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.

Transcript

1. 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
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 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
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. 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
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