Dana Ernst
February 25, 2012
640

# 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.

## Dana Ernst

February 25, 2012

## Transcript

1. Permutation Puzzles
Math Teachers’ Circles
February 28, 2012
Dana C. Ernst
Plymouth State University
Email: [email protected]
Web: http://danaernst.com
1

2. Permutations
2

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

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?

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.

6. 3
The Pebble Problem

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.

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.
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●

9. 4
The Pebble Problem (Continued)

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.

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.
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●

12. 5
The Pebble Problem (Continued)

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.

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.
● ● ● ● ●
● ● ● ˒ ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●

15. 6
The Pebble Problem (Continued)

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.

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.
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●
● ● ● ● ●

18. 7
The Pebble Problem (Continued)

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

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

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.
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●

22. 8
The Pebble Problem (Continued)

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

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

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?

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?

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?

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?

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?

30. 9
SpinpossibleTM

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

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

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.

34. 10
SpinpossibleTM (Continued)

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

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)

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)

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)

39. 11
SpinpossibleTM (Continued)

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

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

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

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

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

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

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

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

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)

49. 12
SpinpossibleTM (Continued)

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

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)

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)

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)

54. 13
SpinpossibleTM (Continued)

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

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

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)

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)

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)

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)

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)

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)

63. 14
SpinpossibleTM (Continued)

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)

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
A 1×5 puzzle
SpinpossibleTM (Continued)

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
A 1×5 puzzle
SpinpossibleTM (Continued)

67. 15
SpinpossibleTM (Continued)

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)

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)

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)

71. 16
SpinpossibleTM (Continued)

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)

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)

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)

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)

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)

77. 19
Some questions
SpinpossibleTM (Continued)

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

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

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

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)

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)

83. 20
Some difficult questions
SpinpossibleTM (Continued)

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

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)

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)

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)

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)
Spinpossible” by Alex Sutherland & Andrew
Sutherland. [http://arxiv.org/abs/1110.6645]

89. 21
SpinpossibleTM (Continued)
Connections to Group Theory

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.

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.

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.

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.

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:

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)

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)

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

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.

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.

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.

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!