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# 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 talk was given by my undergraduate research students Dane Jacobson and Michael Woodward on March 2, 2013 at the Southwestern Undergraduate Research Conference (SUnMaRC) at the University of New Mexico. March 02, 2013

## Transcript

1. The mathematics of the game Spinpossible
Dane Jacobson & Michael Woodward
Directed by D.C. Ernst
Northern Arizona University
Mathematics & Statistics Department
University of New Mexico
March 2, 2013
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 1 / 15

2. Introduction to Spinpossible
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.
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 2 / 15

3. Introduction to Spinpossible (continued)
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
In this case, we were able to optimally solve the scrambled board in 3 moves in a way
that is not unique.
Comment
• We will make a distinction between a tile labeled i versus position (i). For
example, in the scrambled board above, the tile labeled 9 is in position (2).
• A board is in the solved position iﬀ every tile i is in position (i) with the correct
orientation.
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 3 / 15

4. Rectangle
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)
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 4 / 15

5. Spin
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 s12
s28
s29
, like function composition (right to left).
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 5 / 15

6. Observations
• 6 spin types: 1 × 1, 1 × 2, 2 × 2, 1 × 3, 2 × 3, 3 × 3 (36 total spins).
• sij
sij
is the same as “doing nothing.”
• sij
smn
= smn
sij
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
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 6 / 15

7. Spin3×3
Deﬁnition
Spin3×3
is the group generated by all possible spins of the 3 × 3 board, where the
group operation is composition of spins.
• 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.
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 7 / 15

8. Spin3×3
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.
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 8 / 15

9. Minimal generating set of Spin3×3
Proposition
Spin3×3
is the Coxeter group of type B9
, equivalently, it is isomorphic to the
symmetry group of the 9-dimensional hypercube.
Corollary
Let
c0
= s11
c1
= s12
c2
= s23
c3
= s36
.
.
.
c8
= s89
Then Spin3×3
has a generating set with these 9 elements.
c0
c1
c2
c3
c4
c5
c6
c7
c8
4
Figure : Coxeter graph of type B9.
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 9 / 15

10. Upper Bound Theorem
Kindergarden algorithm
2 3 3
2 3 3
2 2 1
This shows that Spinpossible has an upper bound of 21 moves to solve any board.
Theorem (Upper Bound)
Every element of Spinm×n
can be expressed as a product of at most 3mn-(m+n).
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 10 / 15

11. Optimal solutions
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, 1) = 1. This number is the the maximum of all the minimal length
solutions in Spin1×1
.
• |Spin1×1| = 2 possible boards.
• Case 1: Solution length 0
1
• Case 2: Solution length 1
1
→ 1
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
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 11 / 15

12. Optimal solutions (continued)
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.
Goal
Find a short proof that k(3, 3) = 9. There are several ways to approach this:
• algorithmically (unlikely)
• by properties of reﬂection on the 9-dimensional hypercube.
• concepts of parity
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 12 / 15

13. Reﬂections
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.
Question
Is this related to k(3, 3) being 9?
Unfortunately...
Counterexample
The spin s13
can be written as c0
c1
c2
c0
c1
c0
, which is of even length, and hence
cannot be a conjugate. Therefore, not every spin is a reﬂection.
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 13 / 15

14. Parity
Deﬁnition
Parity plays a role in the number of spins required to orient a tile in its correct
position.
1. If tile i is upside down in any position (j), then tile i will require an odd number
of spins.
2. 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: s12
s13
(even)
• Tile 2 spin path: s12
s13
(even)
• Tile 3 spin path: s13
(odd)
Can we analyze parity in order to maximize overlap and, in turn, minimize the
number of moves?
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 14 / 15

15. Conclusion
Open Problems
• Determine k(4, 4).
• Analyze the distribution of solution lengths in Spinm×n
.
• Determine which spin types sij
are equivalent.
• Give bounds on the number of boards with unique solutions in Spinm×n
.
• Determine whether limn→∞
k(n, n)/n2 exists, and if so, its value.
• Show k(2, 2) = 5.
Thank You!
Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 15 / 15