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11 Triple Loyd
TTHHEE PPUUZZZZLLIINNGG SSIIDDEE OOFF CCHHEESSSS Jeff Coakley TRIPLE LOYDS: BLACK PIECES number 11 September 22, 2012 The “triple loyd” is a puzzle that appears every few weeks on The Puzzling Side of Chess. It is named after Sam Loyd, the American chess composer who published the prototype in 1866. In this column, we feature positions that include black pieces. A triple loyd is three puzzles in one. In each part, your task is to place the black king on the board.to achieve a certain goal. Triple Loyd 07 w________w áKdwdwdwd] àdwdwdwdw] ßwdwdw$wd] ÞdwdRdwdw] Ýwdwdwdwd] Üdwdwdwdw] Ûwdwdpdwd] Údwdwdwdw] wÁÂÃÄÅÆÇÈw Place the black king on the board so that: A. Black is in checkmate. B. Black is in stalemate. C. White has a mate in 1. For triple loyds 1-6 and additional information on Sam Loyd, see columns 1 and 5 in the archives. As you probably noticed from the first puzzle, finding the stalemate (part B) can be easy if Black has any mobile pieces. The black king must be placed to take away their moves. Triple Loyd 08 w________w áwdwdBdwd] àdwdRdwdw] ßwdwdwdwd] Þdwdwdwdw] Ýwdw0Ndwd] ÜdwdPhwdw] ÛwdwGwdwd] Údwdw$wdK] wÁÂÃÄÅÆÇÈw Place the black king on the board so that: A. Black is in checkmate. B. Black is in stalemate. C. White has a mate in 1. The next triple loyd sets a record of sorts. It contains thirty-one pieces. Only the black king is missing. Triple Loyd 09 w________w árhbdwdwH] àgpdpdw0w] ßqdp!w0B0] Þ0ndw0PdN] ÝPdw4Pdwd] ÜdRdPdwdP] Ûw)PdwGPd] ÚdwdwIwdR] wÁÂÃÄÅÆÇÈw Place the black king on the board so that: A. -
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 6= NP): one-player 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. -
Klondike (Sam Loyd)
Back From The Klondike by Sam Loyd The original puzzle was in black and white and had 1 challange: Start from that heart in the center and go three steps in a straight line in any one of the eight directions, north, south, east or west, or northeast, northwest, southeast or southwest. When you have gone three steps in a straight line, you will reach a square with a number on it, which indicates the second day's journey, as many steps as it tells, in a straight line in any of the eight directions. From this new point when reached, march on again according to the number indicated, and continue on, following the requirements of the numbers reached, until you come upon a square with a number which will carry you just one step beyond the border, when you are supposed to be out of the woods and can holler all you want, as you will have solved the puzzle. Joseph Eitel!Page 1 of 6!amagicclassroom.com Back From the Klondike Version 2 This maze was among the many puzzles Loyd created for newspapers beginning in 1890. In this version some squares are yellow or green. This allowed better explanations how the puzzle worked. The numbers in each square in the grid indicate how many squares you can travel in a straight line horizontally, vertically, or diagonally from that square. For example, if you start in the red square in the center, your first move can take you to any one of the three yellow squares. The green squares are some of the possible squares you could land on for the second move. -
Lagrange 1 of 22 the Great Mathematical Puzzelist Samuel Loyd Was Born in Philadelphia, Pennsylvania in 1841 (O'conner). He Wa
LaGrange 1 of 22 The great mathematical puzzelist Samuel Loyd was born in Philadelphia, Pennsylvania ’Conner). He was the youngest of nine children. At the age of three, his family moved in 1841 (O to New York where he attended public school (Carter). Sam Loyd got his start in the puzzle business with chess. At 14 years of age, Loyd started attending chess club with two of his older “New brothers. On April 14th, that same year, Loyd had his first chess problem published in the ” In 1856, the “New York Clipper” published another of his chess York Saturday Courier. ’Conner). At first, all of Loyd’s puzzles were hobbies. problems for which he won a prize (O After school he studied engineering and earned a license in steam and mechanical engineering (Loyd). For a time, Loyd supported himself as a plumbing contractor and the owner of a chain of ’Conner). Plus, music stores. He was also a skilled cartoonist and a self taught wood engraver (O he was skilled in conjuring, mimicry, ventriloquism and silhouette cutting (Gardner). Eventually, Loyd left plumbing behind and focused on mathematical puzzles. He attended chess tournaments, wrote and edited mechanical journals along with his magazine “Sam Loyd’s Puzzle Magazine” (O’Conner). For a while, he also edited the magazine entitled “Chess Monthly” and the chess page of “Scientific American” (Gardner). P. T. Barnum’s Trick ’Conner). This sale alone Donkey was invented by Sam Loyd and sold to Barnum in 1870 (O grossed $10,000 for Loyd (Carter). In 1878 Loyd published his one and only hard cover book Chess Strategy, which included all the chess problems published in Scientific American plus ’Conner). -
Color Blocks
Vol-3Issue-6 2017 IJARIIE-ISSN(O)-2395-4396 COLOR BLOCKS Varun Reddy1, Nikhil Gabbeta2, Sagar Naidu3, Arun Reddy4 1 Btech, Computer Science and Engineering, SRM University, Tamilnadu, India 2 Btech, Computer Science and Engineering, SRM University, Tamilnadu, India 3 Btech, Computer Science and Engineering, SRM University, Tamilnadu, India 4 Btech, Computer Science and Engineering, SRM University, Tamilnadu, India ABSTRACT Smart Puzzle app is a classic sliding puzzle game based on Fifteen Puzzle. It allows you to choose among three levels of complexity which are: 3x3 (8- puzzle), 4x4 (15-puzzle), and 5x5 (24-puzzle).In addition to standard square boards with numbers, user can choose a board with a picture painted on it. There are three stock pictures installed with the puzzle, and user can add his/her own pictures by pressing Menu button when the picture selection page is displayed. The picture user select will be split into equal tiles. Once you select the picture, the solved puzzle is displayed for 3 seconds. Then it is randomly shuffled, and user have to move the tiles to their initial locations. When user solve the puzzle, the app will display that picture again along with the number of moves you have made. To change the puzzle's settings during the game, press Menu button. Note that when user change any settings, the game starts over. When user switch apps or exit during the game, your puzzle is stored and resumed next time user run Smart puzzle. Keyword:-Smart Puzzle; Color Blocks; 1.intoduction 1.1 Overview The COLOR BLOCKS is a sliding puzzle that consists of a frame of numbered square tiles in random order with one tile missing. -
Variations of the 15 Puzzle
VARIATIONS OF THE 15 PUZZLE A thesis submitted to the Kent State University Honors College in partial fulfillment of the requirements for University Honors by Lisa Rose Hendrixson May, 2011 Thesis written by Lisa Rose Hendrixson Approved by , Advisor , Chair, Department of Mathematics Accepted by , Dean, Honors College ii TABLE OF CONTENTS LIST OF FIGURES. .iv ACKNOWLEDGEMENTS . v CHAPTERS I INTRODUCTION . 1 II THE HISTORY OF THE 15 PUZZLE . 3 III MATHEMATICS AND THE 15 PUZZLE . 6 IV VARIATIONS OF THE 15 PUZZLE . 14 V CONCLUSION . 22 BIBLIOGRAPHY . 23 iii LIST OF FIGURES • Figure 1.The 15 Puzzle . 2 • Figure 2.Pictoral representation of the above permutation. 8 • Figure 3.The permutation multiplied by the transposition (7; 8) ..................9 • Figure 4.Two odd length cycles inside the 15 Puzzle . 10 • Figure 5.The two cycles, put together. 11 • Figure 6.Permutting 3 blocks cyclically. 12 • Figure 7.Bipartite graph of the 15 Puzzle. 13 • Figure 8.Starting position for the first variation. 15 • Figure 9.Creating a single transposition inside the variation. 16 • Figure 10.Showing the switch of the blank spaces. 16 • Figure 11.Puzzle with a fixed block. 17 • Figure 12.Bipartite graph of a puzzle wth a glued-down block. 21 iv ACKNOWLEDGEMENTS I would like to thank Dr. Donald White for all his help and support during the process of writing this thesis. Without his dedication, it would not have been possible. Also, I would like to thank Dr. Mark Lewis, Dr. Elizabeth Mann, and Dr. Sara Newman for their willingness to serve on my defense committee and for all of their helpful comments and support along the way. -
Synthetic Games
S\TII}IETIC GAh.fES Synthetic Garnes Play a shortest possible game leading tCI ... G. P. Jelliss September 1998 page I S1NTHETIC GAI\{ES CONTENTS Auto-Surrender Chess BCM: British Chess Magazine, Oppo-Cance llati on Che s s CA'. ()hess Amafeur, EP: En Part 1: Introduction . .. .7 5.3 Miscellaneous. .22 Passant, PFCS'; Problemist Fairy 1.1 History.".2 Auto-Coexi s tence Ches s Chess Supplement, UT: Ultimate 1.2 Theon'...3 D3tnamo Chess Thernes, CDL' C. D" I,ocock, GPJ: Gravitational Chess G. P. Jelliss, JA: J. Akenhead. Part 2: 0rthodox Chess . ...5 Madrssi Chess TGP: T. G. Pollard, TRD: 2. I Checknrates.. .5 Series Auto-Tag Chess T. R. Dar,vson. 2.2 Stalernates... S 2.3 Problem Finales. I PART 1 I.I HISTOR,Y 2.4 Multiple Pawns... l0 INTRODUCTIOT{ Much of my information on the 2.,5 Kings and Pawns".. l1 A'synthetic game' is a sequence early history comes from articles 2.6 Other Pattern Play...13 of moves in chess, or in any form by T. R. Dar,vson cited below, of variant chess, or indesd in any Chess Amsteur l9l4 especially. Part 3. Variant Play . ...14 other garne: which simulates the 3.1 Exact Play... 14 moves of a possible, though Fool's Mste 3 .2 Imitative Direct. l 5 usually improbable, actual game? A primitive example of a 3.3 Imitative Oblique.. " l6 and is constructed to show certain synthetic game in orthodox chess 3.4 Maximumming...lT specified events rvith fewest moves. is the 'fool's mate': l.f3l4 e6l5 3.5 Seriesplay ...17 The following notes on history 2.g4 Qh4 mate. -
Pebble Motion on Graphs with Rotations: Efficient Feasibility Tests
1 Pebble Motion on Graphs with Rotations: Efficient Feasibility Tests and Planning Algorithms Jingjin Yu, Daniela Rus Abstract We study the problem of planning paths for p distinguishable pebbles (robots) residing on the vertices of an n-vertex connected graph with p ≤ n. A pebble may move from a vertex to an adjacent one in a time step provided that it does not collide with other pebbles. When p = n, the only collision free moves are synchronous rotations of pebbles on disjoint cycles of the graph. We show that the feasibility of such problems is intrinsically determined by the diameter of a (unique) permutation group induced by the underlying graph. Roughly speaking, the diameter of a group G is the minimum length of the generator product required to reach an arbitrary element of G from the identity element. Through bounding the diameter of this associated permutation group, which assumes a maximum value of O(n2), we establish a linear time algorithm for deciding the feasibility of such problems and an O(n3) algorithm for planning complete paths. I. INTRODUCTION In Sam Loyd’s 15-puzzle Loyd (1959), a player arranges square blocks labeled 1-15, scrambled arXiv:1205.5263v4 [cs.DS] 31 Jul 2014 on a 4 × 4 board, to achieve a shuffled row major ordering of the blocks using one empty swap cell (see, e.g., Fig. 1). Generalizing the grid-based board to an arbitrary connected graph over n vertices, the 15-puzzle becomes the problem of pebble motion on graphs (PMG). Here, up to n−1 uniquely labeled pebbles on the vertices of the graph must be moved to some desired goal config- uration, using unoccupied (empty) vertices as swap spaces.1 Since the initial work by Kornhauser Jingjin Yu and Daniela Rus are the Computer Science and Artificial Intelligence Lab at the Massachusetts Institute of Technology. -
Zig-Zag Numberlink Is NP-Complete
Zig-Zag Numberlink is NP-Complete The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Adcock, Aaron, Erik D. Demaine, Martin L. Demaine, Michael P. O’Brien, Felix Reidl, Fernando Sanchez Villaamil, and Blair D. Sullivan. “Zig-Zag Numberlink Is NP-Complete.” Journal of Information Processing 23, no. 3 (2015): 239–245. As Published http://dx.doi.org/10.2197/ipsjjip.23.239 Publisher Information Processing Society of Japan Version Original manuscript Citable link http://hdl.handle.net/1721.1/100008 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Zig-Zag Numberlink is NP-Complete Aaron Adcock1, Erik D. Demaine2, Martin L. Demaine2, Michael P. O'Brien3, Felix Reidl4, Fernando S´anchez Villaamil4, and Blair D. Sullivan3 1Stanford University, Palo Alto, CA, USA, [email protected] 2Massachusetts Institute of Technology, Cambridge, MA, USA, fedemaine,[email protected] 3North Carolina State University, Raleigh, NC, USA, fmpobrie3,blair [email protected] 4RWTH Aachen University, Aachen, Germany, freidl,[email protected] October 23, 2014 Abstract When can t terminal pairs in an m × n grid be connected by t vertex-disjoint paths that cover all vertices of the grid? We prove that this problem is NP-complete. Our hardness result can be compared to two previous NP-hardness proofs: Lynch's 1975 proof without the \cover all vertices" constraint, and Kotsuma and Takenaga's 2010 proof when the paths are restricted to have the fewest possible corners within their homotopy class. -
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 . -
Sam Loydʼs Most Successful Hoax Jerry Slocum
1 Sam Loydʼs Most Successful Hoax Jerry Slocum Martin Gardner called Sam Loyd “Americaʼs Greatest Puzzlist”. And Loyd is famous for the numerous wonderful puzzles that he invented, Figure 1. Two Great Puzzles and the Cyclopedia of 5000 Puzzles by Sam Loyd such as the Trick Mules (above left), Get Off the Earth Puzzle Mystery (above center) and the thousands of delightful puzzles included in his Cyclopedia (above right) as well as the engaging stories he used to pose his puzzles. However Loyd also has a reputation for taking credit for puzzles created by others. Henry Dudeney complained numerous times that Loyd did not give him credit for puzzles that Dudeney had created. And sometimes his delightful stories accompanying his puzzles were beyond a mere exaggeration, they were a hoax, completely false. Loydʼs book about the Tangram puzzle called, The 8th Book of Tan, included a extensive, but bogus history of the puzzle that claimed that it was 4,000 years old. In fact the puzzle is actually about 200 years old. Figure 2. The 8th Book of Tan by Sam Loyd 2 Although Sir James Murray, Editor of the Oxford English Dictionary, exposed Loydʼs false Tangram history in 1910, 7 years after the book was published, Loydʼs hoax is still occasionally reported in publications and web sites An interview with Sam Loyd in the Lima (Ohio) Daily Times on January 13, 1891, provided Loyd an opportunity to plug a new Puzzle of his, named “Blind Luck”. Loyd also mentioned in the interview, for the first time that he was the inventor of: Pigs-in-Clover, which had been a puzzle craze in 1889, just 2 years earlier. -
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.