When a box rests on a round sheet of wood on the ground, it exerts an average pressure p on the wood. If the wood is replaced by a sheet that has half the diameter of the original piece, what is the new average pressure? I think the answer is 4p but I don't know why.

Pressure is (weight of the box) / (area under the bottom of the box). Neither of those quantities depends on the area of whatever is under the box. You could move the box from the deck of a battleship and put it on the kitchen table. The pressure it exerts doesn't change.