Student's Overlapping Rectangles
24 students in Calvin's Master Session each choose a rectangle of area 24 in the Cartesian plane. All of the rectangles have sides parallel to the axis and their corners are at lattice points. Calvin picks three students from the class and draws their rectangles on the board. He finds that there is at least one 1 by 1 square contained in the intersection of the three rectangles. In fact, no matter how he chooses three students from the class, their rectangles always intersect in at least one 1 by 1 square. If all the rectangles are drawn together on the same set of axes, what is the largest possible area they can cover?
Details and assumptions
The 1 by 1 square(s) of intersection need not be the same for 2 distinct triplets of students.
Students may have chosen the same rectangles as each other. For example, if everyone picked the same rectangle, it would clearly satisfy the conditions.