# Push your luck?

**Discrete Mathematics**Level 5

Here's an example game. I toss the coins, you get \(\$1\) on six coins, so the prize is now at \(\$6\). If you quit now, you collect \(\$6\); if you continue, I will toss the remaining four coins. Suppose that you continue; among the four coins, two more come up \(\$1\), adding to your prize pool to \(\$8\). You could have quit with \(\$8\), but no, you decide to continue again, with only two coins left. I toss them, and they come out \(\text{GAME OVER}\) for both. Game over, you lose, you don't get the eight dollars.

Let \(E\) be the expected amount of money you can get by playing optimally. What is \(\lfloor 10^4E \rfloor\)?

(You may use a calculator or similar to do calculations. You may also try simulating it, but I'm pretty sure you won't get it simply with simulation.)