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The Handshake Problem

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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. 

This is a great post!
Posted by: LaikenJ_65910 at 1/9/2014 1:58 PM

I also love this task. I have used it with children as well as with my pre-service elementary mathematics teachers (as you said, "It's a classic"). I find this task is a great opportunity for students to explore the idea of connecting different representations. My students often use t-charts, diagrams and formulas to help them answer this problem and I really push them to explain how the patterns they see in the chart, their formal and diagrams all relate to each other in a mathematical sense. Its particularly cool when they connect the "divide by 2" in their formula with their pictures. i love how this task helps me help my students see that mathematics is supposed to make sense! Great post, Ralph!
Posted by: AndrewT_47214 at 2/6/2014 11:14 AM

I like how for younger students you can adjust the number of people involved in the problem. To start, students can act out the problem. This concrete experience can get their problem solving minds thinking. Then students can be asked to represent their answers through a drawing or chart. To extend thinking, pose a question like: How would your answer change if one more person came along?
Posted by: PamelaG_76413 at 2/8/2014 1:22 PM

Great ideas, Pamela. It's often interesting to see young children start to shake hands "randomly", then realize they are having trouble counting, and coming up with an "organizer" on their own. The "one more person" question is excellent for getting them to realize that they don't have to "repeat" all the previous handshakes.
Posted by: RalphC_79522 at 2/10/2014 1:21 PM

I too have this problem in my "go to" activity list and have blogged about it as well: I love how it is such a good problem when first introducing Polya's problem solving process and different heuristics that can be used (like act it out, use a smaller number, make a table, etc.) and introducing the concept of Nth term. I actually was fortunate enough to do this activity with my students in Rwanda when I was there as a Fulbright Scholar and was pleasantly surprised to see that it "translated" beautifully! They loved it as much as my U.S. students do! It's great to see the students really understand why, in the direct formula, you divide by two ("because me shaking you is the same as you shaking me") and why you subtract one ("because you can't shake hands with yourself"). Funny that this blog is the first one NCTM posted because I also just did this activity with my Geometry students (university class of pre-service teachers) as a precursor to the follow-up problem, "Given a polygon of 30 sides, how many diagonals does it have?"
Posted by: DianaP_82657 at 2/19/2014 12:07 PM

Hi, Diana, and thanks SO much for posting! (nice to know folks are reading the blog! :) You are clearly a "serious" blogger, and have been at it a long time, with some great stuff on your blog!--thanks for sharing that information. Great to know that your Rwanda students engaged with the activity, also, so it seems to have international appeal! It was also a "great minds think alike" moment :) when you mentioned following up with the "how many diagonals" problem with your pre-service students, as I always did the same thing with my pre-service students--the "why" of the formula for the handshake problem, and the "why" of the formula for the diagonals having such clear parallels! Hope you continue to follow the TCM Blog!
Posted by: RalphC_79522 at 2/21/2014 10:25 AM

This could also go hand-in-hand (pun intended!) with the staircase problem, which asks how many blocks you need to build a staircase that goes n steps high. (Picture a simple staircase from the side made up of square blocks -- the first step has one, the second has two, etc.) The answers in both problems are the triangular numbers, and it is a good challenge to be able to explain how these two problems are alike, and why the answers are the same. What happens every time you add a new person to the handshake problem? What happens every time you add another step to the staircase problem?
Posted by: EllenM_51550 at 2/24/2014 2:37 PM

I love this problem, and also have used it with a wide range of learners (from early elementary through college level, and also for pre- and in-service teachers). I am always amazed at the variety of representations that students will use to think through the problem, and love seeing the satisfaction on someone's face when they are able to generalize for any number of people at the party. And who can resist telling Gauss' story about his motivation for the generalization!
Posted by: CeciliaA_14223 at 2/25/2014 1:39 PM

My "down-under" (Aussie) version of this is always the first lesson when I meet a new class of students or preservice teachers. But a little differently.
I ask everyone to draw four straight lines that never intersect, then four that intersect in one point, two points, three points,...,seven points (of course, 2 and 7 are impossible but I don't tell them that). Then I ask a volunteer to draw the first solution, ask their name, shake their hand when they are correct and tell them to stay where they are. Then we have a volunteer for one intersection, and again I ask their name, shake their hand, but also ask the other person standing at the board to shake their hand and congratulate them. This continues until we have eight students plus myself at the front (I accept "impossible" as the answer for 2 and 7 intersections). Then I ask everyone to sit back down and ask the class how many handshakes there were. And we go on to discuss solution methods etc.
What's nice about this is that:
(1) I learn the names of 8 students
(2) The other students learn some names too
(3) It's participatory as it gets students out of their seat
(4) The first question (no intersections) is very accessible
(5) Some of the initial questions are quite challenging, but students can use one answer to work out another.
(6) Impossible questions are actually relatively common in mathematics
(7) Problems that look different are actually the same - the maximum number of intersections of n straight lines is the number of handshakes between n people. I ask students to go home and think about what handshakes have to do with intersecting lines.
(8) It leads to all sorts of other interesting connections with triangular numbers.
(9) Perhaps most important (for me) is that the students have no idea what's going on when I ask them to stay at the front and shake the hand of the person coming up. The element of mystery is very engaging!
Posted by: StephenT_97169 at 2/25/2014 5:59 PM

WOW!!! What FANTASTIC comments/ideas!!! It is so great and so exciting to see folks commenting on the blog. I've already revealed that I am completely new at this, so, to be safe, I'll apologize in advance if I inadvertently violate any blog rules/etiquette. Ellen--thanks so much for the connection to the staircase problem--great idea. I hope you'll come back and visit the blog when the "solution/discussion" post goes up (next week, I think), as one of the links I'll share then makes this connection in a beautiful and visual way! Cecilia--thanks so much for "reaffirming" my claim that one major appeal of this problem is its accessibility to such a wide range of learners. Stephen--what a unique and interesting approach to the problem! As long as I've been using this problem (and that's a LONG time! :) , I was aware of the "algebraic equivalence" of the number of handshakes and the maximum number of intersections of n straight lines (and several other contexts), but had never thought about the intersecting lines as an illustration of the handshakes--shame on me, but proof that you can learn something new every day! :) Keep those comments coming--this is great!!
Posted by: RalphC_79522 at 2/26/2014 11:40 AM

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