Ichess 00:

Overview for Parents and Teachers

Combinatorial Game Theory, CGT, is a branch of mathematics which is popular among some math professors and graduate students.  The heart of CGT is the decomposition of sums of positions into simpler positions.  These powerful techniques provide precise and rigorous analyses of numerous endgame positions in Dots-and-Boxes, Amazons, Go, and many other games, several of which are

introduced and explored in other videos on this website. 


CGT requires virtually no prerequisites.   For this reason, introductions to it can enrich the mathematical experience of high school and junior high school students, and even some students in the upper elementary grades.  


Historically, CGT began in 1901, when Charles Bouton, a Harvard math professor, published his solution to a popular 19th century game called NIM. In the next half-century, P. M. Grundy and others extended this theory to a larger class called impartial games.  In the late twentieth century

it was extended to a much broader class of games.


Bouton's theorem remains the fundamental result of impartial game theory. The conventional pedagogical approach pulls this result out of thin air, and then focusses entirely on its proof, which turns out to be relatively straightforward. In contrast, this sequence of videos takes a different approach. It focuses much more on discovery.  It begins with ichess 1 and ichess 2, each of

which features a single green chessman whose mobility is restricted so that he can be moved only in certain directions. The two players take turns moving this same chessman until he becomes immobilized on or near a certain corner or edge of the board.  The winner is the last player who has been able to move him.


Each video is short.  Most of them are under ten minutes, but some important facts are covered rather quickly. I recommend that novices see no more than one new video per day, rewinding and watching some portions again and again. Repetition is a powerful tool for learning. I strongly recommend it.  


Ichess 3-10 continue the analyses of ichess 1 and ichess 2 in increasing depth, leading to the discovery of Bouton's theorem and some of its consequences.  

These results have broader ramifications.  They play important roles in many games which are not impartial. These include Dots & Boxes and Hackenbush. After ichess 10, some viewers may choose to interrupt the ichess sequence and switch over to the advanced videos on those games.  Ichess 11-14 are included for those viewers who instead yearn for more advanced problems on impartial chess.