About this Course
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Beginner Level


Subtitles: English

Skills you will gain

Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Beginner Level


Subtitles: English

Syllabus - What you will learn from this course

3 hours to complete

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 (Total 43 min), 4 readings, 4 quizzes
10 videos
Proof by Example1m
Impossibility Proof2m
Impossibility Proof, II and Conclusion3m
One Example is Enough3m
Splitting an Octagon1m
Making Fun in Real Life: Tensegrities10m
Know Your Rights5m
Nobody Can Win All The Time: Nonexisting Examples8m
4 readings
1 practice exercise
Tiles, dominos, black and white, even and odd6m
5 hours to complete

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 (Total 90 min), 6 readings, 12 quizzes
16 videos
Narrowing the Search6m
Multiplicative Magic Squares5m
More Puzzles9m
Integer Linear Combinations5m
Paths In a Graph4m
N Queens: Brute Force Search (Optional)10m
N Queens: Backtracking: Example (Optional)7m
N Queens: Backtracking: Code (Optional)7m
16 Diagonals (Optional)3m
Subset without x and 100-x4m
Rooks on a Chessboard2m
Knights on a Chessboard5m
Bishops on a Chessboard2m
Subset without x and 2x6m
6 readings
N Queens: Brute Force Solution Code (Optional)10m
N Queens: Backtracking Solution Code (Optional)10m
16 Diagonals: Code (Optional)10m
Slides (Optional)1m
3 practice exercises
Is there...20m
Number of Solutions for the 8 Queens Puzzle (Optional)20m
Maximum Number of Two-digit Integers2m
6 hours to complete

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 (Total 111 min), 3 readings, 8 quizzes
13 videos
Coin Problem4m
Hanoi Towers7m
Introduction, Lines and Triangles Problem10m
Lines and Triangles: Proof by Induction5m
Connecting Points12m
Odd Points: Proof by Induction5m
Sums of Numbers8m
Bernoulli's Inequality8m
Coins Problem9m
Cutting a Triangle8m
Flawed Induction Proofs9m
Alternating Sum9m
3 readings
Two Cells of Opposite Colors: Hints10m
5 practice exercises
Largest Amount that Cannot Be Paid with 5- and 7-Coins10m
Pay Any Large Amount with 5- and 7-Coins20m
Number of Moves to Solve the Hanoi Towers Puzzle30m
Two Cells of Opposite Colors: Feedback
3 hours to complete


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 (Total 53 min), 2 readings, 9 quizzes
10 videos
Basic Logic Constructs10m
If-Then Generalization, Quantification8m
Reductio ad Absurdum4m
Balls in Boxes4m
Numbers in Tables5m
Pigeonhole Principle2m
An (-1,0,1) Antimagic Square2m
2 readings
4 practice exercises
Examples, Counterexamples and Logic14m
Numbers in Boxes5m
How to Pick Socks5m
Pigeonhole Principle10m
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Top reviews from Mathematical Thinking in Computer Science

By ADMar 26th 2019

The teachers are informative and good. They explain the topic in a way that we can easily understand. The slides provide all the information that is needed. The external tools are fun and informative.

By JVOct 16th 2017

I really liked this course, it's a good introduction to mathematical thinking, with plenty of examples and exercises, I also liked the use of other external graphical tools as exercises.



Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

Michael Levin

Computer Science

Vladimir Podolskii

Associate Professor
Computer Science Department

About 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....

About 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 communicamathematics, engineering, and more. Learn more on www.hse.ru...

About the Introduction to Discrete Mathematics for Computer Science Specialization

Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and financial analysts (it is not a coincidence that math puzzles are often used for interviews). We cover the basic notions and results (combinatorics, graphs, probability, number theory) that are universally needed. To deliver techniques and ideas in discrete mathematics to the learner we extensively use interactive puzzles specially created for this specialization. To bring the learners experience closer to IT-applications we incorporate programming examples, problems and projects in our courses....
Introduction to Discrete Mathematics for Computer Science

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