Brainteaser - Freeze or Roll
Freeze or Roll
Probability
You begin by rolling two fair six‑sided dice to create their initial face values.
On each turn you first choose at least one die that is still unfrozen and freeze it, locking in its current face value. It is allowed to freeze both dice. After freezing you re‑roll every die that remains unfrozen.
The game ends as soon as both dice are frozen. Your score is the sum of the two frozen face values.
Devise a strategy that maximises the expected final sum. State the optimal rule for deciding when to freeze a die and the corresponding expected score.
On each turn you first choose at least one die that is still unfrozen and freeze it, locking in its current face value. It is allowed to freeze both dice. After freezing you re‑roll every die that remains unfrozen.
The game ends as soon as both dice are frozen. Your score is the sum of the two frozen face values.
Devise a strategy that maximises the expected final sum. State the optimal rule for deciding when to freeze a die and the corresponding expected score.