Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq 1 \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.

He has \$1 with which to bet with. On each round, he makes a double-or-nothing bet with any amount up to his entire fortune, and including zero, on one of the coins turning up either heads or tails.

Upon betting on each of the coins and the result being shown, it disintegrates into thin air, and Nate can only bet on the remaining coins in subsequent rounds.

Question: How can Nate maximise his expected value from this game after $n$ flips, and what is the maximal expected value?

Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which.

He has \$1 with which to bet with. On each round, he makes a double-or-nothing bet with any amount up to his entire fortune, and including zero, on one of the coins turning up either heads or tails.

Upon betting on each of the coins and the result being shown, it disintegrates into thin air, and Nate can only bet on the remaining coins in subsequent rounds.

Question: How can Nate maximise his expected value from this game after $n$ flips, and what is the maximal expected value?