Island of LiarsKill the jokers! - Part 1Kill the jokers! - Part 2Another village of liars and truth-tellersLiar and the Truth teller with 6 inhabitantsThree truth teller boys on an islandWho led the mutiny?What did he say he said? Liars with a twist!How many liars are in the room?Liar liar pants on fireRound tabel conference with liars?
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Island of Liars
Kill the jokers! - Part 1Kill the jokers! - Part 2Another village of liars and truth-tellersLiar and the Truth teller with 6 inhabitantsThree truth teller boys on an islandWho led the mutiny?What did he say he said? Liars with a twist!How many liars are in the room?Liar liar pants on fireRound tabel conference with liars?
$begingroup$
A king goes to an island, where there are 5823 people who always tell the truth and 8723 who always lie.
The king is special guest for the party celebrating 2053 year.
On the party, liars and truth tellers dance in couples (everyone can dance with everyone), some people leave the dance, others join and some don't even participate in the dances as they are not dance lovers.
After the party, the king asked everyone who had danced on the party, how many truth tellers did he dance with and put the numbers into his diary.
Surprisingly, when he checked his diary later on, he found out he had all the numbers from 0 to 956 typed in uniquely.
How many truth tellers danced on the party, assuming everyone knows who is truth teller and who is liar?
mathematics liars
edited 13 hours ago
jafe
24.1k469237
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
$endgroup$
add a comment |
$begingroup$
A king goes to an island, where there are 5823 people who always tell the truth and 8723 who always lie.
The king is special guest for the party celebrating 2053 year.
On the party, liars and truth tellers dance in couples (everyone can dance with everyone), some people leave the dance, others join and some don't even participate in the dances as they are not dance lovers.
After the party, the king asked everyone who had danced on the party, how many truth tellers did he dance with and put the numbers into his diary.
Surprisingly, when he checked his diary later on, he found out he had all the numbers from 0 to 956 typed in uniquely.
How many truth tellers danced on the party, assuming everyone knows who is truth teller and who is liar?
mathematics liars
edited 13 hours ago
jafe
24.1k469237
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
$endgroup$
3
$begingroup$
Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks!
$endgroup$
– PiIsNot3
13 hours ago
1
$begingroup$
1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
Is the King lying? What does a truth-teller say about dancing with the King?
$endgroup$
– Weather Vane
13 hours ago
$begingroup$
Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing.
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
$begingroup$
A king goes to an island, where there are 5823 people who always tell the truth and 8723 who always lie.
The king is special guest for the party celebrating 2053 year.
On the party, liars and truth tellers dance in couples (everyone can dance with everyone), some people leave the dance, others join and some don't even participate in the dances as they are not dance lovers.
After the party, the king asked everyone who had danced on the party, how many truth tellers did he dance with and put the numbers into his diary.
Surprisingly, when he checked his diary later on, he found out he had all the numbers from 0 to 956 typed in uniquely.
How many truth tellers danced on the party, assuming everyone knows who is truth teller and who is liar?
mathematics liars
edited 13 hours ago
jafe
24.1k469237
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
$endgroup$
A king goes to an island, where there are 5823 people who always tell the truth and 8723 who always lie.
The king is special guest for the party celebrating 2053 year.
On the party, liars and truth tellers dance in couples (everyone can dance with everyone), some people leave the dance, others join and some don't even participate in the dances as they are not dance lovers.
After the party, the king asked everyone who had danced on the party, how many truth tellers did he dance with and put the numbers into his diary.
Surprisingly, when he checked his diary later on, he found out he had all the numbers from 0 to 956 typed in uniquely.
How many truth tellers danced on the party, assuming everyone knows who is truth teller and who is liar?
mathematics liars
mathematics liars
edited 13 hours ago
jafe
24.1k469237
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
edited 13 hours ago
jafe
24.1k469237
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
edited 13 hours ago
jafe
24.1k469237
edited 13 hours ago
jafe
24.1k469237
edited 13 hours ago
jafe
24.1k469237
24.1k469237
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
asked 13 hours ago
Kradec na kysmetKradec na kysmet
526
526
3
$begingroup$
Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks!
$endgroup$
– PiIsNot3
13 hours ago
1
$begingroup$
1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
Is the King lying? What does a truth-teller say about dancing with the King?
$endgroup$
– Weather Vane
13 hours ago
$begingroup$
Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing.
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
3
$begingroup$
Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks!
$endgroup$
– PiIsNot3
13 hours ago
1
$begingroup$
1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
Is the King lying? What does a truth-teller say about dancing with the King?
$endgroup$
– Weather Vane
13 hours ago
$begingroup$
Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing.
$endgroup$
– Kradec na kysmet
13 hours ago
3
3
$begingroup$
Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks!
$endgroup$
– PiIsNot3
13 hours ago
$begingroup$
Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks!
$endgroup$
– PiIsNot3
13 hours ago
1
1
$begingroup$
1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
Is the King lying? What does a truth-teller say about dancing with the King?
$endgroup$
– Weather Vane
13 hours ago
$begingroup$
Is the King lying? What does a truth-teller say about dancing with the King?
$endgroup$
– Weather Vane
13 hours ago
$begingroup$
Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing.
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I think that the answer is
Just $1$ truthteller danced.
Reasoning
Suppose more than one truthteller dances, say $N$. Now consider the graph of dancers just involving truthtellers, where an edge represents "danced with". Within this graph, either the lowest degree of a node is $0$, where then the highest possible is $N-2$, or the lowest degree is $1$, where the highest possible is $N-1$. In either case, by the Pigeonhole Principle, there will be at least two truthtellers who report the same number to the king. Of course, if there are zero truthtellers who danced then one of the liars would have to have told the truth - saying $0$.
Proof that this works
Suppose there is $1$ truthteller, he will report $0$. There just needs to be one liar who doesn't dance with the truthteller who reports $1$. Then everyone else is verifiably lying.
answered 13 hours ago
hexominohexomino
44.7k4136215
$endgroup$
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think that the answer is
Just $1$ truthteller danced.
Reasoning
Suppose more than one truthteller dances, say $N$. Now consider the graph of dancers just involving truthtellers, where an edge represents "danced with". Within this graph, either the lowest degree of a node is $0$, where then the highest possible is $N-2$, or the lowest degree is $1$, where the highest possible is $N-1$. In either case, by the Pigeonhole Principle, there will be at least two truthtellers who report the same number to the king. Of course, if there are zero truthtellers who danced then one of the liars would have to have told the truth - saying $0$.
Proof that this works
Suppose there is $1$ truthteller, he will report $0$. There just needs to be one liar who doesn't dance with the truthteller who reports $1$. Then everyone else is verifiably lying.
answered 13 hours ago
hexominohexomino
44.7k4136215
$endgroup$
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
$begingroup$
I think that the answer is
Just $1$ truthteller danced.
Reasoning
Suppose more than one truthteller dances, say $N$. Now consider the graph of dancers just involving truthtellers, where an edge represents "danced with". Within this graph, either the lowest degree of a node is $0$, where then the highest possible is $N-2$, or the lowest degree is $1$, where the highest possible is $N-1$. In either case, by the Pigeonhole Principle, there will be at least two truthtellers who report the same number to the king. Of course, if there are zero truthtellers who danced then one of the liars would have to have told the truth - saying $0$.
Proof that this works
Suppose there is $1$ truthteller, he will report $0$. There just needs to be one liar who doesn't dance with the truthteller who reports $1$. Then everyone else is verifiably lying.
answered 13 hours ago
hexominohexomino
44.7k4136215
$endgroup$
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
$begingroup$
I think that the answer is
Just $1$ truthteller danced.
Reasoning
Suppose more than one truthteller dances, say $N$. Now consider the graph of dancers just involving truthtellers, where an edge represents "danced with". Within this graph, either the lowest degree of a node is $0$, where then the highest possible is $N-2$, or the lowest degree is $1$, where the highest possible is $N-1$. In either case, by the Pigeonhole Principle, there will be at least two truthtellers who report the same number to the king. Of course, if there are zero truthtellers who danced then one of the liars would have to have told the truth - saying $0$.
Proof that this works
Suppose there is $1$ truthteller, he will report $0$. There just needs to be one liar who doesn't dance with the truthteller who reports $1$. Then everyone else is verifiably lying.
answered 13 hours ago
hexominohexomino
44.7k4136215
$endgroup$
I think that the answer is
Just $1$ truthteller danced.
Reasoning
Suppose more than one truthteller dances, say $N$. Now consider the graph of dancers just involving truthtellers, where an edge represents "danced with". Within this graph, either the lowest degree of a node is $0$, where then the highest possible is $N-2$, or the lowest degree is $1$, where the highest possible is $N-1$. In either case, by the Pigeonhole Principle, there will be at least two truthtellers who report the same number to the king. Of course, if there are zero truthtellers who danced then one of the liars would have to have told the truth - saying $0$.
Proof that this works
Suppose there is $1$ truthteller, he will report $0$. There just needs to be one liar who doesn't dance with the truthteller who reports $1$. Then everyone else is verifiably lying.
answered 13 hours ago
hexominohexomino
44.7k4136215
answered 13 hours ago
hexominohexomino
44.7k4136215
answered 13 hours ago
hexominohexomino
44.7k4136215
answered 13 hours ago
hexominohexomino
44.7k4136215
44.7k4136215
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
That's it, that was fast even with all the baits I put.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
@Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions.
$endgroup$
– hexomino
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
I will admit I was laughing while typing all those random numbers :D
$endgroup$
– Kradec na kysmet
13 hours ago
add a comment |
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3
$begingroup$
Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks!
$endgroup$
– PiIsNot3
13 hours ago
1
$begingroup$
1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all.
$endgroup$
– Kradec na kysmet
13 hours ago
$begingroup$
Is the King lying? What does a truth-teller say about dancing with the King?
$endgroup$
– Weather Vane
13 hours ago
$begingroup$
Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing.
$endgroup$
– Kradec na kysmet
13 hours ago