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Mathematics of the game Spinpossible

Dane Jacobson & Michael Woodward, Directed by Dana C. Ernst

Department of Mathematics & Statistics, Northern Arizona University

.

.Game Summary

.

.

The game Spinpossible is played on a 3×3 board. A scrambled board consists

of the numbers 1–9 arranged on the board, where each tile of the board

contains a single number and that number can be either right side up or up

side down.

The object of the game is to convert a scrambled board into the solved board

by applying a sequence of spins, where a spin consists of rotating an m × n

subrectangles by 180◦.

?

−

→ · · · ?

−

→

To win, you must return the scrambled board to the solved board using the

minimum possible number of spins.

.

.Example

.

.

Let’s play with an example:

.

.

2

.

9

.

1

.

4

.

6

.

5

.

7

.

3

.

8

→

.

.

2

.

8

. 3

.

4

.

5

.

6

.

7

.

1

.

9

→

.

.

2

.

1

. 3

.

4

.

5

.

6

.

7

.

8

.

9

→

.

.

1

. 2

. 3

.

4

.

5

.

6

.

7

.

8

.

9

.

.Deﬁntion

.

.

A rectangle Rij

denotes the rectangle having position (i) in the upper left

corner and position (j) in lower right corner. If i = j, then Rii

is the rectangle

consisting only of position (i).

.

.

(i)

.

.

.

.

.

.

.

(j)

. .

.

.

.

.

.

.

(i)

.

.

.

.

.Example

.

.

R13

is the entire top row of a board:

.

.

(1)

.

. (3)

.

.

.

.

.

.

.

.Deﬁnition

.

.

Given a rectangle R, we denote the corresponding spin of R by sR

. In particular,

if R = Rij

, then we may write sR

= sij

.

.

.Example

.

.

.

.

2

.

9

.

1

.

4

.

6

.

5

.

7

.

3

.

8

s29

−

→

.

.

2

.

8

. 3

.

4

.

5

.

6

.

7

.

1

.

9

s28

−

→

.

.

2

.

1

. 3

.

4

.

5

.

6

.

7

.

8

.

9

s12

−

→

.

.

1

. 2

. 3

.

4

.

5

.

6

.

7

.

8

.

9

This solution is expressed as s12s28s29

, like function composition (right to left).

.

.Comments

.

.

▶ 6 spin types: 1 × 1, 1 × 2, 2 × 2, 1 × 3, 2 × 3, 3 × 3 (36 total spins).

▶ sijsij

is the same as “doing nothing.”

▶ sijsmn

= smnsij

iﬀ Rij

and Rmn

are disjoint or have a common center.

.

.

1

. 2

. 3

.

4

.

5

.

6

.

7

.

8

.

9

→

.

.

1

.

8

. 3

.

6

.

5

.

4

.

7

.

2

.

9

←

.

.

1

. 2

. 3

.

4

.

5

.

6

.

7

.

8

.

9

.

.Deﬁnition

.

.

Spin3×3

is the group generated by all possible spins of the 3 × 3 board, where

the group operation is composition of spins.

.

.Comments

.

.

▶ Every b ∈ Spin3×3

can be expressed as a product of spins (not necessarily

unique).

▶ Every product of spins yields a scrambled board.

▶ Conversely, every scrambled board is determined by a product of spins.

▶ Consequently, there is a 1-1 correspondence between elements of Spin3×3

and

scrambled boards.

.

.Properties

.

.

▶ |Spin3×3| = 9!29 = 185, 794, 560 possible boards.

▶ If b ∈ Spin3×3

, then b−1 will return the corresponding scrambled board to the

solved board. In other words, b−1b corresponds to the solved board.

▶ Finding a solution to a scrambled board is equivalent to ﬁnding a minimal

length expression for b−1.

.

.Theorem (Upper Bound)

.

.

Every element of Spinm×n

can be expressed as a product of at most 3mn-(m+n).

.

.Deﬁnition

.

.

We deﬁne the number k(m, n) to be the maximum of all the minimal length

solutions in Spinm×n

.

.

.Example

.

.

Claim: k(1, 2) = 3. |Spin1×2| = 8, but an exhaustive approach yields the

following maximally scrambled board in Spin1×2

.

.

.

2

. 1 → .

.

1

.

2

→ .

.

1

.

2

→ .

.

1

. 2

.

.Theorem

.

.

It has been shown computationally that k(3, 3) = 9. That is, every board in

Spinpossible can be unscrambled in at most 9 moves.

.

.Proposition

.

.

Spin3×3

is the Coxeter group of type B9

, equivalently, it is isomorphic to the

symmetry group of the 9-dimensional hypercube.

.

.Deﬁnition

.

.

A reﬂection is a conjugation of a Coxeter generator.

.

.Theorem

.

.

It is well-known that the longest word in terms of reﬂections of the 9-

dimensional hypercube is 9.

.

.Comment

.

.

The spin s13

can be written as c0c1c2c0c1c0

, which is of even length, and hence

cannot be a conjugate. Therefore, not every spin is a reﬂection.

.

.Deﬁnition

.

.

▶ If tile i is upside down in any position (j), then tile i will require an odd

number of spins.

▶ If tile i is rightside up in any position (j), then tile i will require an even

number of spins.

.

.Example

.

.

.

.

3

. 1

. 2 s13

−

→ .

.

2

.

1

. 3 s12

−

→ .

.

1

. 2

. 3

▶ Tile 1 spin path: s12s13

(even)

▶ Tile 2 spin path: s12s13

(even)

▶ Tile 3 spin path: s13

(odd)

.

.Open Problems

.

.

▶ Show k(2, 2) = 5.

▶ Determine k(4, 4).

▶ Determine which spin types sij

are equivalent.

▶ Analyze the distribution of solution lengths in Spinm×n

.

Email: Dane [[email protected]], Michael [mw342] Typeset using L

A

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