# Unraveling the Mystery of 45 Liars and Truthers

• K Sengupta
In summary, there were a minimum of 11 liars at the dinner, and most likely 23. There are 11 possible pairs of liars and truth tellers, and the answer is determined by trying all the combinations.
K Sengupta
We know that Truthers always tell the truth but Liars will always speak falsely.

An island comprises entirely of Liars and Truthers..

Precisely 45 people attended a dinner in the island. Each one of them sat at a big round table.

After the dinner was over, each of the attendees was asked about their neighbours, and each stated that they were seated between one Liar and one Truther.

It transpired later that precisely two of the Truthers were mistaken in their statements.

Determine the number of Liars And Truthers attending the dinner.

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Sorry, my first answer was wrong:

K Sengupta

It looks quite possible that there was only two Truthers,
that they were mistaken,
and that they were not sitting close to each other.

I am sure if there is only one solution to this problem.

Michel

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Sengupta,

I tried for 9 people, and found 6 truther and 3 liars.
For more people I don't know.

Michel

Deleted it sorry my first answer was rubbish kinda misread the question thus it was stupid.

Spoiler:-

I have it I think, 15 liars and 30 truthers.

Pattern starts LTLTTT and then LTTLTT til the end.

I think that works? Two mistaken truthers at the start.
1____________________________________________45
LTLTTTLTTLTTLTTLTTLTTLTTLTTLTTLTTLTTLTTLTTLTT

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I think that there might be 14 liars and 31 truth tellers.I think this is the answer.

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Use
They are eating steak with au gratin potatos and a side salad with a balsamic vinegrette[/color$$]$$Just like this, which reminded me I should of spoilered my answer,oops

ten of the best characters

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A solution

T=truth teller
L= lier
T'=mistaken truth teller

It is assumed that T and L both see T' as a T (they don't know of mistake)

possible adjacent seating for mistaken truth teller T' are
LT'L and TT'T

all other seats must be LTT LTT LTT LTT LTT etc

so a solution is LTT ..LT'L TT'T ..LTT LTT LTT LTT UNTIL 45 SEATS
so this gives 15 L and 30 T with two of the Ts being T'

Vamfun don't forget to use the spoiler colour see post no.6 above.

ie

Spoiler:

Well that's two of us with that solution, I think it must be the right one, if you try other combinations it doesn't work, so I'm hazarding a guess it's the only solution, although?

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Schrodinger's Dog said:
Well that's two of us with that solution, I think it must be the right one, if you try other combinations it doesn't work, so I'm hazarding a guess it's the only solution, although?

Yep! I'll see if I can prove it:

For starters, we must state a couple assumptions:
1) The liars all told falsehoods (NONE of the liars were next to both a truth teller and a liar)
2) Exactly two of the truth tellers were mistaken, the rest were correct in stating that they sat next to both a liar and a truth teller.

Now, there's *one* possibility that's really easy, which is:

Solution A) FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

But because we know that there are at least two truth tellers who attended the dinner, this answer is wrong.

We can also easily deduce that no two liars sat next to each other. The possibilities would be as follows:

...TFFT... <-- this would make both middle liars correct, and is disqualified
...TFFF... <-- this would make the leftmost liar correct, disqualified
...FFFT... <-- this would make the rightmost liar correct, disqualified
...FFFF... <-- this is plausible, but leads ONLY to Solution A, which is wrong.

Therefore, no two liars sat next to each other.

From that, we know that there are at MOST 22 liars, which means at LEAST 23 truth tellers. And that means that there are at least 21 truth tellers who were correct in their statements. As a result, we know that there are a minimum of 11 liars, because the 21 truth tellers all correctly reference at least one liar who was known to be at the dinner, but half of the accurate truth tellers may have been referencing the same liars.

So! Between 11 and 22 liars, proven.

We've shown that no liars can be seated next to each other. This means that *EVERY* liar must be seated in between TWO truth tellers. And, the vast majority of those truth tellers must in turn be seated next to *another* truth teller because the majority of truth tellers are correct in their assesments.

The only thing left are the anomolies. The anomalous truth tellers can either be seated as TTT or FTF. The remainder of people MUST be situated in the pattern: ...TTFTTFTTF... The pattern itself is therefore a multiple of 3.

Hence, we are left with possibilities:
insert a TTT and a TTT
insert a TTT and a FTF
insert a FTF and a FTF

But notice that by inserting a TTT into the pattern, we must NECESSARILY be adding *one* person (a truth teller) to a single phase of the pattern. And conversely, by inserting an FTF into the pattern, we must necessarily be REMOVING one person (a truth teller) from the pattern. So, let's reanalyze our possibilities:

insert a TTT and a TTT - wrong, this would add 2, and our total would not be a multiple of 3.

insert a TTT and a FTF - possible

insert a FTF and a FTF - wrong, this would remove 2, and our total would not be a multiple of 3.

As a result, we must take our base pattern of TTFTTFTTF, and insert a TTT and an FTF, in order to keep the total at 45. Where we add them is irrelevant.

If we start with our base pattern, we get that there must be twice as many truth tellers as liars, or 30 truth tellers and 15 liars. Then, we put in the necessary TTT, and we get 31 truth tellers and 15 liars. Finally, we put in the necessary FTF, and we go back to having 30 truth tellers and 15 liars.

Therefore, the solution of 30 truth tellers and 15 liars is unique.

Bam!

DaveE

Superb. Well done I can't see a flaw in your logic.

Thanks... not sure how to do it but here goes.

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That's it, if you can edit your last post with the answer with this colour so it's spoilered. I'm not sure, but I suspect you could ask a mentor or admin to do that if not?

## 1. What is the purpose of "Unraveling the Mystery of 45 Liars and Truthers"?

The purpose of this study is to understand the psychological and behavioral patterns of individuals who exhibit lying and truth-telling behaviors. By examining these patterns, we hope to gain insights into the motivations and mechanisms behind these behaviors.

## 2. How were the 45 individuals chosen for this study?

The individuals were carefully selected through a rigorous screening process, which involved evaluating their past behaviors and responses to certain situations. We wanted to ensure that we had a diverse group of individuals who demonstrated both lying and truth-telling tendencies.

## 3. What methods were used to collect data for this study?

We utilized a combination of surveys, interviews, and behavioral observations to collect data from the 45 individuals. This allowed us to gather both self-reported information and direct observations of their behaviors.

## 4. What were some of the key findings from this study?

One of the key findings was that individuals who exhibited more lying behaviors tended to have lower levels of empathy and higher levels of narcissism. Additionally, we found that there were certain triggers or situations that were more likely to elicit lying or truth-telling behaviors in these individuals.

## 5. How can the results of this study be applied in real-world situations?

The findings from this study can be used to better understand and manage individuals who exhibit lying and truth-telling tendencies. This can be valuable in various settings such as relationships, workplaces, and legal proceedings. Additionally, this study can also contribute to the development of interventions or treatments for individuals who struggle with pathological lying or truth-telling behaviors.

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