# counting hands shake

Mr. and Mrs. Brown gave a party for their friends they have not seen for a long time. Three couples came. During the party, some of the people were so happy to see each other again, that they even shook hands. (None of the men shook hands with their own wives.) Later on, Mr. Brown asked everybody how many people he/she shook hands with. He received seven different answers. How many guests did Mrs. Brown shake hands with?

• The $3$ couples don't include Mr. and Mrs. Brown right? – Abraham Zhang Nov 12 '14 at 9:23

There are in total $8$ persons in the party and none of them shakes hands with their own partner. So getting $7$ different answers by asking $7$ persons means the answers are respectively $0,1,2,3,4,5,6$.

Suppose $A$ shake hands $6$ times, then everyone else except A's partner shake hand with $A$, so the persons who answers $0$ is A's partner.

Now suppose $B$ shake hands $5$ times, which means $B$ didn't shake hands with his/her partner and A's partner, but with everyone else. So the person who answers $1$ is B's partner, since everybody else(except A, B and their partners) shake hands at least two times with $A$ and $B$.

Now suppose $C$ shakes hands $4$ times, since $C$ didn't shake hands with his own partner and $A,B$'s partners, $C$ shaked hands with everyone else. So the persons who answers $2$ has to be $C$'s partner, since everybody else(except A, B, C and their partners) shake hands at least three times with A, B and C.

Finally, since we have identified that the persons answering $6$ and $0$ are one couple, the persons answering $5$ and $1$ are one couple, the persons answering $4$ and $2$ are one couple, the persons who answers $3$ is Mrs Brown.

• Right. One might add that Mr. Brown himself shook hands with the same $3$ persons. – Marc van Leeuwen Nov 12 '14 at 9:46
• @MarcvanLeeuwen Yes indeed, that's another information we can get – Petite Etincelle Nov 12 '14 at 9:55
• This logic doesn't seem to work. The two people who shook hands exactly 3 times (Mr. and Mrs. Brown according to your answer) are just another couple. Therefore, Mr. and Mrs. Brown could also be the pair 6-0, 5-1 or 4-2. As per the problem statement, there is nothing to differentiate the hosts from the guests. – user139000 Nov 12 '14 at 10:22
• @pew Mr Brown is the one who asks question, he is not part of the seven persons who give answers – Petite Etincelle Nov 12 '14 at 10:23
• @PetiteEtincelle: Thank you, I get it now. Great analysis! – user139000 Nov 12 '14 at 10:25