Problem 394
Eating pie
Jeff eats a pie in an unusual way.
The pie is circular. He starts with slicing an initial cut in the pie along a radius.
While there is at least a given fraction F of pie left, he performs the following procedure:
- He makes two slices from the pie centre to any point of what is remaining of the pie border, any point on the remaining pie border equally likely. This will divide the remaining pie into three pieces.
- Going counterclockwise from the initial cut, he takes the first two pie pieces and eats them.
When less than a fraction F of pie remains, he does not repeat this procedure. Instead, he eats all of the remaining pie.
For x ≥ 1, let E(x) be the expected number of times Jeff repeats the procedure above with F = 1/x.
It can be verified that E(1) = 1, E(2) ≈ 1.2676536759, and E(7.5) ≈ 2.1215732071.
Find E(40) rounded to 10 decimal places behind the decimal point.
吃馅饼
杰夫用一种不同寻常的方式吃馅饼。
馅饼是圆形的，他先沿着一条半径在馅饼上切一刀。
如果剩下的馅饼不少于某个给定的比例F，他就进行如下步骤：
- 他在剩下的馅饼边缘上等概率地选择两个点，然后从馅饼的中心到这两个点切两刀。这样剩下的馅饼就分为三份。
- 他从切这两刀之前切的位置处开始，按逆时针拿走前两份馅饼然后吃掉它们。
如果剩下的馅饼不足上述比例F，他就不再重复这些步骤，而是把剩下的馅饼全部拿走吃掉。
对于x ≥ 1，取F = 1/x，记E(x)是杰夫重复上述步骤的期望次数。
可以验证E(1) = 1，E(2) ≈ 1.2676536759，以及E(7.5) ≈ 2.1215732071。
求E(40)，并四舍五入到小数点后10位小数。