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There are three great deck builder card games: Dominion, Slay the Spire, and Magic the Gathering. There are four different characters in Slay the Spire. My favorite is the Silent character who often uses poison to defeat the monsters.

One method of attacking with poison in Slay the Spire is to play the “Noxious Fumes+” card. The card self destructs after it is played (it cannot be played again until the you enter the next room in your quest.) Nothing happens on the turn in which it is played. The next turn, the monster experiences 3 damage from the poison. The next turn after that, the monster experiences 5 additional poison damage for a total of 8 poison damage. The next turn after that 7 additional poison damage for a total of 15 poison damage. And so on.

A few weeks ago, I wrote that if you play an ordinary Noxious Fumes card, every opponent monster with h health points or less will be dead in $\sqrt{2 h}$ turns. (See Footnote [1] for details.)

What about the upgraded Noxious Fumes+? Well amazingly, if you play a Noxious Fumes+, then the monster dies in $\sqrt{h}$  turns or less.

In fact, if

• the monster has h health points when the Noxious Fumes+ card is played,

• the monster can’t heal,

• the monster experiences no poison damage on the turn that the Noxious Fumes+ is played,

• it experiences 3 poison damage on the turn after Noxious Fumes+ is played, 5 poison damage on the next turn, 7 on the next, and so on for t=floor($\sqrt{h}$) turns, and

• the monster experiences only poison damage,

then the monster will die exactly

t = floor($\sqrt{h}$)

turns after the Noxious Fumes+ is played.

Notation: floor(x) is the largest whole number less than or equal to x. It is x rounded down. For example, floor(3) = 3, floor(4.8) = 4, and floor(π) = 3.

Example 1

Suppose there is a minion that has 8 hit points when the Noxious Fumes+ is played. Suppose that the minion is not healed, it has no poison on the turn that the Noxious Fumes+ is played, and you do not take any other offensive actions against that minion (nor do you play any area of effect cards). Then, that minion will die in exactly

t = floor($\sqrt{8}$) = floor(2.8284) = 2 turns.

One turn after the Noxious Fumes+ is played, it experiences 3 points poison damage, and the following turn it dies with 5 additional points poison damage.

Example 2

Suppose there is a monster that has 25 hit points when the Noxious Fumes+ is played. Suppose that the monster is not healed, it has no poison on the turn that the Noxious Fumes+ is played, and you do not take any other offensive actions against that monster (nor do you play any area of effect cards). Then, that monster will die in exactly

t = floor(√25) = floor(5) = 5 turns.

The total poison damage over the next 4 turns is 3+5+7+9 = 24. On the fifth turn after the Noxious Fumes+ is played, it experiences 11 additional poison damage and dies.

Footnote [1] If the monster can’t heal or avoid poison damage, and it experiences 2 poison damage on the next turn after an ordinary Noxious Fumes is played, 3 poison damage on the next turn, 4 on the next, and so on, then it will be dead within $\sqrt{2 h}$ turns after the Noxious Fumes is played. With a few more calculations, you can get a better, 100% accurate, but more complicated formula for the number of turns to death if all the damage comes from the Noxious Fumes card.