A logic-loving and merciful king gathers a meeting of all of his death row inmates. The room can only fit 11 max (and all prisoners know this and know that all prisoners know this), so including the king, it means that the number of prisoners is 10 at most, but each prisoner can only see 2 other prisoners since they are all in a line.
"There are no less than three of you," the king confirms. "And no more than ten- however, I will not say the exact number of prisoners. Now, since I am a kind and merciful king, I have devised a little game that will test your skills to the limit. If you succeed, all of you will be released. But if somebody makes a false guess, everyone is executed. If nobody makes a guess within 5000 hours, everyone is executed anyway. You cannot risk the lives of you and your fellow prisoners, so you may only make a guess that you know is 100% correct... even if the chances are 99.9%, and even if it's the 4999th hour and everyone dies the next hour anyway. That is, you must give me reasoning with your guess to show that it is, for sure, 100% correct. Is that clear?"
The prisoners nodded obediently, excited at their chance for freedom.
"Good," continued the king. "Now for the game... but first, I must warn you. I do not know if there is a successful method. This is a little test of mine, to test your skills. Maybe it's possible, maybe it's not. Maybe you can think of a way, maybe not. But thank me for being such a merciful king and giving you such a chance."
"Thank you," they mumbled.
"Now," said the king, smiling a little. "Here is the game: I have a room, nearby, which has three knobs. The knob's original use is long gone, but you can still see their settings: 0, 1 or 2. Each hour starting 1 hour from now, a random prisoner will be selected, with no regards to previous selections, and sent into the room. The prisoner will be watched, and thus is only allowed to change one knob. The knob can be set to any of the three values, though there is no between-states, the knob clicks between those three values only (0,1,2). And remember, you are being watched. There can be no modification of the room in any other way, it will be reset. The only thing you can do is to pick ONE knob and change it. The knobs are clearly labeled A, B and C, and these labels will not change in any way. The knobs will not change in any way other than a prisoner adjusting it to 0,1,2, and nobody else will tamper with the knobs. You guys win the game if someone, anyone, can guess when every single prisoner has been in the room. It has to be foolproof, 100% correct. If the guess is made [correctly], everyone goes free... seem impossible? However," the king said. "You will be allowed to plan your strategy through anonymous letters via me. I will make sure there is no way to discern how many prisoners there are via the letters - that could be anything from 3 to 10, nobody knows. But in the letters, you CAN discuss strategy, so that everybody knows what to do."
The prisoners attempt a few ideas, but can't agree on anything.
You are the one of the prisoners. Do you have a brilliant plan that will save everyone?
1) Only a 100%-sure guess can be made, ie, a guess is made correctly, OR no guess is made until the 5000th hour, at which the prisoners are executed
2) Assume that the prisoners can keep track of time in their cells
3) No changing anything except those three knobs to three values each (0,1,2). And you can only change ONE knob in your visit
4) the prisoners cannot communicate in any way except for the knobs
5) There are two possible starting positions: (the king tells them [truthfully] which it is)
5a) the knobs start at all-zero (000)
5b) the knobs start randomly, ie, each knob has a 1/3 chance to be 0, 1/3 to be 1, and 1/3 to be 2
Question
unreality
A logic-loving and merciful king gathers a meeting of all of his death row inmates. The room can only fit 11 max (and all prisoners know this and know that all prisoners know this), so including the king, it means that the number of prisoners is 10 at most, but each prisoner can only see 2 other prisoners since they are all in a line.
"There are no less than three of you," the king confirms. "And no more than ten- however, I will not say the exact number of prisoners. Now, since I am a kind and merciful king, I have devised a little game that will test your skills to the limit. If you succeed, all of you will be released. But if somebody makes a false guess, everyone is executed. If nobody makes a guess within 5000 hours, everyone is executed anyway. You cannot risk the lives of you and your fellow prisoners, so you may only make a guess that you know is 100% correct... even if the chances are 99.9%, and even if it's the 4999th hour and everyone dies the next hour anyway. That is, you must give me reasoning with your guess to show that it is, for sure, 100% correct. Is that clear?"
The prisoners nodded obediently, excited at their chance for freedom.
"Good," continued the king. "Now for the game... but first, I must warn you. I do not know if there is a successful method. This is a little test of mine, to test your skills. Maybe it's possible, maybe it's not. Maybe you can think of a way, maybe not. But thank me for being such a merciful king and giving you such a chance."
"Thank you," they mumbled.
"Now," said the king, smiling a little. "Here is the game: I have a room, nearby, which has three knobs. The knob's original use is long gone, but you can still see their settings: 0, 1 or 2. Each hour starting 1 hour from now, a random prisoner will be selected, with no regards to previous selections, and sent into the room. The prisoner will be watched, and thus is only allowed to change one knob. The knob can be set to any of the three values, though there is no between-states, the knob clicks between those three values only (0,1,2). And remember, you are being watched. There can be no modification of the room in any other way, it will be reset. The only thing you can do is to pick ONE knob and change it. The knobs are clearly labeled A, B and C, and these labels will not change in any way. The knobs will not change in any way other than a prisoner adjusting it to 0,1,2, and nobody else will tamper with the knobs. You guys win the game if someone, anyone, can guess when every single prisoner has been in the room. It has to be foolproof, 100% correct. If the guess is made [correctly], everyone goes free... seem impossible? However," the king said. "You will be allowed to plan your strategy through anonymous letters via me. I will make sure there is no way to discern how many prisoners there are via the letters - that could be anything from 3 to 10, nobody knows. But in the letters, you CAN discuss strategy, so that everybody knows what to do."
The prisoners attempt a few ideas, but can't agree on anything.
You are the one of the prisoners. Do you have a brilliant plan that will save everyone?
1) Only a 100%-sure guess can be made, ie, a guess is made correctly, OR no guess is made until the 5000th hour, at which the prisoners are executed
2) Assume that the prisoners can keep track of time in their cells
3) No changing anything except those three knobs to three values each (0,1,2). And you can only change ONE knob in your visit
4) the prisoners cannot communicate in any way except for the knobs
5) There are two possible starting positions: (the king tells them [truthfully] which it is)
5a) the knobs start at all-zero (000)
5b) the knobs start randomly, ie, each knob has a 1/3 chance to be 0, 1/3 to be 1, and 1/3 to be 2
Can you make it work with 4b, let alone 4a?
Please use spoilers, thank you
edit: minor typo
Edited by unrealityLink to comment
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