Sharring problem A1 from the 2017 Putnam with kids

We had a snow day today and I finally got around to sharing a neat problem from the 2017 Putnam Exam with the boys.

When I first saw the problem I thought it would be absolutely terrific to share with kids:

I started off the project today by having them read the problem and spending a little bit of time playing around:

After the initial conversation the boys, I triehd to start getting a bit more precise. The first sequence of numbers they knew was in the set was 2, 7, 12, 17, . . . .

They were not sure if 4 was in the set or not. My first challenge problem to them was to show that if 4 was in the set, then 3 would be in the set.

My next challenge question was whether or not 1 would be in the set.

Now we moved on to one of the number theory aspects of the problem – is 5 in the set?

During this conversation my younger son noticed that we had found a number that was 1 mod 5.

Finally, we talked through how you could find 6 from the number my son noticed in the last video.

I’m really happy with how this project went. This problem is not one (obviously) that I would expect the kids to be able to solve on their own, but most of the steps necessary to solve the problem are accessible to kids. It was really neat to hear their ideas along the way.

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