Trade show information exchange math/logic problem $n+1$ companies each send 2 representatives to a trade show. Example Inc. sends Bob and Kim, filling up the ranks of some $2n + 2$ attendees. Some people exchanged information. One did not exchange information with oneself and two people representing the same company did not exchange information with each other. At the end of the show, Bob asked each person other than himself as to how many people did they exchange information with. Interestingly, they all gave a different answer, i.e., each person other than Bob had exchanged information with a different number of people.
(a) If $n = 4$, how many people did Kim exchange information with?
(b) Generalize your solution for arbitrary $n$.
I'm mostly just stuck with a; I'm told getting it will essentially unlock b. So for $n=4$, I first thought that Bob must've received $9$ distinct answers from 0-8, since 0 would be the minimum and 8 the maximum (as one could only ask a max of 2n people). The problem with this is that if someone exchanged info with 0 people, then no one could've exchanged info with 8, limiting me to 0-7, one answer short. Similarly, if I just use 1-8, I face the same problem. Even if I find distinct answers, I'm at a loss as to how I can know which one is Kim's.
I can't really see a way past this, I'm assuming it may have something to do with specific answer sets being allowed, but it's just speculation. Any help or insight would be much appreciated.
 A: Ill add an answer for completeness - As the number of exchanges for each person excluding Bob is distinct and there are $9$ such people, the number of exchanges for these people is some permutation of $\{0,1,2,\cdots,8\}$ because the minimum number of exchanges for a person is $0$, maximum is $8$ since a person cannot exchange with themself or their company partner and since we have 9 possible values and 9 people, each value must occur. Now look at the person with $8$ exchanges $P_8$. They must have exchanged info with everyone but their company partner, and the only person who they havent exchanged with is the person with $0$ exchanges $P_0$. So $P_8$ and $P_0$ must be partners. Similarly, the person with $7$ exchanges $P_7$ has exchanged information with all but $2$ people and one of them must be their partner. One of them is $P_0$ obviously, and the other is $P_1$ because $P_1$ has already exchanged with $P_8$ and can't exchange anymore. But $P_0$ is already paired up so $P_1$ and $P_7$ are partners. Continuing this way we can pair up $(P_0, P_8),(P_1,P_7),(P_2,P_6),$ and $(P_3,P_5)$ where $P_i$ is the person with $i$ exchanges. So we are left with $P_4$ with $4$ exchanges who must be Kim.
A: Still new to both the subject and the website, but wanted to post some of what I did because I know it's likely someone will search for this in the future: quasi's comment is correct and drawing out the solution on a piece of paper with that as a basis should be enough to make the answer clear; it also helps to remember that there are no limitations on how many people Bob's exchanged information with. From there, proving for b should be easy.
I'd have it drawn in pairs, starting with a $0/8$ and using dots/notches/whatever to mark interactions should reveal a pattern one can solve the problem with.
