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?

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MAKING CONVINCING ARGUMENTS
0:00:00 Promo video
0:01:15 Proofs
0:05:03 Proof by Example
0:06:54 Impossiblity proof
0:09:29 Impossibility proof, 2 and conclusion
0:13:22 One example is Enough
0:16:49 Splitting an octagon
0:18:13 Making Fun in real life Tensegrities (optional)
0:28:38 Know Your Rights
0:34:28 Nobody can win All the time Nonexisting Examples

HOW TO FIND AN EXAMPLE?
0:42:55 Magic Squares
0:46:22 Narrowing the search
0:53:09 Multiplicative Magic Squares
0:58:26 More Puzzles
1:07:48 Integer linear Combinations
1:13:04 Paths in a Graph
1:17:39 Warm-up
1:23:02 Subset without x and 100-x
1:27:12 Rooks on a chessboard
1:29:52 Knights on a Chessboard
1:35:02 Bishop on a chessboard
1:37:45 Subset without x and 2x
1:44:09 N Queens Brute Force Search
1:54:41 N Queens Backtracking Example
2:02:01 N Queens Backtracking Code
2:09:35 16 Diagonals

RECURSION AND INDUCTION
2:13:19 Recursion
2:23:05 Coin Problem
2:27:51 Hanoi Towers
2:35:17 Introduction,Lines and Triangles Problem
2:45:33 Lines and Triangle Proof by Induction
2:51:33 Connection Points
3:04:33 Odd Points Proof by induction
3:10:03 Sums of Numbers
3:18:55 Bernouli's Inequality
3:26:57 Coins Problem
3:36:42 Cutting a Triangle
3:45:23 Flawed Induction Proofs
3:54:51 Alternating Sum
4:04:00 Examples
4:10:44 Counterexamples
4:15:44 Basic Logic Constructs
4:26:43 If-Then Generalization, Quantification
4:34:54 Reductio ad Absurdum
4:39:26 Balls in Boxes
4:43:32 Numbers in Tables
4:49:06 Pigeonhole Principle
4:51:21 An (-1,0,1) Antimagic Square
4:53:53 Handshakes

INVARIANTS
4:57:00 Double Counting
4:59:50 Homework Assignment'problem
5:01:28 Invariants
5:04:18 More Coffee
5:08:37 Debugging Problem
5:11:12 Termination
5:15:24 Atthur's Books
5:23:45 Even and odd Numbers
5:28:21 Summing up Digits
5:32:58 Switching Signs
5:36:17 Advance Signs Switching

SOLVING A 15-PUZZLE
5:44:48 The rules of 15-puzzle
5:47:59 Permutations
5:57:07 Proof the Diffucult part
6:05:46 Mission Impossiple
6:08:14 Classify a Permutation as Even Odd
6:13:25 Bonus Track Fast Classification
6:15:20 Project The Task
6:16:52 Quiz Hint Why Every Even Permutation is Solvable

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.

⭐ Important Notes ⭐
⌨️ This course is created in collaboration with University of California SAN DIEGO

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