Problem 339

Peredur fab Efrawg

*"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."*

en.wikisource.org

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.

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