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Flea Circus

Problem 213

Published on Saturday, 18th October 2008, 10:00 am; Solved by 1576; Difficulty rating: 60%

A 30×30 grid of squares contains 900 fleas, initially one flea per square.
When a bell is rung, each flea jumps to an adjacent square at random (usually 4 possibilities, except for fleas on the edge of the grid or at the corners).

What is the expected number of unoccupied squares after 50 rings of the bell? Give your answer rounded to six decimal places.