This is my solution to the “Kill the Dragon!” puzzle. Improvements, in both the bounds and formality of the argument, are definitely possible.
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Kill the Dragon!
A hapless lost dragon has accidentally landed in a nearby lake. You, the kingdom’s sworn dragon-slayer, have set out on this foggy night to kill it. You are armed with your trusty trebuchet, which can catapult a fiery projectile to any location on the lake. When the projectile hits the water, it will explode in a lethal circle of Greek fire, killing everything within a radius of metres from the point of impact. Especially dragons.
The fire, however is short-lived, and is extinguished instantaneously. This means the dragon, who swims slowly at a constant speed of metres per minute, can safely doggy-paddle into a previously scorched area. You can launch one missile per minute.
It’s so foggy, that you can’t tell whether you’ve killed the dragon, which is too tired to leave the lake, and you can’t be bothered to fetch a boat to check. If the lake is a circle of radius metres, is it possible to aim your volleys strategically to be sure that you will eventually kill the dragon, no matter how it moves?
For which radius is it possible, and for which is it impossible? What about other shapes of lakes?