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The Handshake Problem

(Geometry) Permanent link

kidshandshakeI have the honor of being the “inaugural blogger” for the new Teaching Children Mathematics (TCM) blog, “Math Tasks to Talk About.” Now, to be clear, what I know about blogging could fit in a thimble with plenty of room still left for your finger. However, the talented staff at NCTM can take whatever I submit and magically make it blog-worthy, so here goes!

My absolute favorite math task to talk about is a classic known as the Handshake problem. Alternatively, you may know it as the How Do You Do? problem or the Meet and Greet problem or one of more than at least a dozen different names. No matter what you call it, this problem is my favorite because it can be easily made accessible and interesting to students at all levels, from first grade through high school!

All right—here’s the problem:

Ten [or however many you want] people are at a party, and you want everyone to meet (shake hands with) everyone else at the party. How many handshakes will it take?  

For students in the early grades, simply reduce the number of people who are shaking hands; and for students in the upper grades, move to a generalization for a large number of people shaking hands.

The other reason I find the problem appealing is that the conditions are few and easy to understand. In exploring the problem, students will discover two things: (1) you wouldn’t shake hands with yourself, and (2) when two people shake hands, it counts as one handshake, not two.

OK—that’s it! Solve away! Clearly, the only way this will become a robust and interesting blog is if there is interaction. Give a version of the problem to your class, talk about the different ways your students approach solving it, and talk about your own strategies/reasoning as you think about the problem. Afterward, come back here and post a comment about how it went. You are also welcome to share sample student work and photos. We need your help to make “Math Tasks to Talk About” a rich problem-solving resource.

I’ll be back in two weeks as a follow-up to this post with solutions and ruminations about the problem. In the meantime, I look forward to hearing from you!


RalphConnellyRalph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees. 

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