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Fedimath Episode 4: An Optic in Coronavirus Cases

CMD's apartment consists of 4 rooms arranged in a straight line: [ ] [ ] [ ] [ ] On each turn, CMD may stay in the same room or walk to an adjacent room.

NegaJeff wants to kill CMD. To accomplish this task, NegaJeff will use COVID bombs which may be detonated in any one of the four rooms of CMD's apartment. NegaJeff cannot see which room CMD is in, nor can CMD see in advance where NegaJeff is going to bomb.

NegaJeff has one Delta bomb and an unlimited number of Omicron bombs. These bombs function differently:

DELTA:
If it misses, then the game ends and CMD survives.
If it hits, then CMD dies.

OMICRON:
If it misses, then NegaJeff cannot bomb during the next 4 turns (and CMD learns this fact).
If it hits, then NegaJeff learns which room CMD is currently in, CMD gets to take one more turn, and then NegaJeff *must* hit CMD with another Omicron bomb. If this second bomb hits, then CMD dies. If not, then CMD acquires a natural immunity (the game ends and CMD survives).

In other words, the Omicron bomb acts like the Pfizer vaccine - it only works after two doses spaced out by one turn.

Your task: Given any real number r < 1/2, explain how NegaJeff can guarantee a kill probability of r or greater.

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Example:

I will show that CMD can guarantee a survival probability of 1/2 by randomizing his position according to a uniform distribution.

If NegaJeff launches the Delta bomb, then NegaJeff wins with probability 1/4.

If NegaJeff launches an Omicron bomb and misses, then NegaJeff learns that CMD is in one of the *other* three rooms. In this case, CMD must use the 4 free turns to re-randomize his position, restoring the uniform distribution.

If NegaJeff launches an Omicron bomb at one of the two outer rooms and it hits CMD, then CMD should use his one turn to move or stay with probability 1/2, and then NegaJeff's follow-up Omicron bomb will hit CMD with probability 1/2.

If NegaJeff launches an Omicron bomb at one of the two middle rooms and it hits CMD, then CMD should use his one turn to move left or right or stay with probability 1/3, and then NegaJeff's follow-up Omicron bomb will hit CMD with probability 1/3.

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Extra credit: What if CMD's apartment is a 3 x 3 grid?

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Additional lore: https://bae.st/notice/AbQ8R9ErvR4CgIPvnc

See Fedimath Episode 3 here: https://cawfee.club/notice/AbBc3onB4b2LTyIBnc

cc @MercurialBlack @scenesbycolleen @ceo_of_monoeye_dating @roboneko @jeffcliff @hidden @udongle @tiskaan @Inginsub

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