Permutation Puzzles

Math Teachers' Circle at University of Nebraska, February 28, 2012.

A permutation of a set of objects is simply an arrangement of those objects into a particular order. In this Math Teachers' Circle we will play and explore several games and puzzles involving permutations. Using hands-on activities, we will examine puzzles of varying levels of difficulty and along the way, we will look for patterns, make and test hypotheses, and discuss possible variations of the puzzles.

Transcript

1. Permutation Puzzles
   Math Teachers’ Circles
   University of Nebraska at Omaha
   February 28, 2012
   Dana C. Ernst
   Plymouth State University
   Email: dana@danaernst.com
   Web: http://danaernst.com
   Twitter: @danaernst & @IBLMath
   1
   

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2. Permutations
   2
   

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3. A permutation of a set of objects is simply an
   arrangement of those objects into a particular order.
   Permutations
   2
   

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4. A permutation of a set of objects is simply an
   arrangement of those objects into a particular order.
   Permutations
   2
   Can you think of examples?
   

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5. A permutation of a set of objects is simply an
   arrangement of those objects into a particular order.
   Permutations
   2
   Can you think of examples?
   In this workshop, we will explore puzzles related to
   permutations of objects in m×n grids.
   

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6. 3
   The Pebble Problem
   

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7. 3
   The Pebble Problem
   The 5×5 puzzle
   Imagine you have 25 pebbles, each occupying one
   square on a 5×5 grid. We will explore 3 variations of a
   permutation puzzle involving this grid.
   

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8. 3
   The Pebble Problem
   The 5×5 puzzle
   Imagine you have 25 pebbles, each occupying one
   square on a 5×5 grid. We will explore 3 variations of a
   permutation puzzle involving this grid.
   ● ● ● ● ●
   ● ● ● ● ●
   ● ● ● ● ●
   ● ● ● ● ●
   ● ● ● ● ●
   

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9. 4
   The Pebble Problem (Continued)
   

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10. 4
    The Pebble Problem (Continued)
    Variation 1
    Move every pebble to a unique adjacent square by
    moving up, down, left, or right. If this is possible,
    describe a solution. If impossible, explain why.
    

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11. 4
    The Pebble Problem (Continued)
    Variation 1
    Move every pebble to a unique adjacent square by
    moving up, down, left, or right. If this is possible,
    describe a solution. If impossible, explain why.
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    

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12. 5
    The Pebble Problem (Continued)
    

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13. 5
    The Pebble Problem (Continued)
    Variation 2
    Suppose that all but one pebble (your choice which
    one) must move to a unique adjacent square by
    moving up, down, left, or right. If this is possible,
    describe a solution. If impossible, explain why.
    

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14. 5
    The Pebble Problem (Continued)
    Variation 2
    Suppose that all but one pebble (your choice which
    one) must move to a unique adjacent square by
    moving up, down, left, or right. If this is possible,
    describe a solution. If impossible, explain why.
    ● ● ● ● ●
    ● ● ● ˒ ●
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    

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15. 6
    The Pebble Problem (Continued)
    

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16. 6
    The Pebble Problem (Continued)
    Variation 3
    Consider Variation 1 again, but this time also allow
    diagonal moves to adjacent squares. If this is possible,
    describe a solution. If impossible, explain why.
    

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17. 6
    The Pebble Problem (Continued)
    Variation 3
    Consider Variation 1 again, but this time also allow
    diagonal moves to adjacent squares. If this is possible,
    describe a solution. If impossible, explain why.
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    ● ● ● ● ●
    

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18. 7
    The Pebble Problem (Continued)
    

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19. 7
    The Pebble Problem (Continued)
    The 6×6 puzzle
    Imagine you have 36 pebbles, each occupying one
    square on a 6×6 grid.
    

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20. 7
    The Pebble Problem (Continued)
    The 6×6 puzzle
    Imagine you have 36 pebbles, each occupying one
    square on a 6×6 grid.
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    

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21. 7
    The Pebble Problem (Continued)
    The 6×6 puzzle
    Imagine you have 36 pebbles, each occupying one
    square on a 6×6 grid.
    Repeat Variations 1, 2, & 3 for 6×6 grid.
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    ● ● ● ● ● ●
    

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22. 8
    The Pebble Problem (Continued)
    

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23. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    

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24. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    Questions
    

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25. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    Questions
    For which n does Variation 1 have a solution? Why?
    

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26. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    Questions
    For which n does Variation 1 have a solution? Why?
    For which n does Variation 2 have a solution? Why?
    

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27. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    Questions
    For which n does Variation 1 have a solution? Why?
    For which n does Variation 2 have a solution? Why?
    For which n does Variation 3 have a solution? Why?
    

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28. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    Questions
    For which n does Variation 1 have a solution? Why?
    For which n does Variation 2 have a solution? Why?
    For which n does Variation 3 have a solution? Why?
    When there is a solution, how many are there?
    Note: I do not know the answer. Difficult?
    

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29. 8
    The Pebble Problem (Continued)
    The general n×n puzzle
    Imagine you have n2 pebbles, each occupying one
    square on n×n grid.
    Questions
    For which n does Variation 1 have a solution? Why?
    For which n does Variation 2 have a solution? Why?
    For which n does Variation 3 have a solution? Why?
    When there is a solution, how many are there?
    Note: I do not know the answer. Difficult?
    What other variations can we come up with?
    

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30. 9
    SpinpossibleTM
    

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31. 9
    SpinpossibleTM
    The following puzzles are inspired by the game
    Spinpossible [http://spinpossible.com].
    

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32. 9
    SpinpossibleTM
    The following puzzles are inspired by the game
    Spinpossible [http://spinpossible.com].
    

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33. 9
    SpinpossibleTM
    The following puzzles are inspired by the game
    Spinpossible [http://spinpossible.com].
    The object of the game is to convert a scrambled
    board into the solved board by rotating m×n subgrids.
    

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34. 10
    SpinpossibleTM (Continued)
    

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35. 10
    The 1×3 game
    SpinpossibleTM (Continued)
    

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36. 10
    The 1×3 game
    Imagine you have a 1×3 grid filled with the numbers
    1, 2, 3 in any order. In addition, any subset of the
    numbers may be turned upside down.
    SpinpossibleTM (Continued)
    

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37. 10
    The 1×3 game
    Imagine you have a 1×3 grid filled with the numbers
    1, 2, 3 in any order. In addition, any subset of the
    numbers may be turned upside down.
    By applying sequences of 180o rotations of 1×1, 1×2,
    or 1×3 subgrids, can you convert the scrambled
    board into the solved grid?
    SpinpossibleTM (Continued)
    

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38. 10
    The 1×3 game
    Imagine you have a 1×3 grid filled with the numbers
    1, 2, 3 in any order. In addition, any subset of the
    numbers may be turned upside down.
    By applying sequences of 180o rotations of 1×1, 1×2,
    or 1×3 subgrids, can you convert the scrambled
    board into the solved grid?
    Let’s try one together.
    SpinpossibleTM (Continued)
    

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39. 11
    SpinpossibleTM (Continued)
    

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40. 11
    Example of 1×3 game
    SpinpossibleTM (Continued)
    

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41. 11
    Example of 1×3 game
    1
    2
    3
    SpinpossibleTM (Continued)
    

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42. 11
    Example of 1×3 game
    1
    2
    3
    SpinpossibleTM (Continued)
    

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43. 11
    Example of 1×3 game
    1
    2
    3
    2
    1
    3
    SpinpossibleTM (Continued)
    

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44. 11
    Example of 1×3 game
    1
    2
    3
    2
    1
    3
    SpinpossibleTM (Continued)
    

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45. 11
    Example of 1×3 game
    1
    2
    3
    2
    1
    3
    2
    1
    3
    SpinpossibleTM (Continued)
    

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46. 11
    Example of 1×3 game
    1
    2
    3
    2
    1
    3
    2
    1
    3
    SpinpossibleTM (Continued)
    

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47. 11
    Example of 1×3 game
    1
    2
    3
    2
    1
    3
    2
    1
    3
    2
    1 3
    SpinpossibleTM (Continued)
    

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48. 11
    Example of 1×3 game
    1
    2
    3
    2
    1
    3
    2
    1
    3
    2
    1 3
    Are there other solutions? Is there a “shorter” one?
    SpinpossibleTM (Continued)
    

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49. 12
    SpinpossibleTM (Continued)
    

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50. 12
    OK, your turn! Try solving the following boards. See if
    you can obtain an optimal solution.
    SpinpossibleTM (Continued)
    

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51. 12
    OK, your turn! Try solving the following boards. See if
    you can obtain an optimal solution.
    2
    3
    1
    Min # Spins: 2
    SpinpossibleTM (Continued)
    

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52. 12
    OK, your turn! Try solving the following boards. See if
    you can obtain an optimal solution.
    2
    3
    1
    Min # Spins: 2
    1
    3 2 Min # Spins: 3
    SpinpossibleTM (Continued)
    

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53. 12
    OK, your turn! Try solving the following boards. See if
    you can obtain an optimal solution.
    2
    3
    1
    Min # Spins: 2
    1
    3 2 Min # Spins: 3
    3
    2
    1
    Min # Spins: 3
    SpinpossibleTM (Continued)
    

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54. 13
    SpinpossibleTM (Continued)
    

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55. 13
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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56. 13
    How many different scrambled boards are there?
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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57. 13
    How many different scrambled boards are there?
    Does every scrambled board have at least one
    solution? More than one solution?
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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58. 13
    How many different scrambled boards are there?
    Does every scrambled board have at least one
    solution? More than one solution?
    Are some boards “more scrambled” than others?
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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59. 13
    How many different scrambled boards are there?
    Does every scrambled board have at least one
    solution? More than one solution?
    Are some boards “more scrambled” than others?
    Does the order of the moves matter?
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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60. 13
    How many different scrambled boards are there?
    Does every scrambled board have at least one
    solution? More than one solution?
    Are some boards “more scrambled” than others?
    Does the order of the moves matter?
    Can you come up with an algorithm that will
    guarantee a solution? Will it be an optimal solution?
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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61. 13
    How many different scrambled boards are there?
    Does every scrambled board have at least one
    solution? More than one solution?
    Are some boards “more scrambled” than others?
    Does the order of the moves matter?
    Can you come up with an algorithm that will
    guarantee a solution? Will it be an optimal solution?
    What is the maximum number of spins required to
    guarantee a solution?
    Questions for the 1×3 board
    SpinpossibleTM (Continued)
    

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62. 13
    How many different scrambled boards are there?
    Does every scrambled board have at least one
    solution? More than one solution?
    Are some boards “more scrambled” than others?
    Does the order of the moves matter?
    Can you come up with an algorithm that will
    guarantee a solution? Will it be an optimal solution?
    What is the maximum number of spins required to
    guarantee a solution?
    Questions for the 1×3 board
    Let’s try out some other size boards.
    SpinpossibleTM (Continued)
    

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63. 14
    SpinpossibleTM (Continued)
    

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64. 14
    The 1×5 game
    The same rules apply, but now we have 5 numbers &
    can rotate subgrids up to 5 squares wide.
    SpinpossibleTM (Continued)
    

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65. 14
    The 1×5 game
    The same rules apply, but now we have 5 numbers &
    can rotate subgrids up to 5 squares wide.
    Try to solve the following 1×5 board. Do not worry
    about whether your solution is optimal.
    A 1×5 puzzle
    SpinpossibleTM (Continued)
    

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66. 14
    1
    2
    3
    5
    4
    The 1×5 game
    The same rules apply, but now we have 5 numbers &
    can rotate subgrids up to 5 squares wide.
    Try to solve the following 1×5 board. Do not worry
    about whether your solution is optimal.
    A 1×5 puzzle
    SpinpossibleTM (Continued)
    

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67. 15
    SpinpossibleTM (Continued)
    

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68. 15
    The 2×2 game
    The solved board has the numbers 1-4 in order from
    left to right & top to bottom. We can rotate subgrids
    of size 1×1, 1×2, 2×1, & 2×2.
    SpinpossibleTM (Continued)
    

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69. 15
    The 2×2 game
    The solved board has the numbers 1-4 in order from
    left to right & top to bottom. We can rotate subgrids
    of size 1×1, 1×2, 2×1, & 2×2.
    Try to solve the following 2×2 board.
    A 2×2 puzzle
    SpinpossibleTM (Continued)
    

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70. 15
    The 2×2 game
    The solved board has the numbers 1-4 in order from
    left to right & top to bottom. We can rotate subgrids
    of size 1×1, 1×2, 2×1, & 2×2.
    Try to solve the following 2×2 board.
    A 2×2 puzzle
    4
    2
    3
    1
    SpinpossibleTM (Continued)
    

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71. 16
    SpinpossibleTM (Continued)
    

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72. 16
    The 3×3 game
    I think you’ve got the hang of it by now. This is the
    standard Spinpossible game.
    SpinpossibleTM (Continued)
    

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73. 16
    The 3×3 game
    I think you’ve got the hang of it by now. This is the
    standard Spinpossible game.
    Some 3×3 puzzles
    Try to solve the following 3×3 boards using the
    minimum number of spins.
    SpinpossibleTM (Continued)
    

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74. 16
    The 3×3 game
    I think you’ve got the hang of it by now. This is the
    standard Spinpossible game.
    Some 3×3 puzzles
    Try to solve the following 3×3 boards using the
    minimum number of spins.
    3
    6 2
    4
    5
    1
    7 8 9
    Min # Spins: 3
    SpinpossibleTM (Continued)
    

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75. 17
    3
    6
    2
    4 5
    1
    7
    8
    9
    Min # Spins: 4
    2
    1 7
    3
    6
    4 5
    8 9
    Min # Spins: 3
    Some 3×3 puzzles (Continued)
    SpinpossibleTM (Continued)
    

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76. 18
    SpinpossibleTM (Continued)
    2 1
    7
    3
    6 4
    5
    8
    9
    Min # Spins: 6
    If you figured out all the others, try this last one.
    Some 3×3 puzzles (Continued)
    

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77. 19
    Some questions
    SpinpossibleTM (Continued)
    

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78. 19
    How many different scrambled boards are there for
    the 1×5 board?
    Some questions
    SpinpossibleTM (Continued)
    

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79. 19
    How many different scrambled boards are there for
    the 1×5 board?
    2×2 board?
    Some questions
    SpinpossibleTM (Continued)
    

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80. 19
    How many different scrambled boards are there for
    the 1×5 board?
    2×2 board?
    3×3 board?
    Some questions
    SpinpossibleTM (Continued)
    

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81. 19
    How many different scrambled boards are there for
    the 1×5 board?
    2×2 board?
    3×3 board?
    m×n board?
    Some questions
    SpinpossibleTM (Continued)
    

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82. 19
    How many different scrambled boards are there for
    the 1×5 board?
    2×2 board?
    3×3 board?
    m×n board?
    Does the order of the moves matter?
    Some questions
    SpinpossibleTM (Continued)
    

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83. 20
    Some difficult questions
    SpinpossibleTM (Continued)
    

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84. 20
    Does every scrambled m×n board have at least one
    solution?
    Some difficult questions
    SpinpossibleTM (Continued)
    

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85. 20
    Does every scrambled m×n board have at least one
    solution?
    It turns out that the answer is yes.
    Some difficult questions
    SpinpossibleTM (Continued)
    

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86. 20
    Does every scrambled m×n board have at least one
    solution?
    It turns out that the answer is yes.
    For the 3×3 board, what is the maximum number of
    spins required to solve any scrambled board?
    Some difficult questions
    SpinpossibleTM (Continued)
    

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87. 20
    Does every scrambled m×n board have at least one
    solution?
    It turns out that the answer is yes.
    For the 3×3 board, what is the maximum number of
    spins required to solve any scrambled board?
    An exhaustive search has found that 9 spins are
    always sufficient (and sometimes necessary), but no
    short proof of this fact is known.
    Some difficult questions
    SpinpossibleTM (Continued)
    

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88. 20
    Does every scrambled m×n board have at least one
    solution?
    It turns out that the answer is yes.
    For the 3×3 board, what is the maximum number of
    spins required to solve any scrambled board?
    An exhaustive search has found that 9 spins are
    always sufficient (and sometimes necessary), but no
    short proof of this fact is known.
    Some difficult questions
    SpinpossibleTM (Continued)
    For more information, see “The mathematics of
    Spinpossible” by Alex Sutherland & Andrew
    Sutherland. [http://arxiv.org/abs/1110.6645]
    

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89. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    

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90. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    

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91. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    Some groups are commutative & some are not.
    

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92. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    Some groups are commutative & some are not.
    Groups may be finite or infinite.
    

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93. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    Some groups are commutative & some are not.
    Groups may be finite or infinite.
    Groups are intimately related to symmetry.
    

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94. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    Some groups are commutative & some are not.
    Groups may be finite or infinite.
    Groups are intimately related to symmetry.
    Two classic examples:
    

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95. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    Some groups are commutative & some are not.
    Groups may be finite or infinite.
    Groups are intimately related to symmetry.
    Two classic examples:
    Integers under the operation of addition.
    (Infinite, commutative, 0 is the identity)
    

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96. 21
    SpinpossibleTM (Continued)
    Connections to Group Theory
    A group is a set together with a binary operation
    satisfying closure & associativity, & possessing an
    identity & inverses.
    Some groups are commutative & some are not.
    Groups may be finite or infinite.
    Groups are intimately related to symmetry.
    Two classic examples:
    Integers under the operation of addition.
    (Infinite, commutative, 0 is the identity)
    Rigid symmetries of a square under composition.
    (Finite with 8 elements, not commutative, “do
    nothing” is the identity)
    

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97. 22
    SpinpossibleTM (Continued)
    Connections to Group Theory (Continued)
    

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98. 22
    SpinpossibleTM (Continued)
    Connections to Group Theory (Continued)
    The set of spins for the 1×2 Spinpossible board is
    really just the symmetry group for square.
    

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99. 22
    SpinpossibleTM (Continued)
    Connections to Group Theory (Continued)
    The set of spins for the 1×2 Spinpossible board is
    really just the symmetry group for square.
    The set of spins for the 1×3 Spinpossible board is the
    symmetry group for cube.
    

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100. 22
     SpinpossibleTM (Continued)
     Connections to Group Theory (Continued)
     The set of spins for the 1×2 Spinpossible board is
     really just the symmetry group for square.
     The set of spins for the 1×3 Spinpossible board is the
     symmetry group for cube.
     In general, the set of spins for the m×n Spinpossible
     board is a Coxeter group of type Bmn, which can be
     thought of as the permutations of mn coins, where
     we allow rearrangement of the coins, as well as
     flipping them over.
     

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101. 22
     SpinpossibleTM (Continued)
     Connections to Group Theory (Continued)
     The set of spins for the 1×2 Spinpossible board is
     really just the symmetry group for square.
     The set of spins for the 1×3 Spinpossible board is the
     symmetry group for cube.
     In general, the set of spins for the m×n Spinpossible
     board is a Coxeter group of type Bmn, which can be
     thought of as the permutations of mn coins, where
     we allow rearrangement of the coins, as well as
     flipping them over.
     Thank you!
     

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