Impartial Chess


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Ichess 1: King, K           

The first video deals with the simplest ichess game, a king. We begin by showing a game actually played by two students. We then analyze the game. The viewer is led to discover how the locations on the board can be partitioned into those from which the previous player can win, as contrasted from those from which the next player can win.


In this video, we deal with a knight. This game is somewhat more complicated than the king, but we again lead the viewer through a complete analysis.


This video considers the sum of two such games in which each handicapped chessman occupies his own separate board. This leads to the insight that a win on each board separately ensures overall victory. This is later seen as an illustration of the fact that zero can be viewed as a mathematical abstraction, which still satisfies the familiar equation, 0 + 0 = 0.


This video introduces the baby rook, a piece whose location and handicaps allow him to move in only one direction. By playing the sum of the baby rook with another piece, the bishop, we discover a concept now called Grundy number after its discoverer, P. M. Grundy. The investigation of its properties leads to a formula for computing it.


When this formula is applied recursively to the lone King, we find two dimensional tables whose values reveal patterns which are periodic. This table provides the winning strategy for the game which is the sum of the King and the baby rook.


The computations of the the Knight's Grundy table turn out to be somewhat more subtle and complicated than the King's. It involves considerable drill, but is motivated by the goal of finding the winning strategy to the game which is the sum of the knight and the baby rook. Some viewers may choose to accept the result of these calculations without checking all of the details.


Ichess 7: K + N Revisited

We are now able to analyze all possible positions in K+N, including those which occurred in the students' game shown in Ichess 3. As in Ichess 4, we discover that the winning strategy when playing the sum of two impartial games is to move to a position which matches their Grundy numbers.


In this video we compute R[x,y], the Grundy table for the grownup rook. At first glance, it might appear more complicated than either K{x,y] or N[x,y]


A close examination of the rook's Grundy table leads us to the binary representation of numbers, and then to the simple formula first discovered by Charles Bouton in 1901.


We find how Bouton's theorem solves the sum of any number of independent green chessmen, as well as the solution of the popular 19th century game called NIM.




This video presents a collection of ichess problems, with a wide range of difficulties. Like real world problems, the difficulty of any problem might not be apparent until one starts working on it. Some problems have elegant solutions, but some might not.



This video presents solutions to the problems involving a single lone chessman.


When multiple green chessmen occupy the same board, complications arise because some of their moves may be blocked by other chessmen. Yet, in some interesting cases, the problem can be decomposed in such a way that these complications are greatly reduced or eliminated.


This video provides a complete analysis of how the Grundy number for a lone knight near the Northern or Eastern edge of a rectangular board may differ from the Grundy number on an infinite quarter-plane. The lesson shows that some simply stated problems need not have simply stated solutions.