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Even - Odd Handshake Problem


Date: 05/11/2000 at 18:06:15
From: Josh Cox
Subject: Handshake problem

Dr. Math,

Every person on the earth has shaken a certain number of hands. Prove 
that the number of persons who have shaken an odd number of hands is 
even.

I understand that this problem is true from experimental research I 
conducted on paper. However, I am unsure how to prove this problem 
works for every situation. Could you please help me to prove it? I 
would really appreciate it.

Thanks!


Date: 05/11/2000 at 18:46:21
From: Doctor Wilkinson
Subject: Re: Handshake problem

Very cute problem, Josh!

If you add up the number of hands each person has shaken, you have to 
end up with an even number, because every handshake gets counted 
twice. Now the sum of any number of even numbers is even, and the sum 
of an odd number of odd numbers is odd, and the sum of an even number 
and an odd number is odd; so the number of persons who have shaken an 
odd number of hands must be even, not odd.

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Discrete Mathematics
High School Number Theory

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