[jazzship]

There are 10 people at a party, some of whom always tell the truth, while the rest always lie. One of them says, "No one in the room is honest." Another person says, "There's not more than one honest person here." A third says, "There aren't more than two honest people here." This continues until the tenth person says, "There are not more than nine honest people here."

How many honest people are there at the party?

[Guest]

One person because the first person is a liar. The second is honest. The third to the tenth are all liars.

Edited February 28, 2010 by monkeymirth

[Guest]

5

The first 5 are liars and the second 5 are honest.

The i'th person makes the claim h <= i-1

where h is the number of honest people.

Thus if the person is a liar then we conclude that h > i-1 for that person.

We know that h is somewhere between 0 and 10.

From the inequalities we see that honest people have indexes

i >= h+1

while liars have indexes

i < h+1

Thus all the liars in the party are ordered before the honest people.

The last liar will have index i = h because this is the greatest integer that satisfies i < h+1.

All party guests with an index lower than this are also liars, so there are h liars all together.

But in the beginning we assumed that there are h honest people.

Thus the number of liars and honest people are equal and they must both be 10/2 = 5.

[Guest]

There are 10 people at a party, some of whom always tell the truth, while the rest always lie. One of them says, "No one in the room is honest." Another person says, "There's not more than one honest person here." A third says, "There aren't more than two honest people here." This continues until the tenth person says, "There are not more than nine honest people here."

How many honest people are there at the party?

The second person was right, because theyonly went up to nine. So the second person would be the honest person

[Guest]

The second person was right, because theyonly went up to nine. So the second person would be the honest person

When you say "the" honest person, you imply that there is only one.

The first person says no one is honest, the second person says there is not more than one honest person.

If you say the second person is the only honest one, that means that the third, fourth, fifth, ...etc are liars.

The third person says there is not more than 2 honest people.

According to you, this is a lie because the third person is liar. If that is a lie than there must be more than 2 honest people which contradicts your prior claim that there is only one honest person.

Therefore the claim that there is only one honest person is false.

Like wise all such claims are false except that there are

5

honest people

Edited March 4, 2010 by mmiguel1

[Guest]

When you say "the" honest person, you imply that there is only one.

The first person says no one is honest, the second person says there is not more than one honest person.

If you say the second person is the only honest one, that means that the third, fourth, fifth, ...etc are liars.

The third person says there is not more than 2 honest people.

According to you, this is a lie because the third person is liar. If that is a lie than there must be more than 2 honest people which contradicts your prior claim that there is only one honest person.

Therefore the claim that there is only one honest person is false.

Like wise all such claims are false except that there are

5

honest people

Have you heard this somewhere, if not how can you be so sure?

[Guest]

Have you heard this somewhere, if not how can you be so sure?

I proved it in my previous post above.

If you assume any number other than my answer you will come to a contradiction.

Assume for now that there actually are 5 honest people, then look at each person's statement and see if they are lying based on this assumption.

1st person says no honest people at all: Liar because 5 > 0

2nd person says no more than 1 honest person: Liar because 5 > 1

3rd person says no more than 2 honest people: Liar because 5 > 2

4th person says no more than 3 honest people: Liar because 5 > 3

5th person says no more than 4 honest people: Liar because 5 > 4

6th person says no more than 5 honest people: Truthteller because 5 <= 5

7th person says no more than 6 honest people: Truthteller because 5 <= 6

8th person says no more than 7 honest people: Truthteller because 5 <= 7

9th person says no more than 8 honest people: Truthteller because 5 <= 8

10th person says no more than 9 honest people: Truthteller because 5 <= 9

Now count the number of truth-tellers: we get 5 which is consistent with our initial assumption. So 5 is a valid answer.

Is it the only valid answer though?

What if you assume another number like 4?

Here is the same analysis but assuming 4 honest people:

1st person says no honest people at all: Liar because 4 > 0

2nd person says no more than 1 honest person: Liar because 4 > 1

3rd person says no more than 2 honest people: Liar because 4 > 2

4th person says no more than 3 honest people: Liar because 4 > 3

5th person says no more than 4 honest people: Truthteller because 4 <= 4

6th person says no more than 5 honest people: Truthteller because 4 <= 5

7th person says no more than 6 honest people: Truthteller because 4 <= 6

8th person says no more than 7 honest people: Truthteller because 4 <= 7

9th person says no more than 8 honest people: Truthteller because 4 <= 8

10th person says no more than 9 honest people: Truthteller because 4 <= 9

If we assume 4 honest people and classify each as truthteller or liar, we count 6 honest people which doesn't make sense because we assumed 4.

In general, if you assume X honest people, you will count 10 - X honest people after such an analysis.

If we want our answer to make sense, then our assumed value: X, must be equal to our counted value: 10 - X;

This gives us an equation on which we can apply some simple algebra:

X = 10 - X

2X = 10

X = 5;

Therefore this is the only answer that yields a consistent value for number of honest people.

This counting analysis is all actually quite unnecessary though. A very quick proof can be found in my earlier post.

Edited March 4, 2010 by mmiguel1

[Guest]

#2

[Guest]

I proved it in my previous post above.

If you assume any number other than my answer you will come to a contradiction.

Assume for now that there actually are 5 honest people, then look at each person's statement and see if they are lying based on this assumption.

1st person says no honest people at all: Liar because 5 > 0

2nd person says no more than 1 honest person: Liar because 5 > 1

3rd person says no more than 2 honest people: Liar because 5 > 2

4th person says no more than 3 honest people: Liar because 5 > 3

5th person says no more than 4 honest people: Liar because 5 > 4

6th person says no more than 5 honest people: Truthteller because 5 <= 5

7th person says no more than 6 honest people: Truthteller because 5 <= 6

8th person says no more than 7 honest people: Truthteller because 5 <= 7

9th person says no more than 8 honest people: Truthteller because 5 <= 8

10th person says no more than 9 honest people: Truthteller because 5 <= 9

Now count the number of truth-tellers: we get 5 which is consistent with our initial assumption. So 5 is a valid answer.

Is it the only valid answer though?

What if you assume another number like 4?

Here is the same analysis but assuming 4 honest people:

1st person says no honest people at all: Liar because 4 > 0

2nd person says no more than 1 honest person: Liar because 4 > 1

3rd person says no more than 2 honest people: Liar because 4 > 2

4th person says no more than 3 honest people: Liar because 4 > 3

5th person says no more than 4 honest people: Truthteller because 4 <= 4

6th person says no more than 5 honest people: Truthteller because 4 <= 5

7th person says no more than 6 honest people: Truthteller because 4 <= 6

8th person says no more than 7 honest people: Truthteller because 4 <= 7

9th person says no more than 8 honest people: Truthteller because 4 <= 8

10th person says no more than 9 honest people: Truthteller because 4 <= 9

If we assume 4 honest people and classify each as truthteller or liar, we count 6 honest people which doesn't make sense because we assumed 4.

In general, if you assume X honest people, you will count 10 - X honest people after such an analysis.

If we want our answer to make sense, then our assumed value: X, must be equal to our counted value: 10 - X;

This gives us an equation on which we can apply some simple algebra:

X = 10 - X

2X = 10

X = 5;

Therefore this is the only answer that yields a consistent value for number of honest people.

This counting analysis is all actually quite unnecessary though. A very quick proof can be found in my earlier post.

That doesn't mean that you are right. We have to wait for the answer, in order to see who is correct.

[superprismatic]

That doesn't mean that you are right. We have to wait for the answer, in order to see who is correct.

Well, mmiguel1 gave several proofs of his contention that 5 is the answer. They were clear and concise. I can't imagine anyone arguing with his very nice explanations. Even if the proposer of the problem gives a different answer, I wouldn't believe it. Nice explanations you gave there, mmiguel1!

[jazzship]

The answer is 5. Miguel gave a nice detailed description above so there's nothing really further to add.

In short, by observation you can see that and statement for each person, x, will result in 10-x truthful

answers. Therefore only 5 people telling the truth can satisfy the question:

x = 10 - x

2x = 10

x = 5

Each person says:

"There's not more than x honest people here."

Which is, Honest people <= x.

For the answer to be person #2, or for one person to be telling the truth, it would need to be worded:

"There's only x honest people here."

Edited March 5, 2010 by jazzship

[Izzy]

5, the last five.

The first statement is clearly a lie. Which automatically makes what the 10th person say true. There are 9 people remaining (including the 10th person), and not more than 9 of them can be telling the truth. "Not more than" implies that it can be less than 9. Like wise, not more than 8, 7, 6, and 5 are all statements that can work. That makes a total of five people, making the statement "not more than 4" a lie.

[Guest]

Thanks superprismatic, jazzship and izzy.

I was hoping to get some more supportive answers than earlier, haha.

Edited March 5, 2010 by mmiguel1