HMMT 十一月 2009 · 冲刺赛 · 第 2 题

英文解析

2. [ 5 ] A knight begins on the lower-left square of a standard chessboard. How many squares couldthe knight end up at after exactly 2009 legal knight’s moves? (A knight’s move is 2 squares eitherhorizontally or vertically, followed by 1 square in a direction perpendicular to the first.)
   Answer: 32 The knight goes from a black square to a white square on every move, or vice versa,
   so after 2009 moves he must be on a square whose color is opposite of what he started on. So he canonly land on half the squares after 2009 moves. Note that he can access any of the 32 squares (thereare no other parity issues) because any single jump can also be accomplished in 3 jumps, so with 2009
   jumps, he can land on any of the squares of the right color.