Combinatorics: The Science of Counting and Arrangements
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How does combinatorics address advanced counting and arrangements problems? What are the combinatorics of magic squares? Must we always enumerate all possibilities or can we reason to count faster? How can combinatorics help us count the number of passwords available to smart phone users? What is Archimedes' Stomachion? How many rounds are needed to determine a winner in an elimination tournament? In a graph how many ways are there to traverse all its edges? What are the two main goals of counting in combinatorics?
These questions come from the first week of Ma Yu Chun's free on-line TsinghuaX 60240013.x Combinatorial Mathematics (https://www.edx.org/course/combinatorial-mathematics-tsinghuax-60240013x-0) on the edX (http://edx.org) platform (Note: in Wikipedia it is reported that Tsinghua University (https://en.wikipedia.org/wiki/Tsinghua_University) is sometimes called the "MIT of China" due to its specialty in engineering and science).
To help guide the discussion, please download and work out answers to the 14 exercises in the problem set for the 19 September 2015 meetup on Combinatorics (http://files.meetup.com/3948532/combinatorics.Sep2015.problems.pdf). Most of the problems are easy, but a few will require careful thought and others will require watching the videos. If you have questions or issues with any of the problems, please post a comment below.
The following 10 videos totaling just over 1½ hours will form the basis for our meetup discussion on combinatorics.
• 2 minute teaser video for Ma Yu Chun's Combinatorial Mathematics: it gives a good overview of the scope of combinatorics
https://www.youtube.com/watch?v=i0egb-QskVs
• 5½ minute Ma Yu Chun video introducing the subject of combinatorics in the context of the whole of mathematics
https://www.youtube.com/watch?v=BZ-O9E4__AM
• 17 minute Ma Yu Chun video introducing the history of combinatorics through the history of magic squares
http://www.youtube.com/watch?v=7FUJOscizOQ
• 6 minute Ma Yu Chun video exploring more history of combinatorics through the example of Archimedes' Stomachion
http://www.youtube.com/watch?v=1ya9PSpH1D4
• 8 minute Ma Yu Chun video exploring smart phone password security as an example of combinatory analysis
http://www.youtube.com/watch?v=U7HCYchSrMw
• 2 minute Ma Yu Chun video introducing the key idea of enumeration in combinatorics through the example of counting the number of rounds needed in an elimination tournament
http://www.youtube.com/watch?v=g1VD2xoMYaE
• 2 minute Ma Yu Chun video solving the elimination tournament problem without enumeration (can you solve the problem before watching the video?)
http://www.youtube.com/watch?v=Glbf65VY9Ns
• 9 minute Ma Yu Chun video introducing the famous Bridges of Königsberg problem
http://www.youtube.com/watch?v=j1LhQ7-3yS8
• 1 minute Ma Yu Chun video summarizing the importance of combinatorics
http://www.youtube.com/watch?v=-_f8MUx9li0
• 41 minute Norman Wildberger video surveying the history of combinatorics
Combinatorics: The Science of Counting and Arrangements