In the far away land of Frugalia - a land where the conservation of money has been elevated to the status of any of Newton's laws - a conundrum is being had. There has been an upsurge in citizens breaking the law: They are buying lottery tickets, paying taxes, choosing higher end products, and otherwise not saving money. A long time ago, such deeds were handled by the police, who were authorized to use lethal force on such scofflaws. However, as of late, changes to the national budget have caused the police to dwindle to just a single officer.
The brilliant economic strategists of Frugalia have come up with a solution: The police force will be supplemented with an army of cardboard cutouts in police uniforms, meant to function very much like scarecrows for criminals. This has been an immensely popular proposal, since people believe it can lower crime rates while costing far less than a traditional police force.
Obviously, the Frugalians wish to purchase as few cardboard policeman as possible, in order to save money. The remaining policeman has the following to say about what the police force must necessarily do:
As we all know, Frugalia is a city made of a $n\times n$ grid of cells, each cell connected to its $4$ neighbors, with the edges wrapping around top to bottom and left to right to make a torus. There is a very strict law that says that there may not be more than one criminal at a time, so we only need to consider having one target for a police chase.
As I am only paid to work $1$ minute a day, I can only do so much in a chase: Each day, I will move one cardboard policeman to an adjacent cell not occupied by either the criminal or another cardboard policeman. Thankfully, due to our network of low quality CCTV cameras, I will always be aware of where the criminal is.
Between my movements, the criminal can make as many moves between adjacent cells as they wish, so long as they never occupy the same cell as a cardboard police officer. That is, they are free to move in any connected portion of the city in which there are no officers. They will surrender to arrest if they are ever surrounded on all four sides.
The town has announced that, despite its frugal nature, it will award a generous bounty of one upvote and the associated 10 imaginary internet points to any individual who can determine the minimum number of cardboard cops that the town needs to ensure the eventual capture of any criminals. Of course - needless to say - the penalty for advising the town to buy more cardboard policeman than is necessary is death, as the wasting of money is a capital offense in Frugalia.