Brain teaser: Hand shakes

Q: Suppose you and your wife meet two other couples.  While greeting,  they shake hands with each other. (So, there are 6 people). But, a husband does not shake hands with his own wife.  After greeting, suppose you asks other people how many times did they hand shake. Surprisingly, all of them (5 people including your wife)  answers differently. In that case, what is the number of hand shakes of your wife?

    당신과 당신의 아내가 다른 2쌍의  부부를 만났다고 하자. (즉, 6명이 만났다.) 인사를 하면서 각자 악수를 한다음, 당신이 모두에게( 당신의 부인을 포함한 5명에게) 몇번이나 악수를 했는지 물었다고 하자.  단, 부부 끼리는 서로 악수를 하지 않는다. 놀랍게도 모두들 다른 숫자를 말했다고 한다면, 이 때 당신의 부인은 몇번 악수를 했을까?  (문제에 주어진 조건은 충분하다. )




A: 답은 2.

    Note that because there are 6 people but the couple does not shake hands with each other,
    maximum possible number of handshakes is 4. Because five people answered differently,
    the numbers must be 0,1,2,3, and 4.

    (1) If your wife's hand shakes is 4, your wife must have hand shakes with all others except you.
         Thus, it is impossible to have zero number of hand-shakes. Thus, it is contradictory.
         So, your wife cannot have 4 handshakes.

     (2) Let's name other people as Mr. A and his wife Mrs A, Mr. B and his wife Mrs. B.
           Because your wife cannot have 4 handshakes, let's suppose Mr.A had 4 handshakes.
           (It does not matter whether it is Mr.A or Mrs A.)
           Then, Mrs. A must have zero handshakes because Mr.A hand shakes with all others.
           Thus, your wife cannot have zero hand shakes.  (Thus, possible numbers are 1,2,3)

     (3) Suppose your wife have 1 hand shakes. This must be with Mr.A.
           Then, suppose Mr. B have 2 handshakes. (It does not matter
            who had two hand shakes) Then, Mrs. B must have 3 handshakes.
           For Mr. B, one is with Mr.A and the other must be with you. (Because
           Mrs. A, your wife and Mrs. B should not handshake with Mr.B)
           Then, Mrs. B cannot have 3 handshakes because there are only you and Mr.A is available
           for her to handshake. Thus, it is impossible for your wife to have 1 handshakes.

     (4) If your wife have 3 handshakes,  it is with Mr. A , Mr. B and Mrs. B.
          Then Mr. B and Mrs. B cannot have 1 handshakes because they already handshake Mr.A
          and your wife. Thus, it is impossible for your wife to have 3 handshakes.

     (5) Thus, only possibility is that your wife have two handshakes.  One of them is with Mr.A
           and the other is with one of Mr.B and Mrs.B. And Mr.B or Mrs. B have 1 and 3 handshakes.
           Because 1 handshake is with Mr.A,  your wife hand shake
           with a person who have 3 handshakes. (This 3 must be with Mr.A, your wife and you.
          Thus you must have 1 handshake.)

     Thus, your wife's handshakes are 2.
     These argument does not depend who was the actual person had 4 handshakes.