. . . .. . . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . . .. . . . .. . . . .. 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 [

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