圆桌放硬币博弈：先手必胜
Coin Game on a Round Table
题目详情
两名玩家轮流在一张圆桌上放置同样大小的硬币，硬币之间不能重叠。无法再放下硬币的玩家输。
证明：先手有必胜策略。
英文原题
Two players take turns placing identical coins on a round table. The coins cannot overlap. The first player who cannot place a coin loses. Prove that the first player has a winning strategy.
解析
先手第一步把硬币放在圆桌的正中心。
之后采用“中心对称应对”策略：
- 若后手在某位置放下一枚硬币，先手就在以圆心为对称中心的对称位置放下一枚硬币。
该策略可行性：圆桌对圆心中心对称，且两枚同样大小硬币若后手那枚不与中心硬币重叠，则其对称位置到中心的距离相同，也不会与中心硬币重叠；若对称位置与桌上某枚硬币重叠，则对称回去会导致后手那步本身就与对应硬币重叠，矛盾。
因此只要后手有合法一步，先手就一定也有合法应对一步。
由于硬币可放的步数是有限的，最终必是后手先无路可走而输，故先手必胜。
英文解析
The first step is to place the coin in the center of the round table.
Then adopt the “central symmetric response” strategy:
- If the rear hand places a coin in a certain position, the first hand places a coin in a symmetrical position with the center of the circle as the center of symmetry.
The feasibility of this strategy: if the circular table is symmetrical to the center of the circle, and the two coins of the same size do not overlap with the center coin, the distance from the symmetrical position to the center will be the same, and it will not overlap with the center coin; if the symmetrical position overlaps with a coin on the table, the symmetrical return will cause the back step itself to overlap with the corresponding coin, which is contradictory.
Therefore, as long as the latter hand has a legal step, the first hand must also have a legal response.
Since the number of steps a coin can be placed in is limited, it must eventually be lost after having no way to go, so the first hand must win.