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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 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.

April 26, 2013

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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 [dmj69@nau.edu], Michael [mw342] Typeset using L A TEX, TikZ, and beamerposter