Mathematics of the game Spinpossible

The game Spinpossible is played on a 3 by 3 board of scrambled tiles numbered 1 to 9, each of which may be right-side-up or up-side-down. The objective of the game is to return the board to the standard configuration where tiles are arranged in numerical order and right-side-up. This is accomplished by a sequence of "spins", each of which rotates a rectangular region of the board by 180 degrees. The goal is to minimize the number of spins used. It turns out that the group generated by the set of spins allowed in Spinpossible is identical to the symmetry group of the 9 dimensional hyper-cube. A number of interesting results about Spinpossible have been shown either computationally or analytically. Using brute-force, Sutherland and Sutherland verified that every scrambled board can be solved in at most 9 moves. In this talk, we will relay our progress on finding a short proof of this fact.

This poster was presented by my undergraduate research students Dane Jacobson and Michael Woodward on on April 26, 2013 at the 2013 NAU Undergraduate Research Symposium at Northern Arizona University.

Transcript

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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:
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   2
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   9
   .
   1
   .
   4
   .
   6
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   5
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   7
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   3
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   8
   →
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   2
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   8
   . 3
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   4
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   5
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   6
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   7
   .
   1
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   9
   →
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   2
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   1
   . 3
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   4
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   5
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   6
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   7
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   8
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   9
   →
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   . 2
   . 3
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   5
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   6
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   7
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   8
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   9
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   .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)
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   .
   (j)
   . .
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   (i)
   .
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   .
   .Example
   .
   .
   R13
   is the entire top row of a board:
   .
   .
   (1)
   .
   . (3)
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   .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
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   2
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   9
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   1
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   4
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   6
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   7
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   3
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   8
   s29
   −
   →
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   2
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   8
   . 3
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   4
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   7
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   1
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   9
   s28
   −
   →
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   1
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   5
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   9
   s12
   −
   →
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   1
   . 2
   . 3
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   4
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   5
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   7
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   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.
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   9
   →
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   8
   . 3
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   6
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   5
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   7
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   2
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   9
   ←
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   1
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   . 3
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   4
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   5
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   7
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   8
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   9
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   .Definition
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   .
   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
   .
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   2
   . 1 → .
   .
   1
   .
   2
   → .
   .
   1
   .
   2
   → .
   .
   1
   . 2
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   .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
   .
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   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
   .
   .
   .
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   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 [dmj69@nau.edu], Michael [mw342] Typeset using L
   A
   TEX, TikZ, and beamerposter
   

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