HMMT 二月 2002 · 团队赛 · 第 8 题

英文解析

8. [35] In this problem only, assume that s = 4 and that exactly one board square, say squarenumber n , is marked with an arrow. Determine all choices of n that maximize the average distance 1
   in squares the first player will travel in his first two turns.
   Solution. Because expectation is linear, the average distance the first player travels in his first twoturns is the average sum of two rolls of his die (which does not depend on the board configuration)
   plus four times the probability that he lands on the arrow on one of his first two turns. Thus we justneed to maximize the probability that player 1 lands on the arrow in his first two turns. If n ≥ 5,
   player 1 cannot land on the arrow in his first turn, so he encounters the arrow with probability atmost 1 / 4. If instead n ≤ 4, player 1 has a 1 / 4 chance of landing on the arrow on his first turn.
   If he misses, then he has a 1 / 4 chance of hitting the arrow on his second turn provided that heis not beyond square n already. The chance that player 1’s first roll left him on square n − 1 orfarther left is ( n − 1) / 4. Hence his probability of benefiting from the arrow in his first two turns is
   1 / 4 + (1 / 4)( n − 1) / 4, which is maximized for n = 4, where it is greater than the value of 1 / 4 thatwe get from n ≥ 5. Hence the answer is n = 4 .