PUMaC 2016 · 组合（B 组） · 第 6 题
PUMaC 2016 — Combinatorics (Division B) — Problem 6
题目详情
- A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess L-shape (2 units parallel to one axis and 1 unit parallel to the other) to one of eight possible locations, chosen at random. After 2016 such turns, what is the expected value of the square of the distance of the knight from the origin?
解析
- Suppose the knight is at ( a, b ) before a turn. Consider the two possible moves ( − 2 , − 1) and 2 2 2 2 ( a − 2) +( b − 1) +( a +2) +( b +1) 2 2 (+2 , +1). We have that = a + b +5. The six other possible moves 2 2 2 can be paired up similarly. Summing all possibilities, the expected value of a + b increases by 5 each turn, so after 2016 such turns, the answer is 2016 · 5 = 10080 . Problem written by Mel Shu. 1