Brain Teasers
A 100 Player Game
100 players are lined up in a circle labeled a number 1-100. Player number 1 can either eliminate player 2 or no one. Then player 2 (assuming they're in) can choose to eliminate player 3. This continues until player 100. Player 100 can choose to eliminate player 1 and the cycle repeats until there are two players left , who are the winners. Assuming everyone played perfectly, who won?
Hint
Start from one player, work up from there.Answer
No one did. Let's say 95 players are eliminated, leaving 5, 19, 23, 78, and 90; well if 5 eliminates 19 then 23 eliminates 78 and 90 eliminates 5, so 5 wouldn't do that. 19, 23, 78, and 90 also have the same problem, so no one would ever win.Every combination of five players results in a stalemate. (The rule for this is: 2*(number of winners)+1=(number of players when a stalemate occurs), so setting the number of winners to five will not fix the problem.)
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Very curious puzzle. By the rules, player one can only eliminate player two. By inference, assume each player can eliminate the player currently on their left. Okay, given that...
What is optimal strategy? If player #1 decides not to eliminate, then every player decides not to eliminate and no one does anything.
If player 1 eliminates two given the even number of players, then what does player 3 do with an odd number of players?
if player 3 does not eliminate, then no one does. It seems there are three possible scenarios.
1) No one ever eliminates anyone.
2) Player 1 eliminates player 2, then no one does anything.
3) Every player eliminates the person to their left, until as Rove points out there are 5 players left.
I do not know which of those scenarios would represent "optimal strategy". If a stalemate is better than being eliminated, I suspect scenario 1 or 2 would be optimal.
"A strange game. The only winning move is not to play" - Joshua (W.O.P.R.)
What is optimal strategy? If player #1 decides not to eliminate, then every player decides not to eliminate and no one does anything.
If player 1 eliminates two given the even number of players, then what does player 3 do with an odd number of players?
if player 3 does not eliminate, then no one does. It seems there are three possible scenarios.
1) No one ever eliminates anyone.
2) Player 1 eliminates player 2, then no one does anything.
3) Every player eliminates the person to their left, until as Rove points out there are 5 players left.
I do not know which of those scenarios would represent "optimal strategy". If a stalemate is better than being eliminated, I suspect scenario 1 or 2 would be optimal.
"A strange game. The only winning move is not to play" - Joshua (W.O.P.R.)
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