We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1 x 2 pieces on a hexagonal grid board of 2n + 1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single empty cell on a board and forms a natural extension of the 15-puzzle to include rotational moves. We analyze the puzzle and completely characterize the cases when the puzzle can always be solved. We also study the complexity of determining whether a given set of colored pieces can be placed on a colored hexagonal grid board with matching colors. We show this problem is NP-complete for arbitrarily many colors, but solvable in randomized polynomial time if the number of colors is a fixed constant.

中文翻译：

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葫芦：带转动的滑块拼图

我们提出了一种名为 Gourds 的新型滑块拼图，其目标是使用滑动、转动和枢轴移动在 2n + 1 个单元格的六边形网格板上重新排列 1 x 2 块和 n 块。这个拼图在棋盘上有一个空单元格，形成了 15 拼图的自然延伸，包括旋转移动。我们分析谜题并完全描述谜题总是可以解决的情况。我们还研究了确定一组给定的彩色碎片是否可以放置在具有匹配颜色的彩色六边形网格板上的复杂性。我们证明这个问题对于任意多种颜色是 NP 完全的，但如果颜色数量是固定常数，则可以在随机多项式时间内解决。