Solve the Handshake Problem: Brian's Answer

• K Sengupta
In summary, Brian shook either 8 or 9 hands at the party, as he and Steve were the only ones with a unique number of handshakes.
K Sengupta
Brian and Steve were invited to a party attended by four other pairs of siblings. So, a total of ten persons were present at the party. During the party various handshakes took place, but no person shook their own hand or the hand of their sibling.

At the end of the party Steve asked each person, including Brian, how many different people they shook hands with, and was surprised to note that every number was different.

How many hands did Brian shake?

At a first glance it seems to me as there would be multiple possible answers. I'll have to think about this some more.

Are you sure all needed information is provided?

Seems to be a bit of a problem, here. If nobody shook their siblings' hands, the highest number of different people that any person could share a handshake with is 8 and the smallest number is 0. If each of the 10 people shook hands with a different number of people, the numbers 0-8 would not allow each of them to have a unique number of handshakes.

turbo-1 said:
Seems to be a bit of a problem, here. If nobody shook their siblings' hands, the highest number of different people that any person could share a handshake with is 8 and the smallest number is 0. If each of the 10 people shook hands with a different number of people, the numbers 0-8 would not allow each of them to have a unique number of handshakes.
0-8 means that there are 9 options. Steve was asking the 9 others, that does not imply he is part of the set of unique handshakes.

MeJennifer said:
0-8 means that there are 9 options. Steve was asking the 9 others, that does not imply he is part of the set of unique handshakes.
True - the problem statement did not say that Steve had to consider himself - only the people that he asked. Thank you.

Edit: Interesting - Brian cannot have shaken 0 hands because if anybody other than Steve (who is not in the 0-8 grouping) had shaken 8 hands, he would have had to have shaken Brian's hand, as well as Steve's. Since Brian cannot have shaken 0 hands if anybody else shook 8 hands, nobody else is in a position to have shaken 8 hands except Brian. He was the high-scorer.

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One thing isn't clear to me-- is Brian Steve's sibling? It doesn't really state this, but I'm not sure if I should assume it to be true. If not, then the possibilities are 1-9 or 0-8. Essentially, the same logic applies, meaning that either way, Brian shook 8 or 9 hands. 8 if they're siblings, and 8 or 9 if they're not. Maybe that's the crux of the problem right there-- assume that they're *not* siblings, and now you actually have to figure out a solution where the shaking works out correctly!

DaveE

Ok, wrong assumptions. I assume that Brian and Steve ARE siblings, making the answer unique. And no, it's not 8. If they AREN'T siblings, then there are multiple solutions, as asserted above. ... Further if they AREN'T siblings, Brian has 6 possibilities for the number of handshakes that he could've had.[/edit]

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I still don't think I'm understanding the logic needed to solve this problem.

I understand that the number of handshakes possible are 0-8. Each one is assigned to each of the people (besides Steve).

The part that gets me is that I don't see ANY distinguishing traits about Brian has compared to all of the other people at the party.

DyslexicHobo said:
The part that gets me is that I don't see ANY distinguishing traits about Brian has compared to all of the other people at the party.

Try actually drawing out a solution. Hint:
Start with whoever shakes 8 hands. You'll notice that the sibling of whoever shakes 8 hands *has* to be the one who shakes 0 hands. Next, arbitrarily choose someone else at the party who shakes 7 hands. Who does his sibling have to be? Etc.
DaveE

Under this conditions the task is a paradox. There will be 3 people minimum with the same amount of handshake - 4. Then the author should rephrase it that he didn't count himself and his sibling. Then his sibling just like himself had 4 handshakes. And yes draw the table 10x10 and match their handshakes giving the #1 person 8 handshakes, #2 - 7 handshakes and so on. Soon you'll see that when you'll reach 4 handshakes they will repeat in 2 more persons and only then will drop to 3, then to 2 and to 1. With all other variations you'll get even more same numbers of handshakes.
Am I right?

smaster said:
Under this conditions the task is a paradox. There will be 3 people minimum with the same amount of handshake - 4. Then the author should rephrase it that he didn't count himself and his sibling. Then his sibling just like himself had 4 handshakes. And yes draw the table 10x10 and match their handshakes giving the #1 person 8 handshakes, #2 - 7 handshakes and so on. Soon you'll see that when you'll reach 4 handshakes they will repeat in 2 more persons and only then will drop to 3, then to 2 and to 1. With all other variations you'll get even more same numbers of handshakes.
Am I right?

Well, no. First of all, there are a minimum of 2 out of 10 people who shook the same number of hands at the party, not 3. But beyond that, the problem is correct because it doesn't say that Steve didn't shake the same number of hands as everyone else, it just says that none of the people Steve asked shook the same number of hands. Hence, someone shook the same number of hands as Steve. That's actually pretty obvious, since the maximum number possible would be 9, and the lowest number possible would be 0, and we know for a fact that both 9 and 0 cannot co-exist in this example. Hence, the range must be 1-9 or 0-8, which each have 9 numbers in them. And since there are 10 people, it's a verifiable fact that at LEAST two of the people shook the same number of hands. It's just that one of those people happens to be Steve, who's doing the asking.

DaveE

K Sengupta said:
How many hands did Brian shake?
4

PS
My original answer was "4", but PF system did not accept it, because at least 4 characters needed, That is why I posted this PS.

PPS
Steve got answers 0 to 8. Somebody said 8. So, there are 8 + 1 = 9 persons, who shook hands at least 1 time. But who shook hands 0 times? There is only one answer. Sibling of a person, who said 8. Thus, we have numbers of handshakes grouped by siblings.

8-0
7-1 (somebody said 7, who said 1? His/her sibling) etc.
6-2
5-3
4-4

There is only one way for Steve to get 9 different answers. He shook hands 4 times and so did his sibling Brian.

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The only possible way i can see this could be true is that some of them can't count, but I haven't read replies yet, so explanations may be there to help...

It's too hard to determine because I know that not everyone likes to shake hands. Various people shaking hands does not mean that everyone shook hands. I am reluctant to shake hands with just anyone as some people have sweaty palms. I know its fussy but you have to take things like that into account.

1. What is the Handshake Problem?

The Handshake Problem involves finding the number of handshakes that occur when a group of people each shake hands with one another. It is a mathematical problem that can help us understand relationships and networking.

2. What is Brian's Answer to the Handshake Problem?

Brian's Answer is a formula that can be used to solve the Handshake Problem. It is N(N-1)/2, where N is the number of people in the group.

3. How does Brian's Answer work?

Brian's Answer works by using the fact that each person in a group will shake hands with every other person, except for themselves. So, if there are N people, each person will shake hands with N-1 other people. However, this counts each handshake twice (once for each person involved), so we divide by 2 to get the total number of handshakes.

4. Can Brian's Answer be used for any number of people?

Yes, Brian's Answer can be used for any number of people. As long as you know the number of people in the group, you can plug it into the formula and get the total number of handshakes.

5. What is the significance of solving the Handshake Problem?

The Handshake Problem has real-world applications in fields such as network analysis, computer science, and sociology. It can help us understand the complexity of relationships and the importance of networking. Solving the Handshake Problem can also improve our problem-solving skills and critical thinking abilities.

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