This morning I had the pleasure to be a mathematical guest in my daughter’s third-grade class, full of inquisitive eight- and nine-year-old girls, and we had a wonderful interaction. Following up on my visit last year (math for seven-year-olds), I wanted to explore with them some elementary ideas in graph theory, which I view as mathematically rich, yet accessible to children. 
My specific aim was for them to discover on their own the delightful surprise of the Euler characteristic for connected planar graphs.
We began with a simple example, counting together the number of vertices, edges and regions. For counting the regions, I emphasized that we count the “outside” region as one of the regions.
Then, I injected a little mystery by mentioning that Euler had discovered something peculiar about calculating $V-E+R$. Could they find out what it was that he had noticed?
Each student had her own booklet and calculated the Euler characteristic for various small graphs, as I moved about the room helping out.
Eventually, the girls noticed the peculiar thing — they kept getting the number two as the outcome! I heard them exclaim, why do we keep getting two? They had found Euler’s delightful surprise!
The teachers also were very curious about it, and one of them said to me in amazement, “I really want to know why we always get two!”
With the students, I suggested that we try a few somewhat more unusual cases, to see how robust the always-two situation really was. But in these cases, we still got two as the result.
The girls made their own graphs and tested the hypothesis.
Eventually, of course, I managed to suggest some examples that do in fact test the always-two phenomenon, first by looking at disconnected graphs, and then by considering graphs drawn with crossing edges.
In this way, we were led to refine the $V-E+R=2$ hypothesis to the case of connected planar graphs.
Now, it was time for proof. I was initially unsure whether I should actually give a proof, since these were just third-graders, after all, and I thought it might be too difficult for the students. But when the teachers had expressed their desire to know why, they had specifically encouraged me to give the argument, saying that even if some students didn’t follow it, there was still value merely in seeing that one can mount an argument like that. Excellent teachers!
The idea of the proof is that $V-E+R=2$ is true at the start, in the case of a graph consisting of one vertex and no edges. Furthermore, it remains true when one adds one new vertex connected by one new edge, since the new vertex and new edge cancel out. Also, it remains true when one carves out a new region from part of an old region with the addition of a single new edge, since in this case there is one new edge and one new region, and these also balance each other. Since any connected planar graph can be built up in this way by gradually adding new vertices and edges, this argument shows $V-E+R=2$ for any connected planar graph. This is a proof by mathematical induction on the size of the graph.
Next, we moved on to consider some three-dimensional solids and their surfaces. With various polyhedra, and the girls were able to verify further instances of $V-E+R=2$.
The girls then drew their own solids and calculated the Euler characteristic. I taught them how to draw a cube and several other solids pictured here; when the shape is more than just a simple cube, this can be a challenge for a child, but some of the children made some lovely solids:
In the end, each child had a nice little booklet to take home. The images above are taken from one of the students in the class.
What a great day!
You can find a kit of the booklet here:
See also the account of my previous visit: Math for seven-year-olds: graph coloring and Eulerian paths.
Maybe it’s time you collect these into “Math for children, that adults can enjoy too!” (because surely next year you’ll have a post about math for nine year olds, and so on).
The topics could cover basic graph theory, perhaps very rudimentary explanation about Hilbert’s grand hotel, basic Ramsey theory or other combinatorics.
If you use a font which looks like crayons, and nice colors, it could really stick.
Thanks for the vote of confidence! In fact, I’ve had just such plans — let’s see whether I can complete it before the kids are teenagers!
In fact there is already a great kid’s book about Hilbert’s grand hotel, “A Cat in Numberland”. Look Inside http://www.amazon.com/The-Cat-Numberland-Ivar-Ekeland/dp/081262744X
That is pretty awesome. The drawings are great, too!
Brilliant….
Thanks so much, I’m glad you enjoyed it.
Ok, now do diff eq for kids (because i’m an adult and really struggling with it and graph theory was even harder to get my brain around)
I wish every concept of math was available in this format – easy to grasp.
Here is some more : http://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain
Something you might want to keep around:
“A High School-Level Course in Feedback Control
A Matlab-Based Introduction Requiring Only Algebra and Trigonometry”
JORGE CORTÉS and WILLIAM B. DUNBAR
https://classes.soe.ucsc.edu/cmpe008/Spring09/LectureNotes/CSM_paper.pdf
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The post made it to the front of Hacker News! See discussion at https://news.ycombinator.com/item?id=9354597
It also made it to the front of Reddit. See several discussions at
http://www.reddit.com/search?q=url%3Ahamkins.org%2Fmath-for-eight-year-olds
I had a question: for the 2D case, the “outside” was counted as one of the regions. For the 3D case, why wasn’t “outside” considered??
If you think about the case of a cube, say, then you can imagine puncturing one of the faces and stretching the cube out so that it lays flat in the plane. The face that you punctured becomes the “outside” region upon doing that. The same is true for all the other polyhedra that we considered — you open up one of the faces and lay the whole thing completely flat, so that the region you opened up becomes the outside region in the plane. One can also imagine the cube drawn on the surface of a sphere, in which case any one of the faces can be viewed as the “outside” region. It is interesting to consider the Euler characteristic for polyhedra that form a torus (like a donut) or more complicated shapes. You no longer always get 2 in these more general cases.
Fantastic!
Thanks, I’m glad you liked it.
I am a 28 year old and i find it equally interesting 🙂
Yes, I think this kind of mathematics is appealing at any age!
These make me happy. thanks.
I like this so much I spent the last 2 hours creating a version of the booklet in French. Will use with my kids and their class ASAP. http://thomas.baudel.name/Mathematiques/graphes-planaires.odt
Great! Bonne chance!
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Thx. I print this right now for my two doughters.
Is a PDF of this available, like for your chromatic numbers class?
Yes, I mention it at the end of the post. You can find it at: https://plus.google.com/u/0/+JoelDavidHamkins1/posts/46ZctASr5US.
Google+ is gone. Any current link for the PDF? Thanks!
Link is at the end of the post.
31 y.o here who found this absolutely fascinating. Please do more when time permits : )
I’m glad to hear that you enjoyed it! I think it is good for any age. I should have called it, Math for eight-year-olds at heart! Or eight-to-eighty-year-olds, or….?
It’s a great title and I wouldn’t change it one bit : ) I guess my parent comment was tangentially hinting at how deficient my schooling was and how much I appreciate you filling in the gaps for me : )
Let me link you guys to a similarly awesome piece on making “Probability” accessible to everyone…
The Raccoon Princess and the Fox Prince: A Bayesian Parable http://cassandraxia.com/projs/raccoon/
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This is absolutely wonderful. I have been convinced for some time now that graph theory is accessible to kids starting at a young age – what a shame that most kids don’t get to encounter it! Here is a post about me introducing a (very basic) graph theory concept to my daughter when she was 4 https://aofradkin.wordpress.com/2013/08/07/graph-theory-with-dolls/ . She is now 6, and I haven’t done too much follow-up graph theory. This post might have just inspired me to come back to it!
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I went through your activity with my kids tonight, and they loved it. Here’s my write up of our time going through it:
https://mikesmathpage.wordpress.com/2015/04/29/going-through-joel-david-hamkinss-graph-theory-for-kids/
Wow!
For a translation into Italian, see Annarita Ruberto’s blog http://www.lanostra-matematica.org/2015/05/la-teoria-dei-grafi-per-i-piccolie-non.html.
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I really appreciate it.
تشکر،عالی بود.
My son says (9 year old): “really cool” . 😉
Thank you so much, this is just what I was looking for. My son (age 9) has come across the Utilities problem (probably because I gave him How Long is a Piece of String to read by Eastaway and Wyndham) and wanted to know WHY K_3,3 isn’t planar (although he didn’t phrase it that way). My Graph Theory is a little rusty, so I was searching for some accessible material I could go through with him and found this, which is spot on, as once we have established Eulers formula he will easily be able to apply it for himself to see why. (As an aside, I’ve never come across someone suggesting giving one house water access by having it’s own well in the garden before, I found that quite creative).
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Many thanks for this once again brilliant activity 🙂
As I already did for the Chromatic numbers leaflet with your permission, I have translated it into French language… and your work has been approved by my kids (6 & 10) 🙂
The french translated leaflet is available on http://fr.toysfab.com/2016/06/la-theorie-des-graphes-pour-enfants/
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Thanks for this wonderful resource,
I will use the graphs (with reference to your web page) in a popular talk for students.
Thank you for this, I would never be able to come up with this by myself. I don’t know if you are aware but your document is on Google+ and now the booklet is no longer available. Would you be able to make this available as a pdf please? I was gutted when I went to download this and couldn’t. I would really appreciate it if you could find time. Thanks.
Ah, I hadn’t anticipated that Google would ever shut down G+. I’ll try to upload it here later. Thanks for the appreciation!
Thank you. Well earned appreciation, I’d say! I’m a homeschooling mum with a virtually non-existent knowledge of maths and I’m determined that my sons do not suffer the same fate as me. It is only by the willingness of mathematicians, like yourself, who share so freely that I can even begin to attempt this. Luckily I feel like I am succeeding as they both enjoy maths and have even used the word ‘cool’, panic and fear do not enter into it! I am learning right along with them. I really cannot thank you enough!
Just wanted to second the request for a updated pdf link since the G+ link no longer works. Thank you so much!
Link is now updated to host the pdf file on this blog.
Thank you!
Might be interesting and related, in particular the book “Infinity and me”:
https://medium.com/however-mathematics/8-picture-mathematics-books-for-your-curious-children-552ac415e7f7?fbclid=IwAR2jsaCBYXE25-VKKwmjvnBHiL0TRFyv0RgzeRIqHi1ljk7GrYCX0I23iLk
Downloaded for a niece. Thanks!
I gave your talk for seven year olds to my 8 years old granddaughter and it was a huge success. She commented that it was such fun to challenge your brain! However, graph theory for eight years old did not impress her. She said, “O, its 2. Than what?” There was no puzzle in it like Konigsberg bridge problem. Thank you for publishing it..
I’m glad your daughter enjoyed the first one. To my way of thinking, the main puzzle of the Euler characteristic is why it should always be 2? Why is that? The puzzle is to figure out why it is always the same. But a secondary puzzle is to look into the three-dimensional cases, and to figure out how to count faces etc. in that case.
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Great ressource. Just used it to keep the kids entertained during Corona virus lockdown here in Germany! Thanks so much.
Our kids enjoyed both graph coloring and the Euler characteristic. I totally appreciate the sketch of the inductive prove. Really easy for the kids to follow!
Keep up the great work and to inspire kids.
Thanks for the vote of confidence! I’m glad that you enjoyed it with your kids.
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Thank you! I was curious about the proof for the 3-D graph characteristic. Can one prove that by induction too? If so, what is the starting point (similar to the single vertex for planar graphs)? Also, I have a sense that the value of the Euler characteristic depends on the dimensionality of the space the network is embedded in and this has to do with the counting of the “outside” regions. Any suggestion will be appreciated.
The higher-dimensional analogues of the Euler characteristic are rather more complicated, and I don’t know of any simple proof by induction that follows the case for finite planar graphs. In the general case, one must pay attention to the genus of the resulting topological object. Perhaps one can prove it by induction by considering a simplicial complex. I’m just not sure, and I would welcome comments from the topologists!
Thank you I am grateful for the resources that you have!
This is very understandable explanation! Thank you very much!
This lesson is so well thought-out. Thank you for posting it! I worked through it with my son, but had him try to figure out on his own what makes a graph impossible. Lots of fun grappling and that “AHA” moment! Here’s a run through of how we played with that idea:
http://justaddplay.com/2020/09/big-math-ideas-impossible-graphs/
Wow! Your account is fantastic—I just love it. I tweeted about your blog post at https://twitter.com/JDHamkins/status/1538296588250095622.
Thank you so much! That is awesome. I can’t wait to go through some of your other kids activities.