Sharing problem #3 from the European Girls’ Math Olympiad with kids

Yesterday I saw the great news that team USA won the European Girls’ Math Olympiad:

Flipping through the problems last night, problem #3 really caught my eye as one that math students might really enjoy because the solution is really cool. Here’s the problem:


This afternoon I thought it would be fun to talk through the problem with the boys. I have no expectation that they would be able to solve this problem – obviously! – but I really did think that a sketch of the solution would be really interesting to them.

I started by talking through the problem to make sure that they would understand it:

Once the boys understood the problem we dove into trying to solve it – where do you even begin – both boys said in the last video that the problem seemed impossible! Starting with some simple configurations with 2, 3, and 4 lines helped us see that the answer to the problem might be “no”.

To wrap up I showed the boys how you solve this problem via a coloring argument. The critical idea is that you can color the regions that are formed by the lines, with no two regions sharing a side having the same color – with just two colors. Once you have the coloring, there’s a fun little “aha” moment when you watch the path the snail takes . . . .

So, a seemingly impossible problem has a really pretty and really instructive solution. I think the coloring idea is something that middle school and high school kids who are interested in math will really enjoy seeing.

Exploring the “Snapchat problem” with my kids

A few weeks ago a friend asked me this question:

“Say [our group]] has 26 people in it, but Snapchat has a limit of 16 in a group. How many groups would need to be created so that everyone is in a group with all of the [people in the original group]?”

I wrote about my solution to the problem here:

A fun math question about Snapchat from a friend

Last night I finally got around to going over the problem with the boys. When I first saw the problem I thought that it would be a really interesting problem for kids to explore. My kids are probably on the young side to fully understand the problem, though, so I started the project by making sure that I explained the problem really carefully. One thing that turned out to be a bit challenging for my younger son was distinguishing between the group of 26 people and the groups of 16 people:

At the end of the last video my older son wanted to start drawing some pictures. I let him draw in the 26 people off camera and then we started exploring how to make the groups. There were several interesting mathematical ideas that came up almost immediately. For instance: if we are going to minimize the number of groups, how many people should we try to put in those groups?

One thing that was really interesting to me here is that my younger son had a really hard time translating the words describing how he wanted to make the groups into what to draw in the picture.

We just paused the last video after we got to about 5 min and continued the discussion. I was a little bit stumped about what to do – my younger son had exactly the right idea and was saying the exact right words, but he just couldn’t figure out how to draw in some lines that represented those words. My plan was to simply let him figure it out.

Most of this video is my younger son struggling to find a way to draw the picture he wants. He eventually got it, though, which made me happy.

So, we’d just found a way to solve the problem with 4 groups, which was really cool – they found that solution faster than I did 🙂 The question now was could we do it with three?

Both kids had some pretty good intuition about why 3 would be impossible. I wanted to show them a slightly more mathematical way to think about it. Although it took a bit longer than I expected, it was fun to explain this proof to them.

I really like this problem and think that lots of kids learning math will really like it. The process you go through in solving this problem is a great example of problem solving in math.