DOCSLIB.ORG

May 29, 2015 Contents

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
May 29, 2015 Contents 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 29, 2015 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 . 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 . 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 Sage . 47 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 . 59 5.7 Exercises . 62 6 Permutations: Products of 2-Cycles 67 6.1 Introduction . 67 6.2 Product of 2-Cycles . 68 6.3 Solvability of Swap . 69 6.4 Exercises . 70 7 Permutations: The Parity Theorem 71 7.1 Introduction . 71 7.2 Variation of Swap . 73 7.3 Proof of the Parity Theorem . 74 7.4 Exercises . 79 8 Permutations: An and 3-Cycles 83 8.1 Swap Variation: A Challenge . 83 8.2 The Alternating Group An ....................................... 83 8.3 Products of 3-cycles . 85 8.4 Variations of Swap: Revisited . 87 8.5 Exercises . 88 CONTENTS iii 9 Mastering the 15-Puzzle 91 9.1 Solvability Criteria . 91 9.2 Proof of Solvability Criteria . 93 9.3 Strategy for Solution . 96 9.4 Exercises . 97 10 Groups 103 10.1 Group: Definition . 103 10.2 Some Everyday Examples of Groups . 105 10.3 Further Examples of Groups . 108 10.4 Exercises . 120 11 Subgroups 125 11.1 Subgroups . 125 11.2 Examples of Subgroups . 126 11.3 The Center of a Group . 127 11.4 Lagrange’s Theorem . 128 11.5 Cyclic Groups Revisited . 129 11.6 Cayley’s Theorem . 130 11.7 Exercises . 131 12 Puzzle Groups 135 12.1 Puzzle Groups . 135 12.2 Rubik’s Cube . 136 12.3 Hungarian Rings . 143 12.4 15-Puzzle . 144 12.5 Exercises . 145 13 Commutators 147 13.1 Commutators . 147 13.2 Creating Puzzle moves with Commutators . 148 13.3 Exercises . 156 14 Conjugates 161 14.1 Conjugates . 161 14.2 Modifying Puzzle moves with Conjugates . 163 14.3 Exercises . 169 15 Mastering the Oval Track Puzzle 173 iv CONTENTS 15.1 Oval Track with T = (1; 4)(2; 3) .................................... 173 15.2 Variations of the Oval Track T move . 185 15.3 Exercises . 186 16 Mastering the Hungarian Rings Puzzle 189 16.1 Hungarian Rings - Numbered version . 189 16.2 Building Small Cycles: Tools for Our End-Game Toolbox . 191 16.3 Solving the end-game . ..
Load more

Recommended publications
  • Abstract Algebra
    Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 lecture notes loosely based on Dummit and Foote's text Abstract Algebra (3rd ed) Prerequisite: Linear Algebra (203) 1 Introduction Pure Mathematics Algebra Analysis Foundations (set theory/logic) G eometry & Topology What is Algebra? • Number systems N = f1; 2; 3;::: g \natural numbers" Z = f:::; −1; 0; 1; 2;::: g \integers" Q = ffractionsg \rational numbers" R = fdecimalsg = pts on the line \real numbers" p C = fa + bi j a; b 2 R; i = −1g = pts in the plane \complex nos" k polar form re iθ, where a = r cos θ; b = r sin θ a + bi b r θ a p Note N ⊂ Z ⊂ Q ⊂ R ⊂ C (all proper inclusions, e.g. 2 62 Q; exercise) There are many other important number systems inside C. 2 • Structure \binary operations" + and · associative, commutative, and distributive properties \identity elements" 0 and 1 for + and · resp. 2 solve equations, e.g. 1 ax + bx + c = 0 has two (complex) solutions i p −b ± b2 − 4ac x = 2a 2 2 2 2 x + y = z has infinitely many solutions, even in N (thei \Pythagorian triples": (3,4,5), (5,12,13), . ). n n n 3 x + y = z has no solutions x; y; z 2 N for any fixed n ≥ 3 (Fermat'si Last Theorem, proved in 1995 by Andrew Wiles; we'll give a proof for n = 3 at end of semester). • Abstract systems groups, rings, fields, vector spaces, modules, . A group is a set G with an associative binary operation ∗ which has an identity element e (x ∗ e = x = e ∗ x for all x 2 G) and inverses for each of its elements (8 x 2 G; 9 y 2 G such that x ∗ y = y ∗ x = e).
    [Show full text]
  • Parity Alternating Permutations Starting with an Odd Integer
    Parity alternating permutations starting with an odd integer Frether Getachew Kebede1a, Fanja Rakotondrajaob aDepartment of Mathematics, College of Natural and Computational Sciences, Addis Ababa University, P.O.Box 1176, Addis Ababa, Ethiopia; e-mail: frigetach@gmail.com bD´epartement de Math´ematiques et Informatique, BP 907 Universit´ed’Antananarivo, 101 Antananarivo, Madagascar; e-mail: frakoton@yahoo.fr Abstract A Parity Alternating Permutation of the set [n] = 1, 2,...,n is a permutation with { } even and odd entries alternatively. We deal with parity alternating permutations having an odd entry in the first position, PAPs. We study the numbers that count the PAPs with even as well as odd parity. We also study a subclass of PAPs being derangements as well, Parity Alternating Derangements (PADs). Moreover, by considering the parity of these PADs we look into their statistical property of excedance. Keywords: parity, parity alternating permutation, parity alternating derangement, excedance 2020 MSC: 05A05, 05A15, 05A19 arXiv:2101.09125v1 [math.CO] 22 Jan 2021 1. Introduction and preliminaries A permutation π is a bijection from the set [n]= 1, 2,...,n to itself and we will write it { } in standard representation as π = π(1) π(2) π(n), or as the product of disjoint cycles. ··· The parity of a permutation π is defined as the parity of the number of transpositions (cycles of length two) in any representation of π as a product of transpositions. One 1Corresponding author. n c way of determining the parity of π is by obtaining the sign of ( 1) − , where c is the − number of cycles in the cycle representation of π.
    [Show full text]
  • Arxiv:2009.00554V2 [Math.CO] 10 Sep 2020 Graph On√ More Than Half of the Vertices of the D-Dimensional Hypercube Qd Has Maximum Degree at Least D
    ON SENSITIVITY IN BIPARTITE CAYLEY GRAPHS IGNACIO GARCÍA-MARCO Facultad de Ciencias, Universidad de La Laguna, La Laguna, Spain KOLJA KNAUER∗ Aix Marseille Univ, Université de Toulon, CNRS, LIS, Marseille, France Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Spain ABSTRACT. Huang proved that every set of more than halfp the vertices of the d-dimensional hyper- cube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs. First, we present three infinite families of Cayley graphs of unbounded degree that contain in- duced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encoun- tered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an d induced matching on a 2d−1 -fraction of the vertices. This is largest possible and answers a question of Lehner and Verret. Second, we consider Huang’s lower bound for graphs with subcubes and show that the corre- sponding lower bound is tight for products of Coxeter groups of type An, I2(2k + 1), and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang’s question.
    [Show full text]
  • Notes on Finite Group Theory
    Notes on finite group theory Peter J. Cameron October 2013 2 Preface Group theory is a central part of modern mathematics. Its origins lie in geome- try (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of solv- ing the equation). These notes are based on a Masters course I gave at Queen Mary, University of London. Of the two lecturers who preceded me, one had concentrated on finite soluble groups, the other on finite simple groups; I have tried to steer a middle course, while keeping finite groups as the focus. The notes do not in any sense form a textbook, even on finite group theory. Finite group theory has been enormously changed in the last few decades by the immense Classification of Finite Simple Groups. The most important structure theorem for finite groups is the Jordan–Holder¨ Theorem, which shows that any finite group is built up from finite simple groups. If the finite simple groups are the building blocks of finite group theory, then extension theory is the mortar that holds them together, so I have covered both of these topics in some detail: examples of simple groups are given (alternating groups and projective special linear groups), and extension theory (via factor sets) is developed for extensions of abelian groups. In a Masters course, it is not possible to assume that all the students have reached any given level of proficiency at group theory.
    [Show full text]
  • Maxmalgebra M the Linear Algebra of Combinatorics?
    Max-algebra - the linear algebra of combinatorics? Peter Butkoviÿc School of Mathematics and Statistics The University of Birmingham Edgbaston Birmingham B15 2TT United Kingdom Submitted by ABSTRACT Let a b =max(a, b),a b = a + b for a, b R := R .Bymax- algebra we⊕ understand the analogue⊗ of linear algebra∈ developed∪ {−∞ for} the pair of operations ( , ) extended to matrices and vectors. Max-algebra, which has been studied⊕ for⊗ more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machine-scheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue-eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial opti- misation problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra. 1. Introduction Let a b =max(a, b) ⊕ and a b = a + b ⊗ for a, b R := R . By max-algebra we understand in this paper the analogue∈ of linear∪ {−∞ algebra} developed for the pair of operations ( , ) extended to matrices and vectors formally in the same way as in linear⊕ ⊗ algebra, that is if A =(aij),B=(bij) and C =(cij ) are matrices with 1 2 elements from R of compatible sizes, we write C = A B if cij = aij bij ⊕ ⊕ for all i, j, C = A B if cij = ⊕ aik bkj =maxk(aik + bkj) for all i, j ⊗ k ⊗ and α A =(α aij) for α R.
    [Show full text]
  • Algebra Course Notes
    Algebra course notes Margaret Bilu December 20, 2019 1 Contents 1 Quantifiers, sets, maps, equivalence relations 3 1.1 Sets, subsets . 3 1.2 Unions, intersections, products . 4 1.3 Maps . 5 1.4 Equivalence relations . 5 2 Integers 7 2.1 Addition and multiplication on the set of integers Z ............. 7 2.2 Divisibility . 8 2.3 Euclidean division . 9 2.4 GCD and Euclid’s algorithm . 9 2.5 Unique factorization of integers . 10 2.6 Congruence classes . 11 2.7 Units in Z=nZ .................................. 13 2.8 Back to congruences . 14 2.9 Conclusion of the chapter . 15 3 Groups 15 3.1 Laws of composition . 15 3.2 Groups . 17 3.3 Subgroups . 19 3.4 Products of groups . 20 3.5 Cyclic groups . 20 3.6 Group homomorphisms . 22 3.7 Isomorphisms . 24 3.8 Classification of groups of small order . 26 3.9 Conclusion of the chapter . 27 4 Permutation groups 28 4.1 Definition . 29 4.2 Cycles . 30 4.3 Parity of a permutation . 32 4.4 Generators of Sn and An ............................ 34 4.5 Dihedral groups . 35 4.6 Conclusion of the chapter . 36 5 Cosets and Lagrange’s theorem 37 5.1 Left and right cosets . 37 5.2 Index of a subgroup . 38 5.3 Lagrange’s theorem . 40 5.4 Some arithmetic applications of Lagrange’s theorem . 42 2 5.5 Cosets and homomorphisms . 43 5.6 Conclusion of the chapter . 45 6 Normal subgroups and quotients of groups 46 6.1 Normal subgroups . 46 6.2 Quotient groups .
    [Show full text]
  • The Alternating Group
    The Alternating Group R. C. Daileda 1 The Parity of a Permutation Let n ≥ 2 be an integer. We have seen that any σ 2 Sn can be written as a product of transpositions, that is σ = τ1τ2 ··· τm; (1) where each τi is a transposition (2-cycle). Although such an expression is never unique, there is still an important invariant that can be extracted from (1), namely the parity of σ. Specifically, we will say that σ is even if there is an expression of the form (1) with m even, and that σ is odd if otherwise. Note that if σ is odd, then (1) holds only when m is odd. However, the converse is not immediately clear. That is, it is not a priori evident that a given permutation can't be expressed as both an even and an odd number of transpositions. Although it is true that this situation is, indeed, impossible, there is no simple, direct proof that this is the case. The easiest proofs involve an auxiliary quantity known as the sign of a permutation. The sign is uniquely determined for any given permutation by construction, and is easily related to the parity. This thereby shows that the latter is uniquely determined as well. The proof that we give below follows this general outline and is particularly straight- forward, given that the reader has an elementary knowledge of linear algebra. In particular, we assume familiarity with matrix multiplication and properties of the determinant. n We begin by letting e1; e2;:::; en 2 R denote the standard basis vectors, which are the columns of the n × n identity matrix: I = (e1 e2 ··· en): For σ 2 Sn, we define π(σ) = (eσ(1) eσ(2) ··· eσ(n)): Thus π(σ) is the matrix whose ith column is the σ(i)th column of the identity matrix.
    [Show full text]
  • Chapter IV. Determinants
    Notes ⃝c F.P. Greenleaf 2014 LAI-f14-det.tex version 8/11/2014 Chapter IV. Determinants. IV.1 The Permutation Group Sn. The permutation group Sn consists of all bijections σ :[1,n] [1,n]where[1,n]= 1,...,n ,withcompositionofoperators → { } σ σ (k)=σ (σ (k)) for 1 k n 1 ◦ 2 1 2 ≤ ≤ as the group operation. The identity element e is the identity map id[1,n] such that e(k)=k,forallk [1,n]. We recall that a group is any set G equipped with a binary operation ( )satisfyingthefollowingaxioms:∈ ∗ 1. Associativity: x (y z)=(x y) z; ∗ ∗ ∗ ∗ 2. Identity element: There is an e G such that e x = x = x e,forallx G; ∈ ∗ ∗ ∈ 3. Inverses: Every x G has a “two-sided inverse,” an element x−1 G such that x−1 x = x x−1 =∈e. ∈ ∗ ∗ We do not assume that the system (G, )iscommutative,withx y = y x;agroupwith this extra property is a commutative group∗ ,alsoreferredtoasan∗ abelian∗ group. Here are some examples of familiar groups. 1. The integers (Z, +) become a commutative group when equipped with (+) as the group operation; multiplication ( ) does not make Z agroup.(Why?) · 2. Any vector space equipped with its (+) operation is a commutatve group, for instance (Kn, +); 3. The set (C×, )=C 0 of nonzero complex numbers equipped with complex multiplication· ( )isacommutativegroup;soisthesubset∼{} S1 = z C : z =1 (unit circle in the· complex plane) because z , w =1 zw ={ z∈ w =1and| | } 1/z =1/ z =1. | | | | ⇒| | | |·| | | | | | 4. General Linear Group. The set GL(n, K)= A M(n, K):det(A) =0 of invertible n n matrices is a group when equipped{ with∈ matrix multiply̸ as} the group operation.× It is noncommutative when n 2.
    [Show full text]
  • Algebra-2: Groups and Rings
    Algebra-2: Groups and Rings Dmitriy Rumynin∗ January 6, 2011 How to use these notes The lecture notes are split into 27 sections. Each section will be discussed in one lecture, making every lecture self-contained. This means that the material in a section may be reshuffled or even skipped for the lecture, although the numeration of propositions will be consistent. The remaining (up to 3) lectures will be spent on revisions and exercises including past exams. These written notes is an official curriculum: anything in them except vistas can appear on the exam. Each section contains exercises that you should do. To encourage you doing them, I will use some of the exercises in the exam. Vista sections are not assessed or examined in any way. Skip them if you are allergic to nuts or psychologically fragile! The vistas are food for further contemplation. A few of them are sky blue, but most are second year material that we don’t have time to cover. You are encouraged to expand one of them into your second year essay. The main recommended book is Concrete Abstract Algebra by Lauritzen. It is reasonably priced (£25 new, £11 used on Amazon), mostly relevant (except chapter 5) and quite thin. The downside of the book is brevity of exposition and some students prefer more substantial books. An excellent UK-style textbook is Introduction to Algebra by Cameron (from £17 on Amazon). Another worthy book is Algebra by Artin (£75 on Amazon for the new edition but you can get older editions for around £45).
    [Show full text]
  • Groups and Group Actions
    Groups and Group Actions Richard Earl Hilary and Trinity Terms 2014 Syllabus Hilary Term (8 lectures) • Axioms for a group and for an Abelian group. Examples including geometric symmetry groups, matrix groups (GLn, SLn,On, SOn,Un), cyclic groups. Products of groups. [2] • Permutations of a finite set under composition. Cycles and cycle notation. Order. Trans- positions; every permutation may be expressed as a product of transpositions. The parity of a permutation is well-defined via determinants. Conjugacy in permutation groups. [2] • Subgroups; examples. Intersections. The subgroup generated by a subset of a group. A subgroup of a cyclic group is cyclic. Connection with hcf and lcm. Bezout’s Lemma. [1.5] • Recap on equivalence relations including congruence mod n and conjugacy in a group. Proof that equivalence classes partition a set. Cosets and Lagrange’s Theorem; examples. The order of an element. Fermat’s Little Theorem. [2.5] Trinity Term (8 lectures) • Isomorphisms, examples. Groups up to isomorphism of order 8 (stated without proof). Homomorphisms of groups with motivating examples. Kernels. Images. Normal sub- groups. Quotient groups; examples. First Isomorphism Theorem. Simple examples de- termining all homomorphisms between groups. [3] • Group actions; examples. Definition of orbits and stabilizers. Transitivity. Orbits parti- tion the set. Stabilizers are subgroups. [2] • Orbit-stabilizer Theorem. Examples and applications including Cauchy’s Theorem and to conjugacy classes. [1] • Orbit-counting formula. Examples. [1] • The representation G → Sym(S) associated with an action of G on S. Cayley’s Theorem. Symmetry groups of the tetrahedron and cube. [1] Recommended Texts • M. A. Armstrong Groups and Symmetry (Springer, 1997) 2 STANDARD GROUP NOTATION • Z, Q, R, C — the integers/rationals/reals/complex numbers under +.
    [Show full text]
  • Permutation Puzzles a Mathematical Perspective
    Permutation Puzzles A Mathematical Perspective Jamie Mulholland Copyright c 2021 Jamie Mulholland SELF PUBLISHED http://www.sfu.ca/~jtmulhol/permutationpuzzles Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License (the “License”). You may not use this document except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc-sa/4.0/. Unless required by applicable law or agreed to in writing, software distributed under the License is dis- tributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. First printing, May 2011 Contents I Part One: Foundations 1 Permutation Puzzles ........................................... 11 1.1 Introduction 11 1.2 A Collection of Puzzles 12 1.3 Which brings us to the Definition of a Permutation Puzzle 22 1.4 Exercises 22 2 A Bit of Set Theory ............................................ 25 2.1 Introduction 25 2.2 Sets and Subsets 25 2.3 Laws of Set Theory 26 2.4 Examples Using SageMath 28 2.5 Exercises 30 II Part Two: Permutations 3 Permutations ................................................. 33 3.1 Permutation: Preliminary Definition 33 3.2 Permutation: Mathematical Definition 35 3.3 Composing Permutations 38 3.4 Associativity of Permutation Composition 41 3.5 Inverses of Permutations 42 3.6 The Symmetric Group Sn 45 3.7 Rules for Exponents 46 3.8 Order of a Permutation 47 3.9 Exercises 48 4 Permutations: Cycle Notation ................................. 51 4.1 Permutations: Cycle Notation 51 4.2 Products of Permutations: Revisited 54 4.3 Properties of Cycle Form 55 4.4 Order of a Permutation: Revisited 55 4.5 Inverse of a Permutation: Revisited 57 4.6 Summary of Permutations 58 4.7 Working with Permutations in SageMath 59 4.8 Exercises 59 5 From Puzzles To Permutations .................................
    [Show full text]
  • Arxiv:1901.03917V3 [Math.CO] 4 Apr 2021 the Generating Set Plays an Important Role in Com- for Any 0 J ` 1, Such That Π B
    Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph Elena V. Konstantinovaa, Alexey N. Medvedevb,∗ aSobolev Institute of Mathematics, Novosibisk State University, Novosibirsk, Russia bICTEAM, Université catholique de Louvain, Louvain-la-Neuve, Belgium Abstract The Bubble-sort graph BSn; n > 2, is a Cayley graph over the symmetric group Symn generated by transpositions from the set (12); (23);:::; (n 1 n) . It is a bipartite graph containing all even cycles of length `, where 4 6 ` 6 n!. We give an explicitf combinatorial− characterizationg of all its 4- and 6-cycles. Based on this characterization, we define generalized prisms in BSn; n > 5, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms. Keywords: Cayley graphs, Bubble-sort graph, 1-skeleton of the Permutahedron, Coxeter presentation of the symmetric group, n-prisms, generalized prisms, Hamiltonian cycle 1. Introduction braid group into the symmetric group so that the Cox- eter presentation of the symmetric group is given by the We start by introducing a few definitions. The Bubble- following n braid relations [11]: sort graph BS = Cay(Sym ; ); n 2, is a Cayley 2 n n B > graph over the symmetric group Symn of permutations bibi+1bi = bi+1bibi+1; 1 6 i 6 n 2; (1) π = [π1π2 : : : πn], where πi = π(i), 1 6 i 6 n, with − the generating set = b Sym : 1 i n bjbi = bibj; 1 6 i < j 1 6 n 2: (2) B f i 2 n 6 6 − − − 1 of all bubble-sort transpositions bi swapping the i-th It is known (see, for example §1.9 in [8] or [9]) that andg (i + 1)-st elements of a permutation π when mul- if any permutation π is presented by two minimal length tiplied on the right, i.e.
    [Show full text]