If you find a mistake, omission, etc., please let me know by e-mail.
Puzzle followups (Abraham et al.: 59 or 60 moves for Z.Khalid's question, depending on whether KQ switch places; Jie et al.: partial progress on my 2005 challenge and the 8-officer task)
The ``Zermelo'' (retrograde analysis) algorithm, and some variations and consequences:
October 4: Graph theory of the chessboard: distance functions, maximal (co)cliques, domination numbers, and distance functions for the King, Queen, Rook, Bishop, and Knight. Board symmetries: 8, if we ignore Castling (as in the KRR/K puzzle) and one-way pawns; 2·8!2 for the Rook; how many for the Bishop? Some chess applications; enumeration of maximal configurations, including some open problems (King domination and cocliques on rectangular boards of arbitrary size). Aside: Euler paths and Euler's formula V-E+F=2; application to non-planarity of complete graph K5, complete bipartite graph K3,3, and the Petersen graph.
[Oct.11 was Columbus Day]
October 18: Dynamical programming for enumerative problems:
Puzzle followups: An enumerative problem with 2005 solutions; attempts to create a problem with 6n-1 solutions in n moves (we already have 2n-1 [B.Young] and can easily generalize to 3n-1, 4n-1, and 5n-1);
Meet-in-the-middle tricks (sometimes with sorting n objects in time n log(n)), and applications: solving Rubik's cube; the sum of two consecutive Fibonacci squares is an even-numbered Fibonacci number (e.g. 52+82=89 -- what's the corresponding result for odd-numbered Fibonaccis?); the sum of squares of entries in the n-th row of Pascal's triangle is Binomial(2n,n) (e.g. 12+42+62+42+12=70).
Combinatorial Game Theory: Nim, CGT values, pawn endgames, the ``upstart [Up-Star] identity'', the mex rule, Misère Nim.
An introduction to various genres of chess problems and notions of soundness; how one might modify the ``Zermelo'' approach to list all sound problems (under various definitions), or those with features like setplay, twins, etc.
Also: Introduction to asymptotics; some asymptotic chess questions.
Catalan numbers and some chess problems where they arise; more problem variants: combinative separation; Schrödinger's Cat mates in 2; Grasshopper and Maximummer.
Parity proof games and their construction; final project discussion.
Theory of Computation seminar on ``Exponentially long chess problems on large boards''.
Also: the golden ratio as an eigenvalue, and its appearance in the formula for Fibonacci numbers as an illustration of solving general linear recurrences. (Asides on the golden ratio in the regular pentagon(!) and on the (-n)th Fibonacci number.)
Initial progress reports on the final projects.
Also: two classic Babson-task problems; King and multi-leaper vs. King; a one-sided proof game with a unique solution that moves all eight officers; introduction to Kasteleyn's permanent-determinant method for enumerating domino tilings of checkerboard chunks [it works more generally for any bipartite planar graph].
Richard Stanley's guest lecture on queue problems.
Also: a few more progress reports; a Mate in 4 problem based on the Jelliss-Beasley Knight hypercube.
More progress reports; also: the sign of a permutation, and applications to Loyd's 15-puzzle, Rubik's cube, and generalizations.