Chase Orton asked all of us, “What is your professional New Year’s resolution?”

I said that I wanted to stay skilled as a math teacher. As much as I’d like to pretend I’ve still got it in spite of my years outside of the classroom, I know there aren’t any shortcuts here: I need to do more math and I need to do more teaching. I have plans for both halves of that goal.

In support of “doing more math,” I’ll periodically post about my recreational mathematics. Please a) critique my work, and b) shoot me any interesting mathematics you’re working on.

When Should You Bet Your Coffee?

Ken Templeton sent me an image from his local coffee shop.

Should you bet your free coffee or not? Under what circumstances?

This question offers such a ticklish application of the Intermediate Value Theorem:

If the bowl only has one other “Free Coffee” card in it, you’d want to bet your own card on the possibility of a year of free coffee. But if the bowl had one million cards in it, you’d want to hold onto your card. So somewhere in between one and one million, there is a number of cards where your decision switches. How do you figure out that number? (PS. I realize the IVT doesn’t hold for discrete functions like this one. Definitions offer us a lot of insight when we stretch them, though.)

I asked some of my fellow New Year’s Eve partygoers this question and one person offered a concise and intuitive explanation for her number, a number I personally had to calculate using algebraic manipulation. Someone else then did his best to translate some logistical and psychological considerations into mathematics. (eg. “Even if I win it all, I won’t likely go get a drink every day. Plus I’m risk averse.”) It was such an interesting conversation. Plus what great friends right?

Here’s my work and the 3-Act Task for download.

How Many Bottles of Coca-Cola Are in That Pool?

When I watched this video, I had to wonder, “How many bottles of Coca Cola did they have to buy to fill that pool?”

I tweeted the video’s creator and asked him for the dimensions of the pool.

“12 feet across by 30 inches high,” he responded.

Even though the frame of the pool is a dodecagon, the pool lining itself seems roughly cylindrical. So I calculated the volume of the pool and performed some unit conversions to figure out an estimate of the number of 2-liter bottles of Coca Cola he and his collaborators would have to buy.

Here’s my work and the 3-Act Task for download.

How Do You Solve Zukei Puzzles?

Many thanks to Sarah Carter who collected all of these Japanese logic puzzles into one handout.

Carter describes the puzzles as useful for vocabulary practice, but I found myself doing a lot of other interesting work too. For instance, justification. The rhombus was a challenging puzzle for me, and this answer was tempting.

So it’s important for me to know the definition of a rhombus — every side congruent — but also to be able to argue from that definition.

And the challenge that tickled my brain most was pushing myself away from an unsystematic visual search towards a systematic process, and then to write that process down in a way a computer might understand.

For instance, with squares, I’d say:

Computer: pick one of the points. Then pick any other point. Take the distance between those two points and check if you find another point when you venture that distance out on a perpendicular line. If so, see if you can complete the square that matches those three points. If you can’t, move to the next pair.

Commenter-friends:

• What are your professional resolutions for 2017?
• What recreational mathematics are you working on lately?
• If any of you enterprising programmers want to make a Zukei puzzle solver, I’d love to see it.

2017 Jan 2. Ask and ye shall receive! I have Zukei solvers from Matthew Fahrenbacher, Jed, and Dan Anderson. They’re all rather different, each with its own set of strengths and weaknesses.

2017 Jan 2. Shaun Carter is another contender.