Poppy party pizza

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This was recently a UNC's Math Challenge Problem, based on a problem in Discrete Mathematics with Ducks by sarah-marie belcastro.


You are making your famous 1 foot square party pizza. After rolling out the dough, you sprinkle on exactly 37 poppy seeds (your secret ingredient). The poppy seeds fall randomly onto the pizza dough (and luckily none roll off).

What is the probability that at least two poppy seeds will land within 3 inches of each other?


This is not really a problem about probability.
Divide the pizza into 36 equal squares, each with side length 2 inches. By the pigeonhole principle, there must be at least one square containing two or more poppy seeds. Note that the maximum distance between any two seeds in the same square is <m>\sqrt{8} < 3</m>, the length of the diagonal of the square. Thus you are guaranteed to have at least two poppy seeds within 3 inches of each other.