Related papers: New directions in enumerative chess problems
Combinatorial Game Theory has also been called `additive game theory', whenever the analysis involves sums of independent game components. Such {\em disjunctive sums} invoke comparison between games, which allows abstract values to be…
In many multiagent environments, a designer has some, but limited control over the game being played. In this paper, we formalize this by considering incompletely specified games, in which some entries of the payoff matrices can be chosen…
Berlekamp proposed a class of impartial combinatorial games based on the moves of chess pieces on rectangular boards. We generalize impartial chess games by playing them on Young diagrams and obtain results about winning and losing…
The game of best choice (or "secretary problem") is a model for making an irrevocable decision among a fixed number of candidate choices that are presented sequentially in random order, one at a time. Because the classically optimal…
This article concerns the resolution of impartial combinatorial games, and in particular games that can be split in sums of independent positions. We prove that in order to compute the outcome of a sum of independent positions, it is always…
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…
Combinatorial Scoring games, with the property `extra pass moves for a player does no harm', are characterized. The characterization involves an order embedding of Conway's Normal-play games. Also, we give a theorem for comparing games with…
A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erd\H{o}s and J. W. Moon. By probabilistic methods, the existence of "unrankable" tournaments was proved. On the other hand, they…
We consider the permutation analogue of Penney's game for words. Two players, in order, each choose a permutation of length $k\ge3$; then a sequence of independent random values from a continuous distribution is generated, until the…
This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results characterizing…
A chessboard has the property that every row and every column has as many white squares as black squares. In this mostly methodological note, we address the problem of counting such rectangular arrays with a fixed (numeric) number of rows,…
Chore division is the problem of fairly dividing some divisible, undesirable bad, such as a set of chores, among a number of players. Each player has their own valuation of the chores, and must be satisfied they did not receive more than…
We study a family of sorting match puzzles on grids, which we call permutation match puzzles. In this puzzle, each row and column of a $n \times n$ grid is labeled with an ordering constraint -- ascending (A) or descending (D) -- and the…
We analyze Solo Chess puzzles, where the input is an $n \times n$ board containing some standard Chess pieces of the same color, and the goal is to make a sequence of capture moves to reduce down to a single piece. Prior work analyzes this…
Reid conjectured that any finite set of non-negative integers is the score set of some tournament and Yao gave a non-constructive proof of Reid's conjecture using arithmetic arguments. No constructive proof has been found since. In this…
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…
The move-minimizing puzzles presented here are certain types of one-player combinatorial games that are shown to have explicit solutions whenever they can be encoded in a certain way as diamond-colored modular or distributive lattices. Our…
We consider the chessboard pebbling problem analyzed by Chung, Graham, Morrison and Odlyzko [3]. We study the number of reachable configurations $G(k)$ and a related double sequence $G(k,m)$. Exact expressions for these are derived, and we…
Chess and chance are seemingly strange bedfellows. Luck and/or randomness have no apparent role in move selection when the game is played at the highest levels. However, when competition is at the ultimate level, that of the World Chess…
We introduce a game on graphs. By a theorem of Zermelo, each instance of the game on a finite graph is determined. While the general decision problem on which player has a winning strategy in a given instance of the game is unsolved, we…