Let n = number of players
The chart showing the values of n and the final winners is shown below in the attached image.
Drawing out each scenario is optional, but it helps visualize what is going on.
Let's play out scenarios for n = 2 through n = 12 to see what the pattern might be.
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n = 2 players
Seat #1 eliminates seat #2
Winner = seat #1
This is fairly trivial and boring as there is only one round and one elimination.
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n = 3 players
Seat #1 eliminates seat #2
Seat #3 eliminates seat #1
Winner = seat #3
Like with n = 2, this game is over in one round. For n = 4 and greater, we'll start to see multiple rounds.
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n = 4 players
Start of Round 1
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
End of Round 1
Seats remaining = {seat #1, seat #3}
Start of Round 2
Seat #1 eliminates seat #3
End of Round 2
Winner = seat #1
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n = 5 players
Start of Round 1.
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #1
End of Round 1.
Seats remaining = {seat #3, seat #5}
Start of Round 2.
Seat #3 eliminates seat #5
End of Round 2
Winner = seat #3
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n = 6
Start of round 1.
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
End of Round 1
Seats remaining = {seat #1, seat #3, seat #5}
Start of round 2
Seat #1 eliminates seat #3
Seat #5 eliminates seat #1
End of Round 2
Winner = seat #5
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Up next is n = 7 players
Start of round 1.
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
Seat #7 eliminates seat #1
End of Round 1
Seats remaining = {seat #3, seat #5, seat #7}
Start of round 2
Seat #3 eliminates seat #5
Seat #7 eliminates seat #3
End of Round 2
Winner = seat #7
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n = 8 players
Start of Round 1
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
Seat #7 eliminates seat #8
End of Round 1
Seats remaining = {seat #1, seat #3, seat #5, seat #7}
Start of round 2
Seat #1 eliminates seat #3
Seat #5 eliminates seat #7
End of Round 2
Seats remaining = {seat #1, seat #5}
Start of Round 3
Seat #1 eliminates seat #5
End of Round 3
Winner = seat #1
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n = 9 players
Start of Round 1
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
Seat #7 eliminates seat #8
Seat #9 eliminates seat #1
End of Round 1
Seats remaining = {seat #3, seat #5, seat #7, seat #9}
Start of round 2
Seat #3 eliminates seat #5
Seat #7 eliminates seat #9
End of Round 2
Seats remaining = {seat #3, seat #7}
Start of Round 3
Seat #3 eliminates seat #7
End of Round 3
Winner = seat #3
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n = 10 players
Start of Round 1
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
Seat #7 eliminates seat #8
Seat #9 eliminates seat #10
End of Round 1
Seats remaining = {seat #1, seat #3, seat #5, seat #7, seat #9}
Start of Round 2
Seat #1 eliminates seat #3
Seat #5 eliminates seat #7
Seat #9 eliminates seat #1
End of Round 2
Seats remaining = {seat #5, seat #9}
Start of Round 3
seat #5 eliminates seat #9
End of Round 3
Winner = seat #5
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n = 11 players
Start of Round 1
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
Seat #7 eliminates seat #8
Seat #9 eliminates seat #10
Seat #11 eliminates seat #1
End of Round 1
Seats remaining = {seat #3, seat #5, seat #7, seat #9, seat #11}
Start of Round 2
Seat #3 eliminates seat #5
Seat #7 eliminates seat #9
Seat #11 eliminates seat #3
End of Round 2
Seats remaining = {seat #7, seat #11}
Start of Round 3
Seat #7 eliminates seat #11
End of Round 3
Winner = seat #7
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n = 12 players
Start of Round 1
Seat #1 eliminates seat #2
Seat #3 eliminates seat #4
Seat #5 eliminates seat #6
Seat #7 eliminates seat #8
Seat #9 eliminates seat #10
Seat #11 eliminates seat #12
End of Round 1
Seats remaining = {seat #1, seat #3, seat #5, seat #7, seat #9, seat #11}
Start of Round 2
Seat #1 eliminates seat #3
Seat #5 eliminates seat #7
Seat #9 eliminates seat #11
End of Round 2
Seats remaining = {seat #1, seat #5, seat #9}
Start of Round 3
Seat #1 eliminates seat #5
Seat #9 eliminates seat #1
End of Round 3
Winner = seat #9
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Admittedly this is a lot of things to keep track of and it's easy to get lost. Hopefully the process is fairly straight forward even if it requires a lot of drawings.
The chart showing the values of n and the final winners is shown below in the attached image.
Each pattern block is color coded to separate the start and stop of each sequence. Note the pattern of {1,3} then {1,3,5} then {1,3,5,7} then {1,3,5,7,9} and so on. The next pattern block would likely be {1,3,5,7,9,11}. This table will help you determine what seat you will want to sit in if you know how many players there are at the start. Of course you won't know ahead of time how many players there are, but when it comes to you picking your seat you can quickly look at the table (either on the paper or just something you memorize) and you can select the ideal seat based on what n is.
