Cleaning up an oversight in our talk about Patrick Honner’s parallelogram

A couple of days ago my son and I talked through this interesting challenge problem from an old blog post of Patrick Honner. Our project / blog post here:

Patrick Honner’s Challenging Parallelogram Problem

The challenge problem is here:

In writing up our project and reviewing the videos of our talk I noticed that we missed something. Watch for about 30 seconds starting at 1:40 (ish) as we play around with the the Zometool set trying to construct a quadrilateral:


I sort of kicked myself for the oversight, but moved on. In the next couple of days, though, I saw this tweet from Cristina Milos:

and then this one from Patrick Honner:

I guess the combination of those two tweets was enough to convince me that my decision to just move on wasn’t the right one. So we took 5 minutes this morning to revisit the parallelogram problem:


I enjoyed hearing my son talk about the new, non-convex shape and am happy to have seen the Milos and Honner tweets that convinced me to revisit this piece of our project from earlier in the week.

I try to learn from watching them struggle

I’ve written several times that my biggest struggle teaching the boys is that I’ve never been through any of this material before. One of the ways this particular struggle manifests itself is that I have a really hard time knowing when problems will be easy or difficult for them. I guess wrong (both ways) all the time.

Today my older son really struggled with this problem:

Problem 11 from the 2006 AMC 10b

The problem is:

Find the tens digit of the sum 7! + 8! + 9! + \ldots + 2006!

I certainly wouldn’t have considered this an easy problem, but since we’ve spent quite a bit of time on similar number theory problems – including last digit problems – I wouldn’t have thought this one would have completely stumped him.

Having recently finished up a chapter on “last digits” with my younger son I thought he’d be interested in helping us work through this problem, too. So we all went through it tonight for a fun Family Math project.

We started talking through it and I asked the kids what they noticed. My younger son thought that the solution probably would not require us to calculate all of the factorials. My older son thought maybe factoring out a 7! from all of the terms might be a way to start.

The next thought that my older son had was that finding the units digit of this sum would be fairly easy. This idea is, at least in my mind, an important first step to solving the problem because the reason that finding the units digit is fairly easy is essentially the same reason that finding the tens digit isn’t too difficult.

Making the transition to the next step was a little difficult, though


I broke the conversation at about 5 min just to start a new video. When we restarted the conversation I reminded them of one of James Tanton’s strategies – do **something**.

My older son suggested that something we could do would be to make a simpler problem. His suggestion for a simpler problem was to find the tens digit of the sum 7! + 8!. Making a simpler problem is often a great way to see what’s going on with the more complicated problem, and the simpler problem chosen by my son got us going down the right path.

We stayed on the path and calculated the units digit of the sums 7! + 8!, 7! + 8! + 9!, and 7! + 8! + 9! + 10!. Once the kids saw what happens with 10!, they were able to see how to the end on this problem.


The lesson I hope the boys take away from this talk is the James Tanton strategy – try **something**. Even, and maybe especially, when it isn’t obvious where to start with a problem, trying something is a great way to learn what works and what doesn’t.

What I hope to take away from this conversation (and really all of our conversations, of course) is a better understanding of the problems that the boys will find easy and the ones they will find to be difficult. Unfortunately, I think that I’m not very far down that path, yet.