Legal Configurations of the 15-Puzzle Andrew Chapple 1 Alfonso Croeze - PowerPoint PPT Presentation

Legal Configurations of the 15-Puzzle Andrew Chapple 1 Alfonso Croeze 1 Mhel Lazo 1 Hunter Merrill 2 1 Department of Mathematics Louisiana State University Baton Rouge, LA 2 Department of Mathematics Mississippi State University Starkville, MS August 24, 2010 Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 1 / 25

Motivation History Invented in the 1860s Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 2 / 25

Motivation History Invented in the 1860s Puzzle description 15puzzleimg.png Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 2 / 25

Motivation History Invented in the 1860s Puzzle description The object of the puzzle 15puzzleimg.png Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 2 / 25

Motivation History Invented in the 1860s Puzzle description The object of the puzzle 15puzzleimg.png Sam Lloyd’s challenge Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 2 / 25

Motivation History Invented in the 1860s Puzzle description The object of the puzzle 15puzzleimg.png Sam Lloyd’s challenge Our objective Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 2 / 25

Basics Permutations Definition A permutation of a set A is a bijection from A onto itself. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 3 / 25

Basics Permutations Definition A permutation of a set A is a bijection from A onto itself. We will denote the set of all permutations of n elements as S n . Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 3 / 25

Basics Permutations Definition A permutation of a set A is a bijection from A onto itself. We will denote the set of all permutations of n elements as S n . Example Consider the set A = { 1 , 2 , 3 , 4 , 5 , 6 } . Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 3 / 25

Basics Permutations Definition A permutation of a set A is a bijection from A onto itself. We will denote the set of all permutations of n elements as S n . Example Consider the set A = { 1 , 2 , 3 , 4 , 5 , 6 } . Then the permutation P , ( 1 2 3 4 5 6 ) P = 4 1 5 2 3 6 changes 1 to 4, 2 to 1, 3 to 5, 4 to 2, 5 to 3, and fixes , or leaves unchanged, the element 6. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 3 / 25

Basics Cycle Notation Permutations can be more compactly written in cycle notation . Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 4 / 25

Basics Cycle Notation Permutations can be more compactly written in cycle notation . Cycles are always read left to right. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 4 / 25

Basics Cycle Notation Permutations can be more compactly written in cycle notation . Cycles are always read left to right. The element 6 does not change and may be omitted. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 4 / 25

Basics Cycle Notation Permutations can be more compactly written in cycle notation . Cycles are always read left to right. The element 6 does not change and may be omitted. Example The cycle notation of ( 1 ) 2 3 4 5 6 P = 4 1 5 2 3 6 Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 4 / 25

Basics Cycle Notation Permutations can be more compactly written in cycle notation . Cycles are always read left to right. The element 6 does not change and may be omitted. Example The cycle notation of ( 1 ) 2 3 4 5 6 P = 4 1 5 2 3 6 is (1 4 2)(3 5)(6) Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 4 / 25

Basics Cycle Notation Permutations can be more compactly written in cycle notation . Cycles are always read left to right. The element 6 does not change and may be omitted. Example The cycle notation of ( 1 ) 2 3 4 5 6 P = 4 1 5 2 3 6 is (1 4 2)(3 5)(6) or (1 4 2)(3 5) Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 4 / 25

Basics Disjoint Cycles Definition Cycles are disjoint if they have no common elements. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 5 / 25

Basics Disjoint Cycles Definition Cycles are disjoint if they have no common elements. The two cycles which compose P are disjoint . Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 5 / 25

Basics Disjoint Cycles Definition Cycles are disjoint if they have no common elements. The two cycles which compose P are disjoint . Disjoint cycles are commutative. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 5 / 25

Basics Disjoint Cycles Definition Cycles are disjoint if they have no common elements. The two cycles which compose P are disjoint . Disjoint cycles are commutative. Nondisjoint cycles are not necessarily commutative: Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 5 / 25

Basics Disjoint Cycles Definition Cycles are disjoint if they have no common elements. The two cycles which compose P are disjoint . Disjoint cycles are commutative. Nondisjoint cycles are not necessarily commutative: (1 2)(2 3)=(1 2 3) (2 3)(1 2)=(3 2 1) Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 5 / 25

Basics Disjoint Cycles Definition Cycles are disjoint if they have no common elements. The two cycles which compose P are disjoint . Disjoint cycles are commutative. Nondisjoint cycles are not necessarily commutative: (1 2)(2 3)=(1 2 3) (2 3)(1 2)=(3 2 1) Example Therefore, we can also write P as P = (3 5)(1 4 2) Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 5 / 25

Basics Transpositions Definition Cycles consisting of two elements are called transpositions . Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 6 / 25

Basics Transpositions Definition Cycles consisting of two elements are called transpositions . Example The transposition (3 5) can also be written as (5 3), as both have the effect of swapping the elements 3 and 5. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 6 / 25

Basics Transpositions Definition Cycles consisting of two elements are called transpositions . Example The transposition (3 5) can also be written as (5 3), as both have the effect of swapping the elements 3 and 5. A transposition is its own inverse Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 6 / 25

Basics Transpositions Definition Cycles consisting of two elements are called transpositions . Example The transposition (3 5) can also be written as (5 3), as both have the effect of swapping the elements 3 and 5. A transposition is its own inverse (3 5)(3 5) = (3)(5) = I Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 6 / 25

Basics Parity Definition A permutation is odd if it can be written as a product of an odd number of transpositions. Otherwise it is even . Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 7 / 25

Basics Parity Definition A permutation is odd if it can be written as a product of an odd number of transpositions. Otherwise it is even . Examples (1 2)(3 4) is even. (1 2)(3 4)(5 6) is odd. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 7 / 25

Basics Parity Definition A permutation is odd if it can be written as a product of an odd number of transpositions. Otherwise it is even . Examples (1 2)(3 4) is even. (1 2)(3 4)(5 6) is odd. The set of all odd permutations of n elements is denoted by O n Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 7 / 25

Basics Parity Definition A permutation is odd if it can be written as a product of an odd number of transpositions. Otherwise it is even . Examples (1 2)(3 4) is even. (1 2)(3 4)(5 6) is odd. The set of all odd permutations of n elements is denoted by O n The set of all even permutations of n elements is denoted by A n Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 7 / 25

Basics Parity Theorem Theorem If 휎 ∈ S n , then 휎 may be written as the product of an even number of transpositions if and only if 휎 can not be written as the product of an odd number of transpositions. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 8 / 25

Basics Parity Theorem Theorem If 휎 ∈ S n , then 휎 may be written as the product of an even number of transpositions if and only if 휎 can not be written as the product of an odd number of transpositions. Lemma The identity I, the permutation which fixes all elements, is even. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 8 / 25

Basics Proof of Parity Theorem Proof. Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 9 / 25

Basics Proof of Parity Theorem Proof. 휎 = 휏 1 휏 2 ⋅ ⋅ ⋅ 휏 s = q 1 q 2 ⋅ ⋅ ⋅ q t Chapple, Croeze, Lazo, Merrill (LSU&MSU) 15-Puzzle August 24, 2010 9 / 25