Extra-Sliding Puzzles: Experiments in Cognitive Modeling and Representation for Problem Solving
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Permutation Puzzles: a Mathematical Perspective Lecture Notes
Permutation Puzzles: A Mathematical Perspective 15 Puzzle, Oval Track, Rubik’s Cube and Other Mathematical Toys Lecture Notes Jamie Mulholland Department of Mathematics Simon Fraser University c Draft date May 7, 2012 Contents Contents i Preface ix Greek Alphabet xi 1 Permutation Puzzles 1 1.1 Introduction . .1 1.2 A Collection of Puzzles . .2 1.2.1 A basic game, let’s call it Swap ................................2 1.2.2 The 15-Puzzle . .4 1.2.3 The Oval Track Puzzle (or TopSpinTM)............................5 1.2.4 Hungarian Rings . .7 1.2.5 Rubik’s Cube . .8 1.3 Which brings us to the Definition of a Permutation Puzzle . 12 1.4 Exercises . 12 2 A Bit of Set Theory 15 2.1 Introduction . 15 2.2 Sets and Subsets . 15 2.3 Laws of Set Theory . 16 2.4 Examples Using Sage . 17 2.5 Exercises . 19 3 Permutations 21 3.1 Permutation: Preliminary Definition . 21 3.2 Permutation: Mathematical Definition . 23 3.2.1 Functions . 23 3.2.2 Permutations . 24 3.3 Composing Permutations . 26 i ii CONTENTS 3.4 Associativity of Permutation Composition . 28 3.5 Inverses of Permutations . 29 3.5.1 Inverse of a Product . 31 3.5.2 Cancellation Property . 32 3.6 The Symmetric Group Sn ........................................ 33 3.7 Rules for Exponents . 33 3.8 Order of a Permutation . 34 3.9 Exercises . 35 4 Permutations: Cycle Notation 37 4.1 Permutations: Cycle Notation . 37 4.2 Products of Permutations: Revisited . 39 4.3 Properties of Cycle Form . 40 4.4 Order of a Permutation: Revisited . -
Gaming in Education: Using Games As a Support Tool to Teach History
Journal of Education and Practice www.iiste.org ISSN 2222-1735 (Paper) ISSN 2222-288X (Online) Vol.8, No.15, 2017 Gaming in Education: Using Games as a Support Tool to Teach History Victor Samuel Zirawaga * Adeleye Idowu Olusanya Tinovimbanashe Maduku Faculty of Applied Science, Cyprus International University, Nicosia, Turkey Abstract The use of current and emerging tools in education is becoming a blistering topic among educators and educational institutions. Gaming in education may be viewed as an interference to learning but its role in education is to increase students’ motivation and engagement, to enhance visual skills, to improve students’ interaction and collaboration abilities with their peers and to enable them to apply gaming values in a real-world situation. Educational technology is being used to simplify and improve learning but using technology in education does not directly impact student achievement as the technology tools have to be in line with the curriculum for them to be effective. This paper discusses the implementation and use of gaming applications in teaching History, a subject which is mainly concerned about facts, by highlighting the role games have in education. The design of games such as word search, crossword, jigsaw puzzle, brain teasers and sliding puzzle using an open source tool called ProProfs is also discussed in this paper. Keywords: Achievement, Educational technology, Gaming, Simulations 1. Introduction A game is a type of play where participants follow defined rules. (Houghton et al., 2013) discusses educational games as the utilization of games to support teaching and learning. Games can be used as a support tool to complement traditional teaching methods to improve the learning experience of the learners while also teaching other skills such as following rules, adaptation, problem solving, interaction, critical thinking skills, creativity, teamwork, and good sportsmanship. -
Permutation Puzzles: a Mathematical Perspective Lecture Notes
Permutation Puzzles: A Mathematical Perspective 15 Puzzle, Oval Track, Rubik’s Cube and Other Mathematical Toys Lecture Notes Jamie Mulholland Department of Mathematics Simon Fraser University c Draft date June 30, 2016 Contents Contents i Preface vii Greek Alphabet ix 1 Permutation Puzzles 1 1.1 Introduction . .1 1.2 A Collection of Puzzles . .2 1.3 Which brings us to the Definition of a Permutation Puzzle . 10 1.4 Exercises . 12 2 A Bit of Set Theory 15 2.1 Introduction . 15 2.2 Sets and Subsets . 15 2.3 Laws of Set Theory . 16 2.4 Examples Using SageMath . 18 2.5 Exercises . 19 3 Permutations 21 3.1 Permutation: Preliminary Definition . 21 3.2 Permutation: Mathematical Definition . 23 3.3 Composing Permutations . 26 3.4 Associativity of Permutation Composition . 28 3.5 Inverses of Permutations . 29 3.6 The Symmetric Group Sn ........................................ 33 3.7 Rules for Exponents . 33 3.8 Order of a Permutation . 35 3.9 Exercises . 36 i ii CONTENTS 4 Permutations: Cycle Notation 39 4.1 Permutations: Cycle Notation . 39 4.2 Products of Permutations: Revisited . 41 4.3 Properties of Cycle Form . 42 4.4 Order of a Permutation: Revisited . 43 4.5 Inverse of a Permutation: Revisited . 44 4.6 Summary of Permutations . 46 4.7 Working with Permutations in SageMath . 46 4.8 Exercises . 47 5 From Puzzles To Permutations 51 5.1 Introduction . 51 5.2 Swap ................................................... 52 5.3 15-Puzzle . 54 5.4 Oval Track Puzzle . 55 5.5 Hungarian Rings . 58 5.6 Rubik’s Cube . -
15 Puzzle Reduced
The 15 Puzzle As pictured below the 15 Puzzle has 15 numbered wooden tiles in a 4x4 box, with one empty space in the bottom right corner. The object was to rearrange the 15 pieces in any order and then use the empty space to slide the tiles back into the original numerical order. When the puzzle appeared around the year 1870 it created as big a craze as the Rubik's Cube did a hundred years later. In later versions it was called the GEM puzzle Joseph Eitel!Page 1 of 29!amagicclassroom.com The FAD of the 1870’s Sliding puzzles started with a bang in the late1800’s. The Fifteen Puzzle started in the United States around 1870 and it spread quickly. Owing to the uncountable number of devoted players it had conquered, it became a plague. In a matter of months after its introduction, people all over the world were engrossed in trying to solve what came to be known as the 15 Puzzle. It can be argued that the 15 Puzzle had the greatest impact on American and European society in 1880 of any mechanical puzzle the world has ever known. Factories could not keep up with the demand for the 15 Puzzle. In offices an shops bosses were horrified by their employees being completely absorbed by the game during office hours. Owners of entertainment establishments were quick to latch onto the rage and organized large contests. The game had even made its way into solemn halls of the German Reichstag. “I can still visualize quite clearly the grey haired people in the Reichstag intent on a square small box in their hands,” recalls the mathematician Sigmund Gunter who was a deputy during puzzle epidemic. -
Photo Title Ages Players Playing Time About
Photo Title Ages Players Playing Time About It seems easy to name 3 ice cream flavors, but can you do it with time twisting down quickly? Just say 5 Second Rule Jr. 6+ 3-6 N/A what comes to mind...if it's not a right answer, maybe it'll be a funny one. A family version of 5 Second Rule. Read a card and give an answer - but you have to do it in 5 seconds. Aboriginal Tucker 12+ 1+ N/A N/A Puzzle (NEW) Think fast! In this quick-paced, open-ended, vocabulary-building game, players must nimbly name a word that matches one of their category cards AND begins with the letter in play on the board. Think fast After Words 8+ 2-4 N/A before the timer runs out, and be the first to discard all of your cards to win in this rapid recall game! Perfect for family game night, AfterWORDS encourages quick-thinking skills, helps build vocabulary, and promotes good sportsmanship for 2-4 players! A fun way for young children to develop letter recognition, matching skills, color identification, and fine Alpha Pops 2+ 1 N/A motor skills. Alphabet Match Me 4 + 1 N/A A matching, memory and word or picture recognition game. Cards Animal Habitats 3+ 1 N/A Match the animal to its correct habitat! Memory Match The animals want to show how good they are at making tall pyramids! Each turn a player rolls the die and either places one or two animals on to the stack of animals, passes one of his or her animals to Animal Upon another player for them to place, or places an animal on the table, extending the base for other players 4+ 2-4 15 min. -
Read Ebook \\ Mechanical Toys / 4XGV1KFZXPM0
KC5LEBOZV84C « eBook > Mechanical toys Mech anical toys Filesize: 2.96 MB Reviews Very beneficial to all category of folks. I really could comprehended every little thing out of this created e publication. I found out this book from my dad and i encouraged this ebook to discover. (Maia O'Hara) DISCLAIMER | DMCA M3XMACUOUKBF / Kindle ^ Mechanical toys MECHANICAL TOYS To get Mechanical toys eBook, make sure you follow the web link beneath and save the file or gain access to other information that are highly relevant to MECHANICAL TOYS ebook. Reference Series Books LLC Okt 2012, 2012. Taschenbuch. Book Condition: Neu. 247x190x10 mm. Neuware - Source: Wikipedia. Pages: 78. Chapters: Mechanical puzzles, Soma cube, Tower of Hanoi, Flexagon, Rubik's Cube, Jigsaw puzzle, Peg solitaire, Combination puzzle, Speedcubing, Toy train, Rubik's Revenge, Fieen puzzle, Professor's Cube, Klotski, Square One, V-Cube 6, V-Cube 7, Pyramorphix, Megaminx, Rubik's Magic, Dogic, Cymbal- banging monkey toy, Helicopter Cube, Pyraminx, Rubik's Snake, Impossible bottle, Pocket Cube, Snake cube, Pyraminx Crystal, Happy Cube, Rush Hour, Impossiball, Skewb Ultimate, BrainTwist, Rubik's Clock, Mechanical toy, Sliding puzzle, Disentanglement puzzle, Secret decoder ring, Equilibrium, Rubik's Magic: Master Edition, Sudoku Cube, Rubik's 360, Puzzle jug, Alexander's Star, Minus Cube, Puzzle ring, Wind-up toy, Skewb Diamond, Baguenaudier, Nintendo tumbler puzzle, Burr puzzle, Egg of Columbus, Magnet Space Wheel, Eastsheen, Hoppers, Snapper Puzzle, Puzzle globe, Think-a-Dot, Rubik's Triamid, Puzzle box, Missing Link, Pyramid puzzle, Diabolical cube, Lock puzzle, Moo box, Yoshimoto Cube, Human knot, Fuddling cup, Mechanical bank, Puzzle jewelry, Gridlock, Libelle. -
Procedural Generation of Narrative Puzzles
Procedural Generation of Narrative Puzzles by Barbara De Kegel, B.Sc. (Hons) Dissertation Presented to the University of Dublin, Trinity College in fulfillment of the requirements for the Degree of Master of Science in Computer Science (Interactive Entertainment Technology) University of Dublin, Trinity College September 2016 Declaration I, the undersigned, declare that this work has not previously been submitted as an exercise for a degree at this, or any other University, and that unless otherwise stated, is my own work. Barbara De Kegel August 30, 2016 Permission to Lend and/or Copy I, the undersigned, agree that Trinity College Library may lend or copy this thesis upon request. Barbara De Kegel August 30, 2016 Acknowledgments Firstly I want to thank my supervisor Mads Haahr for his advice and the interesting discussions we had over the course of this dissertation. I also want to thank the IET class, in particular Patrick and Dan, for their camaraderie and their patience with my endless questions, and Shane, for supporting me at the end. Finally I want to thank my family for their endless support throughout my education. Barbara De Kegel University of Dublin, Trinity College September 2016 iv Procedural Generation of Narrative Puzzles Barbara De Kegel University of Dublin, Trinity College, 2016 Supervisor: Mads Haahr Narrative puzzles involve exploration, logical thinking and progressing a story. This project proposes a system for the procedural generation of such puzzles for use in story-rich games or games with large open worlds. An extended type of context-free grammar forms the basis for both the generation algorithm and the puzzle solving. -
Algorithmic Combinatorial Game Theory
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P = NP): one-player 6 puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
Exploring Evangelical Video Game Worlds Vincent Gonzalez A
Born-Again Digital: Exploring Evangelical Video Game Worlds Vincent Gonzalez A dissertation submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Ph.D. in the Religious Studies department in the College of Arts and Sciences. Chapel Hill 2014 Approved by: Yaakov Ariel Jason Bivins Jonathan Boyarin David Morgan Barry Saunders Randall Styers ©2014 Vincent Gonzalez ALL RIGHTS RESERVED ii ABSTRACT Vincent Gonzalez: Born-Again Digital: Exploring Evangelical Video Game Worlds (Under the direction of Yaakov Ariel) Evangelical Christians have been creating video games for over thirty years, outpacing the efforts of all other religions. By the count that guides the present study, 773 games were made for religious audiences through 2010, of which 474 identify their affiliation as only “Christian,” or “biblical.” Like other artifacts of digital religion, these games allow us to see the entanglement of people’s theological and technological universes. However, unlike many other aspects of digital religion, religious video gaming’s novel artistic forms, cultural critiques, and theological possibilities largely blossomed beneath notice. Evangelical video game culture, thus, presents the creative production of a historically significant avant garde whose critical perspective has been neglected outside its own community. In particular, Evangelical video gaming transforms the concerns that connect it to other digital cultures – “violence,” for instance, or “immersion” – by attending to the moral status of the player-in-play. This study combines the methods of Religious Studies, Science and Technology Studies, and Cultural Studies to show how the popular artifacts of digital religion can shed light upon their cultural context. -
Homage to a Pied Puzzler
i i i i Homage to a Pied Puzzler i i i i i i i i i i i i i i i i Homage to a Pied Puzzler Edited by Ed Pegg Jr. Alan H. Schoen Tom Rodgers A K Peters, Ltd. Wellesley, Massachusetts i i i i CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150225 International Standard Book Number-13: 978-1-4398-6500-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reason- able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. -
Download English-US Transcript (PDF)
MITOCW | MITCMS_608F10lec18-mp3 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Before we move on to the rest of the topics, maybe just some administrative stuff about the next assignment. So the idea is to model a system, right? So the idea is to pick some sort of dual system. In the past people have picked stuff like applying for college because they knew all about that. People have talked about any type of idea, you know, running a nuclear reactor or something. Specifically there's a-- but hey, any of you nuclear science majors?-- oh, perfect. One of my favorite Apple II games was actually nuclear reactor simulation, and it always melted down. That's why it was fun. But pick a real world system to start modeling. Obviously, you're trying to find the fun. Obviously, you're trying to make something that people are going to enjoy playing. But the idea is to start from a slightly different place from where we had in the first assignment. Now a couple of you guys already did that for the first assignment, though. I mean things like picking up girls at a party. You know, modeling real systems and things. Still haven't played yet, a genealogy game yet. So, we're going to play next March. So [UNINTELLIGIBLE]. The survival horror game, I would say it's not a real world system. -
Math Games: Sliding Block Puzzles 11/07/2007 07:05 PM
Math Games: Sliding Block Puzzles 11/07/2007 07:05 PM Search MAA Membership Publications Professional Development Meetings Organization Competitions Math Games sliding-block Puzzles Ed Pegg Jr., December 13, 2004 The December issue of The Economist contains an article with a prominent question. Has an inventor found the hardest possible simple sliding-block puzzle? It goes on to describe the Quzzle puzzle, by Jim Lewis. The article finishes with "Mr Lewis claims that Quzzle, as he dubs his invention, is 'the world's hardest simple sliding-block puzzle.' Within the terms of his particular definition of 'simple,' he would seem to have succeeded." The claim is wrong. Quzzle is not the world's hardest simple sliding-block puzzle, no matter how you define "hard." Sometimes, hard means lot and lots of moves. Junk Kato has succinctly described a record setting series for the most moves required with n pieces. The basic underlying principle is Edouard Lucas' Tower of Hanoi puzzle. Move the red piece to the bottom slot. Figure 1. Junk's Hanoi by Junk Kato. From Modern sliding-block Puzzles. Often, "hard" maximizes complexity with a sparse number of pieces. Usually, the fewer the pieces, the better the puzzle. Two people at the forefront of making really hard sliding-block puzzles with just a few pieces are Oskar van Deventer and James Stephens. James started with Sliding-Block Puzzles. The two of them collaborated to create the ConSlide Puzzle and the Bulbous Blob Puzzle. Oskar even went so far as to make a prototype of the excellent Simplicity puzzle.