Solitaire Games. MATH 171 Freshman Seminar for Mathematics Majors. J. Robert Buchanan. Department of Mathematics. Fall 2010

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1 Solitaire Games MATH 171 Freshman Seminar for Mathematics Majors J. Robert Buchanan Department of Mathematics Fall 2010

2 Standard Checkerboard

3 Challenge 1 Suppose two diagonally opposite corners of the checkerboard have been removed. Can the remaining region be covered by a set of 31 dominoes?

4 Challenge 2 Suppose two adjacent corners of the checkerboard have been removed. Can the remaining region be covered by a set of 31 dominoes?

5 Challenge 3 1 Is it possible to cover all but one square of an 8 by 8 checkerboard by using 21 straight trominoes? 2 If the covering is possible, which squares must be left uncovered? Straight Trominoe

6 Tricolor Checkerboard

7 Fifteen Puzzle The Fifteen Puzzle was invented in the 1870s by Sam Loyd. The puzzle consists of a flat box containing 15 movable pieces numbered 1 through 15. The starting configuration resembles:

8 Challenge 4 By sliding any piece adjacent to the blank space, rearrange the pieces to resemble: No piece may be removed from the box at any time.

9 Comments and Questions Is this puzzle solvable? Which rearrangements of the pieces are possible?

10 Useful Problem Solving Principle When faced with a problem you do not know how to solve, try solving a simpler, related problem.

11 Useful Problem Solving Principle When faced with a problem you do not know how to solve, try solving a simpler, related problem. Consider a 2 2 version, we will call the Three Puzzle: Which rearrangements of the pieces are possible?

12 Ordering Scheme Ignoring the blank, we may list the pieces in the order given by the pattern:

13 Ordering Scheme Ignoring the blank, we may list the pieces in the order given by the pattern: The possible rearrangements can be listed using ordered triples: (1, 2, 3), (2, 3, 1), and (3, 1, 2).

14 Permutations Definition Any ordering of n distinct items {1, 2, 3,...,n} is called a permutation. There are n(n 1)(n 2)(n 3) (3)(2)(1) = n! permutations of n items.

15 Permutations Definition Any ordering of n distinct items {1, 2, 3,...,n} is called a permutation. There are n(n 1)(n 2)(n 3) (3)(2)(1) = n! permutations of n items. How many permutations of {1, 2, 3} are possible? Which ordered triples are not attainable as rearrangements of the starting position in the Three Puzzle?

16 Inversions Definition In any permutation of the numbers {1, 2, 3,..., n} an inversion occurs whenever a larger number precedes a smaller number.

17 Inversions Definition In any permutation of the numbers {1, 2, 3,..., n} an inversion occurs whenever a larger number precedes a smaller number. Example The permutation (1, 5, 2, 4, 3) contains four inversions.

18 Inversions Definition In any permutation of the numbers {1, 2, 3,..., n} an inversion occurs whenever a larger number precedes a smaller number. Example The permutation (1, 5, 2, 4, 3) contains four inversions. How many inversions are in each of the following permutations? 1 (1, 2, 3, 4, 5, 6) 2 (7, 1, 4, 6, 3, 2, 5) 3 (5, 1, 6, 8, 2, 4, 7, 3)

19 Even and Odd Permutations Definition A permutation is said to be even if it contains an even number of inversions, and is said to be odd otherwise. The even-ness or odd-ness of the permutation is referred to as the permutation s parity.

20 Even and Odd Permutations Definition A permutation is said to be even if it contains an even number of inversions, and is said to be odd otherwise. The even-ness or odd-ness of the permutation is referred to as the permutation s parity. Are the following permutations even or odd? 1 (6, 5, 4, 3, 2, 1) 2 (3, 8, 1, 4, 5, 6, 7, 2) 3 (3, 7, 4, 2, 8, 6, 1, 5)

21 Permutations and Parity (1 of 2) Question: what effect does interchanging two adjacent numbers in a permutation have on the parity? (..., a, b,...) (...,b, a,...)

22 Permutations and Parity (1 of 2) Question: what effect does interchanging two adjacent numbers in a permutation have on the parity? (..., a, b,...) (...,b, a,...) Answer: the two permutations have opposite parities.

23 Permutations and Parity (2 of 2) Task: compare the parities of the following two permutations. (..., a, b 1, b 2,..., b r,...) (...,b 1, b 2,..., b r, a,...)

24 Permutations and Parity (2 of 2) Task: compare the parities of the following two permutations. (..., a, b 1, b 2,..., b r,...) (...,b 1, b 2,..., b r, a,...) Answer: they have the same parity if r is even and opposite parity if r is odd.

25 Five Puzzle Consider the 2 3 version of the puzzle:

26 Analysis of the Five Puzzle How many permutations of {1, 2, 3, 4, 5} are possible?

27 Analysis of the Five Puzzle How many permutations of {1, 2, 3, 4, 5} are possible? Using the permutation listing pattern, what effect does a horizontal move have on the parity of the permutation?

28 Analysis of the Five Puzzle How many permutations of {1, 2, 3, 4, 5} are possible? Using the permutation listing pattern, what effect does a horizontal move have on the parity of the permutation? What effect does a vertical move have on the parity of the permutation?

29 Analysis of the Five Puzzle How many permutations of {1, 2, 3, 4, 5} are possible? Using the permutation listing pattern, what effect does a horizontal move have on the parity of the permutation? What effect does a vertical move have on the parity of the permutation? What is the parity of the starting arrangement?

30 Analysis of the Five Puzzle How many permutations of {1, 2, 3, 4, 5} are possible? Using the permutation listing pattern, what effect does a horizontal move have on the parity of the permutation? What effect does a vertical move have on the parity of the permutation? What is the parity of the starting arrangement? What is the parity of any rearrangement of the starting arrangement?

31 Analysis of the Five Puzzle How many permutations of {1, 2, 3, 4, 5} are possible? Using the permutation listing pattern, what effect does a horizontal move have on the parity of the permutation? What effect does a vertical move have on the parity of the permutation? What is the parity of the starting arrangement? What is the parity of any rearrangement of the starting arrangement? How many different rearrangements of the starting arrangement are possible?

32 Challenge Given a starting arrangement of following rearrangements can be achieved?, which of the (A) (B) (C)

33 Ordering of Fifteen Puzzle Ignoring the blank, use the following ordering for the Fifteen Puzzle: Starting permutation: (1, 2, 3, 4, 8, 7, 6, 5, 9, 10, 11, 12, 15, 14, 13)

34 Homework 1 How many permutations are there for {1, 2, 3,...,15}? 2 What is the parity of the starting permutation? 3 What effect on parity does a horizontal movement of the pieces have? 4 What effect on parity does a vertical movement of the pieces have? (Consider all cases.) 5 Is the solution to the Fifteen Puzzle achievable? Why or why not? 6 How many achievable permutations exist for the Fifteen Puzzle given its starting permutation?

35 Hi Q Consider a board containing 33 holes in a pattern as below. Suppose a peg is in each hole except for the center hole.

36 Hi Q Objective Whenever two adjacent (either horizontally or vertically) holes are occupied and the next hole in the same line is empty, the two pegs from the occupied holes may be removed and one of them placed in the third hole. In other words, one peg jumps over the other and lands in the vacant hole. The peg that was jumped is removed. 1 Can only one peg be left? 2 If this is accomplished, in which holes can the final peg be found?

37 Hi Q Coloring and Parity Suppose we color the squares red, black, and white. What are the counts of occupied red, black, and white squares?

38 Jumps and Parity What effect does a jump have on the counts of occupied red, black, and white squares?

39 Final Position and Parity If only one peg is left on the board, what will be the counts of the occupied red, black and white squares?

40 Final Position and Parity If only one peg is left on the board, what will be the counts of the occupied red, black and white squares? Which of these is compatible with the parity of the starting counts?

41 Final Position and Parity If only one peg is left on the board, what will be the counts of the occupied red, black and white squares? Which of these is compatible with the parity of the starting counts? Which squares can be left occupied last?

42 Numbering the Squares A jump can be denoted by an ordered pair like (46, 44) meaning the peg in square 46 jumps to square 44 and the peg in 45 is removed.

43 Homework Find a sequence of jumps that leaves a single square occupied.

Permutations. = f 1 f = I A

Permutations. = f 1 f = I A Permutations. 1. Definition (Permutation). A permutation of a set A is a bijective function f : A A. The set of all permutations of A is denoted by Perm(A). 2. If A has cardinality n, then Perm(A) has