I would suggest that at least one meal a day, they all eat together. Breakfast, maybe, they eat alone in their rooms, and possibly same with supper. Give them each their own suite of rooms; you don't want to stick them all together like one big family. Not if the king is trying to avoid them killing each other. The princess should also have her own suite of rooms.
The castle presumably has a steward/seneschal (and he probably has several people working under him). That's probably the man who will be arranging all the events to keep the princes occupied. I would suggest that extra servants were hired to help keep things running smoothly.
Even if there's an event planned for every morning, afternoon, and evening (while leaving time for the tests), I doubt that the princes should be required to participate in all of them. They would probably
want to, especially if they thought they might get brownie points by doing well in whatever activity it happens to be, but there still might be times when someone's wandering around the castle. Perhaps there's one servant specifically assigned to each prince, who basically goes everywhere with him? Not much privacy, yeah, but they might be used to that.
There's my thoughts on the matter...hope it helps.
