Consider $n^2-1$ unit-square blocks in an $n \times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 \times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical $1 \times 2$ and horizontal $2 \times 1$ movable blocks and 4-color Subway Shuffle.

中文翻译：

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带有固定块的 1 x 1 高峰时间是 PSPACE 完整的

考虑 $n \times n$ 方板中的 $n^2-1$ 个单位方块，其中每个方块都被标记为水平可移动（仅）、垂直可移动（仅）或不可移动——Rush Hour 的一种变体只有 $1 \times 1$ 汽车和固定块。我们证明，通过面向 2 色的 Subway Shuffle 从非确定性约束逻辑中减少，决定给定块是否可以到达棋盘的左边缘是 PSPACE 完全的。相比之下，多项式时间算法用于确定给定块是否可以移动一个空间，或者每个块何时不可移动或可以水平和垂直移动。我们的结果回答了 Tromp 和 Cilibrasi 提出的一个 15 年前的开放问题，