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Then the puzzle can be cleared by going through the sequence $1\rightarrow 3 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 3 \rightarrow 4$, for a total of 7 moves. For instance, if there four squares with the patterns: 1 2 3 4 By switching the order in which you visit the squares, you can complete the puzzle in a greater or lesser number of moves. But there is an interesting combinatoric/graph theory problem as well. The challenge in the actual game is visual, being able to spot matching patterns across squares. Simultaneously traditional and contemporary, the glossy white tiles of New York City’s subways have invaded the bathrooms and kitchens of homes across the country. The number of instances of any pattern $p$ in the puzzle across all squares is always even, thus guaranteeing that a solution exists. The public and private life of subway tiles. If you end your move on an empty tile, you can start again from any. If the second square is empty after you move to it, you may select any square with at least one pattern and continue from there. A selection of puzzles and illustrations created for The New York Times game Vertex. You must then find a square with at least one of $b$ or $d$ in it, and so on until all the squares are empty.
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So if you start at a square with patterns $\$. When you choose that next square, the overlapping patterns disappear from both squares. The rules of the game are that you start at any square and then move to any other square that contains at least one of the patterns in your start square. Each square contains 4 of the $p$ patterns (although this can be generalized as well). Start with $m$ squares (in the official version, this is 30, in a 6x5 grid), and a set of $p>4$ possible patterns (typically this is a dozen or so, but the precise number doesn't matter). The New York Times has a daily puzzle named Tiles that works as follows.