重叠硬币覆盖桌面
Overlapping Coins
题目详情
一张长方形桌面上放了 100 枚半径为 1 的硬币，硬币互不重叠，并且已经无法再放入任何一枚半径为 1 的硬币而不发生重叠（即为极大 packing）。
利用这组特定摆放，构造一种“400 枚半径为 1 的硬币”的摆放，使其覆盖整个桌面（允许硬币相互重叠）。
覆盖指：桌面上任意一点都在至少一枚硬币下方。
A rectangular table has 100 coins with unit radius, placed on it such that none of the coins overlap, and it is impossible to place any more coins on the table without causing an overlap. Using this specific configuration, find a special configuration of 400 coins which covers the table with overlaps.
Covering means for every point on table there is a coin above it.
Hint
Create coins of radius 2 from the center of all coins. Notice that these coins fill up entire table, they are just bigger than what we are given.
解析
把每枚硬币中心作为圆心，画半径为 2 的圆。
由于原配置已“极大”，若这些半径 2 的圆不能覆盖桌面，就存在一处空隙可放入一枚半径 1 的硬币（与所有中心距离至少 2），矛盾。
因此 100 个半径 2 的圆覆盖桌面。把平面缩放 1/2，则得到 100 个半径 1 的圆覆盖“原桌面的四分之一”。把这组配置复制到 4 个象限并拼回原桌面，即得到 400 枚半径 1 硬币的覆盖配置。
Original Explanation
Solution
Consider just one of these coins, with center P. It follows that the center Q of any other coin cannot lie within the coin of radius 2 with center P because it must be at least 2 units away. Thus, we construct all of these coins of radius 2, concurrent with each of the coins of radius 1. If the set of coins of radius 2 did not cover the rectangle entirely, then we could place a coin of radius 1 in this region, contradiction. Thus, the set of coins of radius 2 entirely covers the rectangle.
We now have 100 coins of radius 2 that entirely covers the rectangle. Scale this by a factor of 1/2 in both planar dimensions. Now we have 100 coins of radius 1 that entirely covers a rectangle that is a quadrant of the original rectangle. By placing four of these sets together, we get 400 coins of radius 1 that entirely covers the original rectangle.