Depth by The Rule

This note corrects a previous treatment of algorithms for the metric DTR, Depth by the Rule

Space-efficient indexing of endgame tables for chess

Chess endgame tables should provide efficiently the value and depth of any required position during play. The indexing of an endgame’s positions is crucial to meeting this objective. This paper updates Heinz’ previous review of approaches to indexing and describes the latest approach by the first and third authors. Heinz’ and Nalimov’s endgame tables (EGTs) encompass the en passant rule and have the most compact index schemes to date. Nalimov’s EGTs, to the Distance-to-Mate (DTM) metric, require only 30.6 × 109 elements in total for all the 3-to-5-man endgames and are individually more compact than previous tables. His new index scheme has proved itself while generating the tables and in the 1999 World Computer Chess Championship where many of the top programs used the new suite of EGTs

Strategies for Constrained Optimisation

The latest 6-man chess endgame results confirm that there are many deep forced mates beyond the 50-move rule. Players with potential wins near this limit naturally want to avoid a claim for a draw: optimal play to current metrics does not guarantee feasible wins or maximise the chances of winning against fallible opposition. A new metric and further strategies are defined which support players’ aspirations and improve their prospects of securing wins in the context of a k-move rule

Data assurance in opaque computations

The chess endgame is increasingly being seen through the lens of, and therefore effectively defined by, a data ‘model’ of itself. It is vital that such models are clearly faithful to the reality they purport to represent. This paper examines that issue and systems engineering responses to it, using the chess endgame as the exemplar scenario. A structured survey has been carried out of the intrinsic challenges and complexity of creating endgame data by reviewing the past pattern of errors during work in progress, surfacing in publications and occurring after the data was generated. Specific measures are proposed to counter observed classes of error-risk, including a preliminary survey of techniques for using state-of-the-art verification tools to generate EGTs that are correct by construction. The approach may be applied generically beyond the game domain

Space-efficient Indexing of Chess Endgame Tables

Chess endgame tables should provide efficiently the value and depth of any required position during play. The indexing of an endgame’s positions is crucial to meeting this objective. This paper updates Heinz’ previous review of approaches to indexing and describes the latest approach by the first and third authors. Heinz’ and Nalimov’s endgame tables (EGTs) encompass the en passant rule and have the most compact index schemes to date. Nalimov’s EGTs, to the Distance-to-Mate (DTM) metric, require only 30.6 × 10^9 elements in total for all the 3-to-5-man endgames and are individually more compact than previous tables. His new index scheme has proved itself while generating the tables and in the 1999 World Computer Chess Championship where many of the top programs used the new suite of EGTs

KQQKQQ and the Kasparov-World Game

The 1999 Kasparov-World game for the first time enabled anyone to join a team playing against a World Chess Champion via the web. It included a surprise in the opening, complex middle-game strategy and a deep ending. As the game headed for its mysterious finale, the World Team re-quested a KQQKQQ endgame table and was provided with two by the authors. This paper describes their work, compares the methods used, examines the issues raised and summarises the concepts involved for the benefit of future workers in the endgame field. It also notes the contribution of this endgame to chess itself

Chess endgames: 6-man data and strategy

While Nalimov’s endgame tables for Western Chess are the most used today, their Depth-to-Mate metric is not the most efficient or effective in use. The authors have developed and used new programs to create tables to alternative metrics and recommend better strategies for endgame play

Analyzing and Computing Complete Solution for Dots and Boxes Game

This thesis improves a process that analyzes all the states of a game of Dots and Boxes. We use retrograde analysis and simulations to create a solution that provides significant performance improvements over our previous best solution. Expanding upon a previous 4x4 solution using rotations, reflections, better optimization, and cloud computing to limit the processing time and gather more data efficiently. We compute a file and the number of states associated with each file and process every state starting with a completely filled board. We optimized the data for cloud computing by running simulations to find the most efficient number of processors and assess potential bottlenecks. The data produced from the results will be able to provide solutions and optimal play for dots and boxes games of different dimensions

Computer-Assisted Proving of Combinatorial Conjectures Over Finite Domains: A Case Study of a Chess Conjecture

There are several approaches for using computers in deriving mathematical proofs. For their illustration, we provide an in-depth study of using computer support for proving one complex combinatorial conjecture -- correctness of a strategy for the chess KRK endgame. The final, machine verifiable, result presented in this paper is that there is a winning strategy for white in the KRK endgame generalized to $n \times n$ board (for natural $n$ greater than $3$). We demonstrate that different approaches for computer-based theorem proving work best together and in synergy and that the technology currently available is powerful enough for providing significant help to humans deriving complex proofs