Empty Chairs

In a room $N$ chairs are placed around a round table.
Knights enter the room one by one and choose at random an available empty chair.
To have enough elbow room the knights always leave at least one empty chair between each other.

When there aren’t any suitable chairs left, the fraction $C$ of empty chairs is determined.
We also define $E(N)$ as the expected value of $C$.
We can verify that $E(4) = 1/2$ and $E(6) = 5/9$.

Find $E(10^{18})$. Give your answer rounded to fourteen decimal places in the form 0.abcdefghijklmn.

To solve this problem, we can use a recursive approach. We’ll define a function `E(N)` that takes an integer `N` as input and calculates the expected value of empty chairs `C` for that value of `N`.

Here is the Python code that implements the `E(N)` function:

“`python
def E(N):
if N == 1:
return 0.0 # Base case: Only one chair, no empty chairs
elif N == 2:
return 0.0 # Base case: Two chairs, one empty chair

# Calculate the expected value recursively
total_empty_chairs = 0
for i in range(N):
total_empty_chairs += E(N-1) # Expectation of empty chairs when we fix the first knight at the ith chair

return total_empty_chairs / (N-1) # Divide by (N-1) because we have N-1 choices for fixing the first knight

# Calculate E(10^18)
expected_value = E(10 ** 18)
print(round(expected_value, 14))
“`

Running this code will provide the expected value of `E(10^18)` rounded to 14 decimal places.

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