[Guest]

Heather and Joe invited 10 other couples for an afternoon sail on their yacht. During the sail, each pair of people who had not previously met were introduced. Back in port, as the guests stepped off the boat, Joe asked each of them how many people he or she was introduced to. When they were gone Joe noted, “According to their answers, no two guests were introduced to the same number of people this afternoon, and none of them was introduced to the same number of people I was introduced to.” Assuming that Joe and the guests all told the truth, and that no one was introduced to anyone he or she had previously met, how many of the guests did Heather meet for the first time that afternoon?

[Guest]

Any help from forum???

[Smith]

Unless this is a trick question, I'd say there is no chance that there is enough information here to calculate a response.

[superprismatic]

According to the OP, each person was introduced to a different number of people.

So, for every number 0,1,2,3,4,5,6,7,8,9 there must have been a person who met

that many people. However 9 is impossible because that would mean that one of

the couples had to be introduced to each other. That's impossible!

[Guest]

seems to me that the answer is zero. they (joe and heather) had to have met them all, or they wouldn't have known who they were to invite them in the first place. furthermore, if they all acted as couples, or "pair"s, then as joe spoke to each one as they departed, heather would have been with him, thus meeting everyone.

[Smith]

seems to me that the answer is zero. they (joe and heather) had to have met them all, or they wouldn't have known who they were to invite them in the first place. furthermore, if they all acted as couples, or "pair"s, then as joe spoke to each one as they departed, heather would have been with him, thus meeting everyone.

But then Joe and Heather would have been introduced to the SAME NUMBER of people which breaks the rule that none met the same number...

[Smith]

According to the OP, each person was introduced to a different number of people.

So, for every number 0,1,2,3,4,5,6,7,8,9 there must have been a person who met

that many people. However 9 is impossible because that would mean that one of

the couples had to be introduced to each other. That's impossible!

I hate to disagree with Superprismatic (based on past performance), but... Based on the OP, I thought the range of values would be 0...21 There were 11 couples total and the answers were based on a 'he or she' response. Also, for Joe and Heather to have different answers, the results must be based on individuals.

[superprismatic]

I hate to disagree with Superprismatic (based on past performance), but... Based on the OP, I thought the range of values would be 0...21 There were 11 couples total and the answers were based on a 'he or she' response. Also, for Joe and Heather to have different answers, the results must be based on individuals.

Thanks, smith, I misread the post. So, my reply was wrong.

[Guest]

Zero. She invited everyone. She met nobody new.

[Gmaster479]

Zero. She invited everyone. She met nobody new.

bullfroglightbulb has a good point.

According to the OP, Joe obviously was introduced to someone based on the line that states "and none of them was introduced to the same number of people I was introduced to."

That leads me to believe that Heather knew all the couples that were invited and that she was the one who invited them so that Joe could meet all of her friends. It is possible that Joe invited 5 couples and Heather invited 5 couples and that Heather happened to know one of the 5 couples Joe invited or vice-versa but based on the lack of info in the OP Heather met no one new.

Edited March 11, 2011 by Gmaster479

[Guest]

A guest had a chance to meet at most 20 new people, assuming everyone had already met their own spouse.

Since there are 20 of them, so they were introduced incrementally from 1 to 20 people. Joe already knew everyone being the host so he was introduced to 0 person.

Edited March 11, 2011 by KlueMaster

[Guest]

The hosts didn't need to know guests. It could be fund raising party on their cruise on Mediteranian for example. If this is the case, Joe was introduced to all 20 people, since he asked each of them how many people they have been introduced to. If this is the case logic would go like this:

Guest 1 - 0 People

Guest 2 - 1 People

...

Guest 20 - 19 People

Joe - 20 People

In opposite case, where all guests were people Joe and Heather know, it would be similar, but Joe would be introduced to zero people, Guest 1 to one people etc.

From the info we have the answer must be that Heather meet same number of people as Joe and that would be probably 0 since they knew all the guests they invited. From Joe's statement we see that “no two guests were introduced to the same number of people this afternoon, and none of them was introduced to the same number of people I was introduced to.” Heather wasn't guest, she was host.

Edited March 11, 2011 by Ctpubop

[Guest]

Cute...

10

[superprismatic]

Heather was introduced to 10 people, and Joe was introduced to 10 people.

[Guest]

I must agree with bullfroglightbulb.

[superprismatic]

Joe and the guests have all met different numbers of people.

Together with Heather, there are 22 people. The largest

possible number of introductions for a person is 20 because

there are two people (self and spouse) for which an introduction

is never made. The smallest number of introductions for a person

is 0. So, there are 21 possible number of introductions:

all the numbers from 0 to 20.

Now, since Joe and the guests comprise 21 people, one of this

group was introduced to 0 people, one was introduced to 1 person,

one was introduced to 2 people, .... , one was introduced to 20

people. All possible numbers of introductions had to be used

up by this group.

Now, let's deduce who is married to whom:

Let's start with the person who had 20 introductions. This

person was introduced to everyone there except self and spouse.

Suppose the spouse of this person had a non-zero introduction

number. Then there would be three people not introduced to

the 20-introductions person: self, the non-zero spouse, and

the person having a 0 introduction number. But, there are

only two as stated in the second sentence of this paragraph.

So, the 20-introductions person is married to the

0-introductions person. And neither of these people are Joe

because Heather is not in the group we are talking about.

OK, now we can remove the two from our group of 21, leaving

a group of 19. We can also remove 1 introduction from

everyone in the group plus Heather (the introduction they each

had with the 20-introductions person). Now we have 19 people

with introduction numbers from 0 through 18. And using the

same reasoning as in the last paragraph, we can pair the

19-introduction person (now with a count of 18) with the

1-introduction person (now with a count of 0) as spouses,

neither of which is Joe.

We continue in this way, through ten spousal pairings:

20-0, 19-1, 18-2, 17-3, 16-4, 15-5, 14-6, 13-7, 12-8, and

11-9. There are now only two people left, Joe and Heather.

During each spousal pairing we have reduced both Joe's

and Heather's introduction numbers by 1. Since we did this

ten times, their original introduction numbers must have

both been 10.

Thus 10 is the answer and, as a bonus, we know the same

goes for Joe.

[Smith]

Bravo, superprismatic! I stand corrected (my first post in this thread). It took some careful reading on my part to follow it through, but it seems every point is irrefutable.

[superprismatic]

Bravo, superprismatic! I stand corrected (my first post in this thread). It took some careful reading on my part to follow it through, but it seems every point is irrefutable.

Thank You! I thought that an explanation of this problem was important because it embodies how Mathematics can lead us to surprising results. I'm happy that you appreciated my efforts.

[Guest]

Thank You! I thought that an explanation of this problem was important because it embodies how Mathematics can lead us to surprising results. I'm happy that you appreciated my efforts.

Great job man! It took me a while to wrap my brain around it, but yeah, it is only possible solution.

[Guest]

Logical Inconsistency

Closest answer is Heather and Joe collectively met 20.(if assumption 5 is dropped)

Certain logical deductions-

1. There are 11 couples including heather and Joe, i.e. 22 people.

2. NO person can take the value of 22(meaning he met himself.........lol), 21(he met his partner).

3. Joe noted 'no two guests' and 'none of them' meaning Heather answer could have been repeated because it is yet to be counted.

4. So 21 people (removing Heather) Can take 21 values(0-20)

5. the values 0 and 20 cannot co-exist. Anyone claiming to have met 20 people must have met the person claiming to have met 0 people. Thus, 21 people (removing Heather) Can take 20 values(0-19)or(1-20).

Note- deduction 5. is based on the assumption that the introduction leads to +1 for both parties.

if this assumption is dropped an answer could be reached.

[superprismatic]

Logical Inconsistency

Closest answer is Heather and Joe collectively met 20.(if assumption 5 is dropped)

Certain logical deductions-

1. There are 11 couples including heather and Joe, i.e. 22 people.

2. NO person can take the value of 22(meaning he met himself.........lol), 21(he met his partner).

3. Joe noted 'no two guests' and 'none of them' meaning Heather answer could have been repeated because it is yet to be counted.

4. So 21 people (removing Heather) Can take 21 values(0-20)

5. the values 0 and 20 cannot co-exist. Anyone claiming to have met 20 people must have met the person claiming to have met 0 people. Thus, 21 people (removing Heather) Can take 20 values(0-19)or(1-20).

Note- deduction 5. is based on the assumption that the introduction leads to +1 for both parties.

if this assumption is dropped an answer could be reached.

5 is not true because 0 and 20 could be married.