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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
TEX, TikZ, and beamerposter