# One of those hand shaking problems

1. Apr 2, 2005

### ashi_mashi

Hi agian....
I have one of those hand shaking problems..it says:
There are 14 couples. There is some shaking of hands. No one shakes hand of thier date. No one shakes hands more than once with any one person. A boy asks each person how many times they shook hands. Each person gave him a different answer. How many times did the boy's date shake hands?Now, i thought it well through and i got the idea that assuming that everyone shook hands, then the total number of hand shakings would be "26+25+24+...+13+13+12..+1+0". (I drew a little sketch ) I thought that since there are two 13's (i'm not 100% sure, but i'm pretty sure;)) then, the boy has to be one of those 13's. And the problem is kinda solved...but, the TOTAL number that everyone shakes hands is the same, that is 26....so, i guess it cant be solved this way...anyway, i'm a little confused...i guess there is another way to solve it, or maybe i shouldnt assume that everyone shakes hands.

Thanks agian.

Last edited: Apr 2, 2005
2. Apr 3, 2005

### honestrosewater

You've gotten it right, with the reasonable assumption that no one shakes their own hand. There are 28 people. The boy asks 27 different people and gets 27 different responses. By the rules, no one can shake hands with more than 26 people, so the responses ranged from 0-26. The person who shook hands with 26 people shook hands with everyone except him/herself and the person who shook hands with 0 people. So {0, 26} must be a couple. By the same reasoning, the rest of the couples are {1, 25}, {2, 24}, ... {13, 13}. Also, by the time you get to 13, 13 has already shaken hands with 14-26, and 0-12 are already paired with their dates, so the boy must be 13's date.