HMMT 二月 2004 · GEN2 赛 · 第 4 题

英文解析

4. A horse stands at the corner of a chessboard, a white square. With each jump, thehorse can move either two squares horizontally and one vertically or two vertically andone horizontally (like a knight moves). The horse earns two carrots every time it landson a black square, but it must pay a carrot in rent to rabbit who owns the chessboardfor every move it makes. When the horse reaches the square on which it began, it canleave. What is the maximum number of carrots the horse can earn without touchingany square more than twice?
   Solution: 0
   The horse must alternate white and black squares, and it ends on the same squarewhere it started. Thus it lands on the same number of black squares ( b ) as whitesquares ( w ). Thus, its net earnings will be 2 b − ( b + w ) = b − w = 0 carrots, regardlessof its path.