## Peredur fab Efrawg

### Problem 339

Published on Sunday, 22nd May 2011, 04:00 am; Solved by 353; Difficulty rating: 70%
*"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep would cross over and become white; and when one of the black sheep bleated, one of the white sheep would cross over and become black."*

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Initially each flock consists of `n` sheep. Each sheep (regardless of colour) is equally likely to be the next sheep to bleat. After a sheep has bleated and a sheep from the other flock has crossed over, Peredur may remove a number of white sheep in order to maximize the expected final number of black sheep. Let E(`n`) be the expected final number of black sheep if Peredur uses an optimal strategy.

You are given that E(5) = 6.871346 rounded to 6 places behind the decimal point.

Find E(10 000) and give your answer rounded to 6 places behind the decimal point.