Effective player guidance in logic puzzles

Pen & paper puzzle games are an extremely popular pastime, often enjoyed by demographics normally not considered to be ‘gamers’. They are increasingly used as ‘serious games’ and there has been extensive research into computationally generating and efficiently solving them. However, there have been few academic studies that have focused on the players themselves. Presenting an appropriate level of challenge to a player is essential for both player enjoyment and engagement. Providing appropriate assistance is an essential mechanic for making a game accessible to a variety of players. In this thesis, we investigate how players solve Progressive Pen & Paper Puzzle Games (PPPPs) and how to provide meaningful assistance that allows players to recover from being stuck, while not reducing the challenge to trivial levels. This thesis begins with a qualitative in-person study of Sudoku solving. This study demonstrates that, in contrast to all existing assumptions used to model players, players were unsystematic, idiosyncratic and error-prone. We then designed an entirely new approach to providing assistance in PPPPs, which guides players towards easier deductions rather than, as current systems do, completing the next cell for them. We implemented a novel hint system using our design, with the assessment of the challenge being done using Minimal Unsatisfiable Sets (MUSs). We conducted four studies, using two different PPPPs, that evaluated the efficacy of the novel hint system compared to the current hint approach. The studies demonstrated that our novel hint system was as helpful as the existing system while also improving the player experience and feeling less like cheating. Players also chose to use our novel hint system significantly more often. We have provided a new approach to providing assistance to PPPP players and demonstrated that players prefer it over existing approaches

SudoQ -- a quantum variant of the popular game

We introduce SudoQ, a quantum version of the classical game Sudoku. Allowing the entries of the grid to be (non-commutative) projections instead of integers, the solution set of SudoQ puzzles can be much larger than in the classical (commutative) setting. We introduce and analyze a randomized algorithm for computing solutions of SudoQ puzzles. Finally, we state two important conjectures relating the quantum and the classical solutions of SudoQ puzzles, corroborated by analytical and numerical evidence.Comment: Python code and examples available at https://github.com/inechita/Sudo

Colorings and Sudoku Puzzles

Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\ obner bases in multivariate polynomial rings. We utilize Gr\ obner bases of an ideal of a multivariate polynomial ring over a finite field to solve coloring, Shidoku, and Sudoku problems. In the last section, we note Gr\ obner bases are also well-suited to hypergraph coloring problems

A constraint solver for software engineering : finding models and cores of large relational specifications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 105-120).Relational logic is an attractive candidate for a software description language, because both the design and implementation of software often involve reasoning about relational structures: organizational hierarchies in the problem domain, architectural configurations in the high level design, or graphs and linked lists in low level code. Until recently, however, frameworks for solving relational constraints have had limited applicability. Designed to analyze small, hand-crafted models of software systems, current frameworks perform poorly on specifications that are large or that have partially known solutions. This thesis presents an efficient constraint solver for relational logic, with recent applications to design analysis, code checking, test-case generation, and declarative configuration. The solver provides analyses for both satisfiable and unsatisfiable specifications--a finite model finder for the former and a minimal unsatisfiable core extractor for the latter. It works by translating a relational problem to a boolean satisfiability problem; applying an off-the-shelf SAT solver to the resulting formula; and converting the SAT solver's output back to the relational domain. The idea of solving relational problems by reduction to SAT is not new. The core contributions of this work, instead, are new techniques for expanding the capacity and applicability of SAT-based engines. They include: a new interface to SAT that extends relational logic with a mechanism for specifying partial solutions; a new translation algorithm based on sparse matrices and auto-compacting circuits; a new symmetry detection technique that works in the presence of partial solutions; and a new core extraction algorithm that recycles inferences made at the boolean level to speed up core minimization at the specification level.by Emina Torlak.Ph.D