HMMT 十一月 2008 · GEN2 赛 · 第 4 题

HMMT November 2008 — GEN2 Round — Problem 4

[ 6 ] Joe B. is frustrated with chess. He breaks the board, leaving a 4 × 4 board, and throws 3 black knights and 3 white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.)

解析

[ 6 ] Joe B. is frustrated with chess. He breaks the board, leaving a 4 × 4 board, and throws 3 black knights and 3 white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.) 9 Answer: We first compute the expected number of checks between a single knight-king pair. If 5 the king is located at any of the 4 corners, the knight has 2 possible checks. If the king is located in one of the 8 squares on the side of the board but not in the corner, the knight has 3 possible checks. If the king is located in any of the 4 central squares, the knight has 4 possible checks. Summing up, 4 · 2 + 8 · 3 + 4 · 4 = 48 of the 16 · 15 knight-king positions yield a single check, so each pair yields 48 1 = expected checks. Now, note that each of the 9 knight-king pairs is in each of 16 · 15 possible 16 · 15 5 1 9 positions with equal probability, so by linearity of expectation the answer is 9 · = . 5 5