2009 IMO Shortlist C6

BGR On a $999 \times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A nonintersecting route of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called cyclic, if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit?

BGR 在 $999 \times 999$ 的棋盘上，跛行车可以按以下方式移动：从任何方格，它可以移动到任何相邻的方格，即具有公共边的方格，并且每次移动都必须是一个转弯，即任何两个连续移动的方向必须垂直。跛行车的非相交路线由一系列成对的不同方格组成，跛行车可以通过允许的移动序列按该顺序访问这些方格。这种不相交的路线称为循环路线，如果跛行车可以在到达路线的最后一个方格后直接移动到路线的第一个方格并重新开始。跛行车的最长可能循环、不相交的路线会访问多少个方格？