恶魔硬币局
Devil's Penny
题目详情
你和恶魔在一张完美圆桌上玩放硬币游戏：轮流放同样大小的硬币，硬币不能重叠，平放桌面。谁先无法放置就输。你先手。
问：如何保证必胜？
You are in a game against devil, on a perfectly round table and with an infinite pile of pennies. He says, "OK, we'll take turns putting one penny down, no overlapping allowed, and the pennies must rest flat on the table surface. The first guy who can't put a penny down loses." You can go first. How will you guarantee victory?
Hint
Try the problem if coins have unit radii, and table has radius 1 & 3.
解析
先手必胜策略：
- 第一枚硬币放在圆桌中心。
- 之后每当对手在某点放一枚，你就在关于圆心的对径点放一枚（与圆心距离相同、方向相反）。
由于圆桌中心对称，这个“镜像应对”始终合法：对手能放，你也能放。最后对手先无处可放，你获胜。
Original Explanation
Solution
You place a penny right in the center of the table. After that, whenever the devil places a penny on the table, mimic his placement by placing a penny diametrically opposite and at same distance from center. If he has a place to place a penny, so will you. The devil will run out of places to put a quarter before you do.