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I recently watched a TV show where 5 contestants had to play a game. The strategies they took didn't look optimal to me. The game is as follows:

There are 100 identical looking buttons that are shared in front of the group of contestants. That is one set of 100 buttons in total. We are told five of the buttons are "bad" and if a contestant presses one of those they are out. At each turn a contestant must press between 1 and 10 buttons that have not been pressed before. They choose which buttons to press and how many. At the end of the game when all the contestants are out, the winner is the contestant who pressed the largest number of buttons.

What is the optimal strategy?


I can see that once you are the last contestant you just press buttons continuously until you are out. At all stages it doesn't matter which buttons you press as you never press a button that has been pressed before by anyone. The only question is how many you should press at each turn.

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  • $\begingroup$ @OlderAmateur yes. They were all in the same room. $\endgroup$
    – Simd
    Nov 27, 2023 at 15:38
  • $\begingroup$ @OlderAmateur The set of buttons is shared. So if another contestant presses a bad button you are less likely to as there are fewer left. It might help to wait until that happens before pressing a lot of buttons? $\endgroup$
    – Simd
    Nov 27, 2023 at 15:57
  • $\begingroup$ @OlderAmateur They take turns (that is player A, then player B etc) so I guess the first question is simply if they all decide to press exactly one button, which one of them is most likely to win. The chance of hitting a bad button depends on how many good and bad buttons have been hit so far by all the contestants. $\endgroup$
    – Simd
    Nov 27, 2023 at 16:29

1 Answer 1

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Since everything is unknown we can have four of the contestants always choose the first available button from the left while our fifth contestant will always choose from the right. Obviously nothing any of the four will decide to do is going to affect how many buttons the fifth contestant will manage to press.

There is no "optimal" strategy.

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  • $\begingroup$ If the first four contestants press a lot of buttons without pressing a bad one, this makes things much worse for the fifth contestant. $\endgroup$
    – Simd
    Nov 27, 2023 at 16:27
  • $\begingroup$ When they decide to press their buttons makes no difference. contestant #5 is pushing buttons from the right and will hit a baddie at exactly the same spot.... $\endgroup$ Nov 27, 2023 at 16:31
  • $\begingroup$ To take an extreme case, if the other contestants press 10 each time and have actually pressed all the good buttons then the fifth contest loses if they are pressing one each time. $\endgroup$
    – Simd
    Nov 27, 2023 at 16:32
  • $\begingroup$ Suppose you're a spectator who can see where the bad ones are. You will immediately know how many the fifth contestant will press what the other four do is irrelevant. $\endgroup$ Nov 27, 2023 at 16:39
  • $\begingroup$ If they each press one button, is one of the 5 more likely to win? Let's say the contestants are called A, B, C, D , E. Clearly if there were 101 contestants the last one can't win. $\endgroup$
    – Simd
    Nov 27, 2023 at 16:40

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