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

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Permutations 2

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

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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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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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3 The Pebble Problem

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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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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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4 The Pebble Problem (Continued)

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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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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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5 The Pebble Problem (Continued)

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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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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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6 The Pebble Problem (Continued)

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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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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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7 The Pebble Problem (Continued)

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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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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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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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8 The Pebble Problem (Continued)

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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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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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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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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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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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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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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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9 SpinpossibleTM

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

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

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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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10 SpinpossibleTM (Continued)

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10 The 1×3 game SpinpossibleTM (Continued)

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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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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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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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11 SpinpossibleTM (Continued)

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11 Example of 1×3 game SpinpossibleTM (Continued)

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

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

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

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

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

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

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

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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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12 SpinpossibleTM (Continued)

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

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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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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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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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13 SpinpossibleTM (Continued)

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

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

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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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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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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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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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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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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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14 SpinpossibleTM (Continued)

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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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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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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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15 SpinpossibleTM (Continued)

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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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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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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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16 SpinpossibleTM (Continued)

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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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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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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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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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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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19 Some questions SpinpossibleTM (Continued)

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

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

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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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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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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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20 Some difficult questions SpinpossibleTM (Continued)

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

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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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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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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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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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21 SpinpossibleTM (Continued) Connections to Group Theory

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

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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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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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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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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!