How many ping pong balls fit into a 12x12 room?

Answer 1

I will assume that the question is to find the maximum number of balls that could be packed into the space and that each ball must be kept intact.

1 inch = 25.4 mm (exactly)

1 mm = 1/25.4 inches = 0.03937007874015748031496062992126

The standard ping pong ball is 40mm in diameter according to the link

40 mm * 0.03937007874015748031496062992126 =

1.5748031496062992125984251968504 inches in diameter

12 feet = 12 feet*12 inches per foot = 144 inches (in reference to the room's dimensions)

We lay the balls end to end along one wall to see how many fit.

144/1.5748031496062992125984251968504 inches = 91.44

At most, 91 whole balls will fit in this space.

We now come to the complication that has to do with how the balls might be packed together efficiently (known as the sphere packing problem). For this solution, I will assume that the balls are not packed in an offset manner (the centers of which would form hexagons), but rather are in a configuration where the centers of the balls form a lattice like structure similar to the integer locations on the usual Cartesian Coordinate system. (One difficulty proving the packed solution is that to be clear, one probably needs the ability to enter images into the answer, something that currently is not allowed.) Therefore, I will simply proceed with the assumption that no offsetting of the balls takes place.

91 balls in each of the 3 directions of the cube gives 91*91*91=753,571 total balls.

Notice that one cannot simply calculate the number of balls in a cubic foot and then multiply by the number of cubic feet in the room. That is because the balls don't fit exactly into the cubic foot and the extra space accumulates as more cubic feet are utilized.

Answer 2

373,000