You are a pirate. One of your crew ran off in the night, swam to your secret island, and dug up your life's treasure. You need to capture her.

You may deploy $n$ pirate ships to patrol the boundary of the island.

(pictured: $n=4$)

Each pirate ship moves exactly as fast as the mutineer. If the mutineer takes even a single safe step into the ocean, then she will escape using her stolen SCUBA kit. Your goal is to capture the criminal scum at the border, or starve her out on the island forever.

How many pirate ships do you need to deploy, if the island is shaped like a square?

The mutineer and pirate ships are all points of zero radius. Everyone can always see everyone else's position. This is not a loophole-finding contest: there are no issues with reaction time, scurvy, et cetera.

You are a pirate. One of your crew ran off in the night, swam to your secret island, and dug up your life's treasure. You need to capture her.

You may deploy $n$ pirate ships to patrol the boundary of the island.

(pictured: $n=4$)

Each pirate ship moves exactly as fast as the mutineer. If the mutineer takes even a single safe step into the ocean, then she will escape using her stolen SCUBA kit. Your goal is to capture the criminal scum at the border, or starve her out on the island forever.

How many pirate ships do you need to deploy, if the island is shaped like a square?

The mutineer and pirate ships are all points of zero radius. The mutineer starts in the center of the island, and you may start your ships wherever you want. Everyone can always see everyone else's position. This is not a loophole-finding contest: there are no issues with reaction time, scurvy, et cetera.

You are a pirate. One of your crew ran off in the night, swam to your secret island, and dug up your life's treasure. You need to capture her.

You may deploy $n$ pirate ships to patrol the boundary of the island.

(pictured: $n=4$)

Each pirate ship moves exactly as fast as the mutineer. If the mutineer takes even a single safe step into the ocean, then she will escape using her stolen SCUBA kit. Your goal is to capture the criminal scum at the border, or starve her out on the island forever.

How many pirate ships do you need to deploy...

(Warm up) ...If the island is shaped like a square?
(Real problem) ...If the island is shaped like a perfect circle?

The mutineer and pirate ships are all points of zero radius. Everyone can always see everyone else's position. This is not a loophole-finding contest: there are no issues with reaction time, scurvy, et cetera.

Be careful! The answer is counterintuitive. It shocked me at first.

You are a pirate. One of your crew ran off in the night, swam to your secret island, and dug up your life's treasure. You need to capture her.

You may deploy $n$ pirate ships to patrol the boundary of the island.

(pictured: $n=4$)

Each pirate ship moves exactly as fast as the mutineer. If the mutineer takes even a single safe step into the ocean, then she will escape using her stolen SCUBA kit. Your goal is to capture the criminal scum at the border, or starve her out on the island forever.

How many pirate ships do you need to deploy, if the island is shaped like a square?

The mutineer and pirate ships are all points of zero radius. Everyone can always see everyone else's position. This is not a loophole-finding contest: there are no issues with reaction time, scurvy, et cetera.