graded: Puzzle: Arthur's Books
Graded: Puzzle: Piece on a Chessboard
Graded: Operations on Even and Odd Numbers
Graded: Puzzle: Summing Up Digits
Graded: Puzzle: Switching Signs
Graded: Recolouring Chessboard
WEEK 6
Solving a 15-Puzzle
In this module, we consider a well known 15-puzzle where one needs to restore order among 15 square pieces in a square box. It turns out that the behavior of this puzzle is determined by mathematics: it is solvable if and only if the corresponding permutation is even. We will learn the basic properties of even and odd permutations. The task is to write a program that determines whether a permutation is even or odd. There is also a more difficult bonus task: to write a program that actually computes a solution (sequence of moves) for a given position assuming that this position is solvable.
8 videos, 3 readings, 1 practice quiz
- Video: Permutations
- Video: Proof: The Difficult Part
- Video: Mission Impossible
- Video: Classify a Permutation as Even/Odd
- Video: Bonus Track: Fast Classification
- 阅读: Slides
- Video: Project: The Task
- 阅读: Even permutations
- 阅读: Bonus Track: Finding The Sequence of Moves
- 练习测验: Bonus Track: Algorithm for 15-Puzzle
- Video: Quiz Hint: Why Every Even Permutation Is Solvable
Graded: Puzzle: 15
Graded: Transpositions and Permutations
Graded: Neighbor transpositions
Graded: Is a permutation even?
FAQs
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Creators
University of California San Diego
UC San Diego is an academic powerhouse and economic engine, recognized as one of the top 10 public universities by U.S. News and World Report. Innovation is central to who we are and what we do. Here, students learn that knowledge isn't just acquired in the classroom—life is their laboratory.
National Research University Higher School of Economics
National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more.
Learn more on www.hse.ru
Pricing
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| Access to course materials | Available |
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| Earn a shareable Course Certificate | Available |
Ratings and Reviews
Great introduction to mathematical thinking and how to apply it to computational problems.
An excelent course, I hope that someday in my country the mathematic wil be taught in this way with puzzle and those thing really made the course interesting. this course really gave me good approaches to proof a lot of problem
learned a lot
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KN
Good course.
I gave a 4-star instead of 5 just because of the pace of the lessons. felt it could be bit slower and with more examples, specially in sections related to logic(if-then-else, reductio ad absurdum & permutations & cycles).