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

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

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

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

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

Observations Comments • 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

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. 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. Dane Jacobson & Michael Woodward Directed by D.C. Ernst The mathematics of the game Spinpossible 7 / 15

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

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

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

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

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

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

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

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