758 research outputs found

    Solitaire Clobber

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    Clobber is a new two-player board game. In this paper, we introduce the one-player variant Solitaire Clobber where the goal is to remove as many stones as possible from the board by alternating white and black moves. We show that a checkerboard configuration on a single row (or single column) can be reduced to about n/4 stones. For boards with at least two rows and two columns, we show that a checkerboard configuration can be reduced to a single stone if and only if the number of stones is not a multiple of three, and otherwise it can be reduced to two stones. We also show that in general it is NP-complete to decide whether an arbitrary Clobber configuration can be reduced to a single stone.Comment: 14 pages. v2 fixes small typ

    PushPush and Push-1 are NP-hard in 2D

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    We prove that two pushing-blocks puzzles are intractable in 2D. One of our constructions improves an earlier result that established intractability in 3D [OS99] for a puzzle inspired by the game PushPush. The second construction answers a question we raised in [DDO00] for a variant we call Push-1. Both puzzles consist of unit square blocks on an integer lattice; all blocks are movable. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover when a block is pushed it slides the maximal extent of its free range. In the Push-1 version, the agent can only push one block one square at a time, the minimal extent---one square. Both NP-hardness proofs are by reduction from SAT, and rely on a common construction.Comment: 10 pages, 11 figures. Corrects an error in the conference version: Proc. of the 12th Canadian Conference on Computational Geometry, August 2000, pp. 211-21

    PushPush is NP-hard in 2D

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    We prove that a particular pushing-blocks puzzle is intractable in 2D, improving an earlier result that established intractability in 3D [OS99]. The puzzle, inspired by the game *PushPush*, consists of unit square blocks on an integer lattice. An agent may push blocks (but never pull them) in attempting to move between given start and goal positions. In the PushPush version, the agent can only push one block at a time, and moreover, each block, when pushed, slides the maximal extent of its free range. We prove this version is NP-hard in 2D by reduction from SAT.Comment: 18 pages, 13 figures, 1 table. Improves cs.CG/991101

    Bust-a-Move/Puzzle Bobble is NP-Complete

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    We prove that the classic 1994 Taito video game, known as Puzzle Bobble or Bust-a-Move, is NP-complete. Our proof applies to the perfect-information version where the bubble sequence is known in advance, and it uses just three bubble colors.Comment: 9 pages, 9 figures. Corrected mistakes in gadget

    Embedding Stacked Polytopes on a Polynomial-Size Grid

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    A stacking operation adds a dd-simplex on top of a facet of a simplicial dd-polytope while maintaining the convexity of the polytope. A stacked dd-polytope is a polytope that is obtained from a dd-simplex and a series of stacking operations. We show that for a fixed dd every stacked dd-polytope with nn vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by O(n2log(2d))O(n^{2\log(2d)}), except for one axis, where the coordinates are bounded by O(n3log(2d))O(n^{3\log(2d)}). The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure
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