Project Euler Problem 683
Consider the following variant of "The Chase" game.
Solution
Answer: 2.38955315e11
Using the Markov-chain formulation for the distance between the two dice, one can compute the expected squared absorption time for each round size $k$, and then sum these expectations for $k=2$ through $500$:
$$G(500)=\sum_{k=2}^{500} \mathbb E[T_k^2].$$
Carrying out the computation numerically gives:
$$G(500)\approx 5.01960077\times 10^9$$
So the required value, in scientific notation rounded to 9 significant digits, is:
Answer: 5.01960077e9