Football Math

Today Brian and I had a football math conversation. Brian was telling me about how the nature of football scoring makes the game interesting in ways that aren't applicable to other sports, like basketball, where scores can increase in increments of 1. And I, in turn, told Brian about the football math problem, which is this:

In a football game you can score either 3 points or 7 points.* 

What is the largest point total that would be impossible to score?

* I know that the scoring is more complicated than that in real life. We just simplify it for the purpose of the problem.

I really like this problem, and I've used versions of it in courses I've taught, although I find it usually takes some explanation for students to understand what it's asking. Basically, if it's only possible to get 3 or 7 points at a time, then some point totals can't ever be formed. For example, 11 is an impossible point total because there is no way to add 3's and 7's together to get 11. So the question is asking for the highest impossible point total. (I'll give the answer at the end, if you're curious.)

But what I liked most about this conversation was that it was a "shared interest" conversation unlike most shared interest conversations that Brian and I have.

One of the things I love about being married to Brian is that we have a lot of shared interests - we have a lot of fun together, and we have a lot of great conversations about the things we like. And one of the other things I love is that we don't like exactly the favorite things, and that we can share and appreciate the things we don't share, or at least sometimes appreciate that we don't share these things.

Infographic: Brian and Amy's Shared and Unshared Interests (Incomplete)

That was what was fun about the brief football math conversation. It wasn't just about me telling Brian about something I like in the realm of mathematics, and it wasn't just Brian telling me things I don't know or fully appreciate in the realm of football. It bridged the two topics, and kind of landed the conversation in the middle of the Venn diagram.

When I first decided to marry Brian, I wasn't yet 100% sure that Brian, who had bypassed math for much of his life, would be able to really appreciate what it is that I do (though I didn't doubt he would respect it). But to my surprise, I've found that math has been more of a bridge-builder than I expected. That's partly a reflection on Brian, who has sat in on my classes just to see what I'm doing, and who listens to me with genuine interest when I talk about my work. But even beyond that, I've also been able to use my math experience to help Brian with what he's doing. Yesterday, for example, Brian sat down with me and his marketing homework and we both worked to decipher formulas and apply them to a Dell case study and run calculations, and I think we both enjoyed and appreciated the shared experience.

My mathematics teaching is very much focused on bridge-building mathematics just by nature of the kind of mathematics I teach. I work with a lot of students who don't care much for mathematics but have to learn it because they'll have to teach it, and so much of my work is about connecting math to what my students are doing and what my students are interested in. My work is taking a subject that is often feared and loathed, or sometimes just tolerated, and to try consciously to bridge the gap between students and subject.

Brian might say similar things about football (the other half of the football math conversation). He's found that having a genuine interest in football (or basketball or other sports) is a great way to begin a conversation or to connect with someone he might not connect with along other dimensions. Watching games on television with my family has helped my siblings and their spouses get to know Brian, and has helped Brian find common ground with them, in the relatively short time that he's been here in Utah. I still struggle to really enjoy football, but if it builds connections, I can certainly get behind that.

The other day a student stopped by my office to get a form signed. She was wearing an Ohio State shirt, and I mentioned that I'd gotten my degree at the University of Michigan. We did that whole obligatory "Oh, you're a Michigan fan" thing, and then had this nice little bonding moment over our rivalry, wherein both of us knew that in the few minutes we interacted, we each got something about the other that most other students wandering the halls of the Talmage Building didn't get. It struck me afterwards as really interesting that, outside the geographic bounds of the schools, this football rivalry was not divisive but unifying. I loved that.

And now that we're back to football, I'll go ahead and end with the answer to the math question. Stop reading if you haven't figured it out yet and want to figure it out yourself.

ANSWER: 11 is the highest impossible point total. There are several ways to figure it out, but one is to realize that although 11 is impossible, 12 is possible (3 + 3 + 3 + 3), as is 13 (7 + 3 + 3) and 14 (7 + 7). Once you have three possible scores in a row, all others are possible because we can just add 3's onto any of those scores:

15 would be 12 + 3

16 would be 13 + 3

17 would be 14 + 3

18 would be 12 + 3 + 3

and so on.