How many marbles must be placed in a square area of $16 in^2$ to ensure that two of the marbles are within $2 \sqrt{2}$ inches of each other?

Question

How many marbles must be placed in a square area of $16 in^2$ to ensure that two of the marbles are within $2 \sqrt{2}$ inches of each other?

Wouldn't even know how to begin this question.

Answer 1

Hint: How big is the square? Imagine dividing it into squares with diagonal $2 \sqrt 2$-how big are those squares? Then think about the pigeonhole principle. This gives an upper bound-it does not prove that a smaller number will not suffice. I don't have an answer for a good lower bound.

Answer 2

The answer is 5. This can be done by showing that 4 does not satisfy the conditions in only one case: the marbles are on the midpoints of the sides.

Placing a fifth marble will result in one of these distances becoming less than $2\sqrt{2}$.