Problem 645

Problem 645

Every Day is a Holiday

On planet J, a year lasts for $D$ days. Holidays are defined by the two following rules.

  1. At the beginning of the reign of the current Emperor, his birthday is declared a holiday from that year onwards.
  2. If both the day before and after a day $d$ are holidays, then $d$ also becomes a holiday.

Initially there are no holidays. Let $E(D)$ be the expected number of Emperors to reign before all the days of the year are holidays, assuming that their birthdays are independent and uniformly distributed throughout the $D$ days of the year.

You are given $E(2)=1$, $E(5)=31/6$, $E(365)\approx 1174.3501$.

Find $E(10000)$. Give your answer rounded to $4$ digits after the decimal point.



  1. 每当一名新皇帝登基后,他的生日被宣告为一个新的休息天;
  2. 如果一年中第$d$天的前一天和后一天都是休息天,则这一天自动成为休息天。


已知$E(2)=1$,$E(5)=31/6$,$E(365)\approx 1174.3501$。