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The "Infamous" Math Handshake Problem

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At the start of the year, the Algebra 2 class looked at one of the “classic” problems in algebra.  It served as an introduction into quadratic equations.  The problem is stated below with the answer following.  Try your best and see if you can shake out the Handshake Problem.

Handshake Problem

You are in a classroom full of people (and naturally being inquisitive with mathematics) start wondering how many handshakes are possible in the classroom.  You can only shake hands with another person once and you can’t shake your own hand as that’s just weird.

Write an equation for the number of people that will calculate the total number of handshakes.  How many handshakes are needed if they are 25 people, 50 people, and 100 people in a classroom?

Answer below…


























One strategy to solving difficult problems in mathematics is to try and look for patterns. Here is a table with 1 person, 2 people, 3 people, 4 people, 5 people, and 6 people. 

Number of people















The pattern becomes 0, 1, 3, 6, 10, and 15.  (As a side note, this pattern is known as triangular numbers).  The pattern might not be obvious, and some additional discussions and hints are usually provided.  The following hint usually provides the equation.


When n = 25 people there are (25)(24)/2 = 300 handshakes.  Likewise, there will be 1,225 handshakes with 50 people, and 4,950 handshakes with 100 people.  Clear as mud?  If so, ask one of the Algebra 2 students for help!

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Freeman Public Schools415 8th StreetP.O. Box 259Adams, NE  68301

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