nini

One thing I've been wanting to do for a while is to create a bunch of Slay the Spire "puzzles". These are essentially snapshots of the game where the puzzle solver needs to deduce the "best" play according to certain constraints.

One type of puzzle challenge might be "find the statistically optimal play" where there is one way to play that is unambiguously superior (statistically) to all other possible plays.

For example, consider this situation:

If you ignore the Incense Burner¹, there is an optimal way to play this fight which loses on average ¹⁄₆₀ HP:

• Turn 1: Play Eruption followed by Strike.

• Turn 2: There are two possible draws for Turn 2:

  • 3 Strikes and 2 Defends

    Victory: Play 3 Strikes.

  • 2 Strikes and 3 Defends

    Play 3 Defends, Miracle, and Strike.

    Turn 3: There are three types of draws:

    • At least 2 Strikes

      Victory: Play 2 Strikes.

    • Eruption and at least 1 Strike

      Victory: Play Eruption and Strike

    • Strike, Vigilance, and 3 Defends

      Play Vigilance and Defend.

      Turn 4: Victory: Play Eruption and 2 Strikes.

    • Strike and 4 Defends

      Play 3 Defends, taking 7 damage.

      Turn 4: Victory: Play 2 Strikes.

Out of all those possibilities, only one of those branches takes any damage, which is the branch where you draw:

Turn 2: 2 Strikes and 3 Defends
Turn 3: 4 Defends

That sequence of draws takes 7 damage, but it's also an extremely rare outcome (only a ¹⁄₄₂₀ chance of happening), so the average amount of HP you lose is ¹⁄₆₀.

All other possible plays lose more life on average, so this is the statistically optimal play.

1. If we add in the Incense Burner then things get more complicated because we might want to delay the fight to charge up the incense burner so that it triggers earlier in the next fight. However, in practice the optimal play would likely still be the same; it would just be extraordinarily hard to provide a mathematically clean proof for why.