## My Winter Break in Recreational Mathematics

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?

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.

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.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Daniel Scher

January 1, 2017 - 6:41 pm -

My latest recreational mathematics obsession are “area mazes” (created, coincidentally, by the same designer as Zukei puzzles). You can see some here:

http://wordplay.blogs.nytimes.com/2015/08/17/naoki-2/?_r=0

The idea is simple: You’re shown a collection of rectangles that share some sides in common. You’re also given some of the side lengths and areas of the rectangles. The challenge is to determine a missing side length or a missing area of a rectangle. The catch? You cannot use decimals or fractions to compute the answer.

The link above contains just a few puzzles, but I’ve purchased five books from Japan that are chock full of these puzzles. These are brilliant puzzles and especially good for educational use. Once you start solving them, you’ll see how the designer has taken a task that sounds utterly mundane—determining an unknown area or length measurement—and turned it into an artful challenge that combines geometric knowledge, logical thinking, and even some early algebraic reasoning.

Oh, and as a shameless plug, here is an interactive puzzle I made a number of years ago that has Zukei-like elements:

http://www.sineofthetimes.org/a-hidden-polygons-puzzle/

2. #### A. Shores

January 1, 2017 - 7:10 pm -

I tend to fall into my recreational math when I read or see something new and then attempt to apply it to something I’m more familiar with. A recent example is after watching these excellent videos on linear algebra (https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) and especially the one on matrix operations I started to play around with shears and decided to shear the unit circle. The result looked like an ellipse. I then spent some time trying to convince myself on that issue (no spoiler here on whether it was or was not an ellipse). I filled up some spare whiteboards in my classroom after hours and during prep. It was a good time.

3. #### Alex

January 2, 2017 - 2:52 am -

My method for the rhombus was to write down every pair of points as a vector. 10 points means 45 vectors.

If two vectors are equal, you have a parallelogram (so that’s that question done). Rhombuses, squares and rectangles are all special cases, so I guess you’d then need to run extra checks on your pool of parallelograms to see which qualify.

I don’t have the programming skills to code that (yet!), but that’s a challenge I’m hoping to take on this year.

4. #### A Black Belt beyond 50

January 2, 2017 - 6:27 am -

Are you looking for more recreational math to work on? I’m not entirely sure what domain or level is appropriate to you so pardon me if I present a problem which is too simple or inappropriate for you. I’m not a fellow mathematician, I’m just a guy who is tired of thinking about a math problem.

As background, sometimes I go to the gym and benchpress weights. Sometimes I stay home and do pushups.

Here’s the question which probably counts a trigonometry since it involves some triangles and signs. Or cosines. Or tangents (gosh, it’s coming back to me even as a I write this).

If I weigh, say X or 200 pounds. How much weight am I lifting with each pushup? When I do pushups, I point my arms straight down from my shoulders. I am reasonably proportioned with a standard arm, body, and length ratio, I think. My shoulder are below by head which is on the same line as my body. I’m on my toes and my feet are a standard size. If you want data, I’m 5′ 10″, lets say 200 pounds, and have size 10 feet. I don’t know how long my arms are or how much my head weighs or how long my neck is. I suppose we could measure some of these things and with a picture of me, estimate some of the necessary proportions

5. #### Joshua

January 2, 2017 - 6:41 am -

http://www.euclidea.xyz puzzles have been my recreational math this week. These are compass and straight-edge construction puzzles. The system appears to be built on geogebra, possibly inspired by euclidthegame.com.

While some are straightforward, I’ve found many to be very challenging, especially trying to find solutions with the fewest operations. Even my young kids have enjoyed playing with the construction tools, drawing beautiful and/or crazily complex designs. In fact, for this open play, the simpler and more restricted interface is actually easier for them to use than Geogebra itself.

Another thing our family is considering recreational math: making a consistent effort to learn Go. So much to recommend this game and the life-and-death challenges are great logic puzzles. See here for a large collection of “beginner” puzzles: http://senseis.xmp.net/?BeginnerExercises

6. #### Barry

January 2, 2017 - 6:53 am -

The “coffee conundrum” offers an especially simple question for launching a discussion about expected value. The difficulty is in realizing this is a question about a random variable — your “net gain” — and then setting up the variable. The random variable is -1 coffee if you lose and +364 coffees if you win (assuming they don’t give players their card(s) back. If they do, but only after the drawing, then you can bring in time value of money and now you are thinking a lot like an actuary/financial risk manager does!)

This dovetails into more complicated situations that I bet many students might not realize are essentially similar — games of chance at a casino, playing the lottery, betting their teacher that two students in the room have the same birthday, etc. My favorite example here is the “Winfall Lottery” fiasco in MA, when three groups of people noted that they had a positive expected gain when playing on certain days and took advantage, making millions of dollars over many years. The group from MIT went further and used combinatorics to figure out how to fill out lottery tickets to minimize the variance on days they played, effectively removing any chance that they might lose money through bad luck. I saw Jordan Ellenberg give a wonderfully entertaining lecture about this — this appears to be the same lecture I saw: https://www.youtube.com/watch?v=2zLoxPEFwec

• #### Barry

January 2, 2017 - 7:08 am -

Oh also, if you call “p” the probability of winning the coffee game, then your expected net gain is a function of p, say E(p). Your observations about there being 1 card and 1000000 cards amount to saying that E(1/2) > 0 and E(1/1000001) =2, so as you say, the Intermediate Value Theorem does not apply. But there IS a continuous function of p, with domain 0 < p < 1, which agrees with your expected value function, namely E(p) = 365p-1. You can apply the Intermediate Value Theorem to this function to obtain a p where your net gain is 0. Then since the function is increasing, you obtain the result you wanted without any loss of rigor. Incidentally, your wording indicates that I should bet if there are fewer than 365 cards in the bowl before I put your own in. Adding my own card changes the probabilities, so with that wording, we should use 364 as the cutoff, no?

7. #### Rob

January 2, 2017 - 7:04 am -

Minor complaint: I read your coffee work to see if my intuition was right. (It was!) But I got confused/scared for a minute when I saw “n=365!” And thought, especially since this was a probability question, that there was some factorial I missed.

How do we encourage our kids to express their excitement without banning one of the most concise written expressions of it? Maybe all them to do something like “n=365. Wow!”

8. #### Paul Jorgens

January 2, 2017 - 8:42 am -

Dan motivated me to write up a problem we ran into at a New Year’s Eve party. My mom was sharing her old 78s with friends and family. There was some disappointment when an old record with Bing Crosby and Louis Armstrong had a piece broken. I was quietly delighted. We wondered how much of the song was left. It became a contest. We made our guesses. Recent college graduates were measuring with a tape measure. We played the music. The best part – I was wrong. I love it when I am wrong and math helps me see the error in my thinking.

I wrote up the problem here. http://www.101qs.com/3899

If any follow the link and take a look, I would encourage you to do the problem first before watching Act3. Don’t miss the opportunity to make a mistake. Dan- thanks for the motivation to write it up.

9. #### Dan Meyer

January 2, 2017 - 3:02 pm -

These are all great. Many, many thanks. This should keep me stocked for recreational mathematics well into the next month.

Just a quick note also to express my thanks to Matthew Fahrenbacher, Jed, and Dan Anderson for devising Zukei puzzle solvers. They’re all rather different, each with its own set of strengths and weaknesses.

10. #### Melvin Peralta

January 3, 2017 - 8:41 am -

Late to the party! I’ve been messing around with continued fractions, which are totally new to me and are easy to play with. There are a couple of great Scientific American blog posts about them here: https://blogs.scientificamerican.com/roots-of-unity/what-8217-s-so-great-about-continued-fractions/ and here: https://blogs.scientificamerican.com/roots-of-unity/don-8217-t-recite-digits-to-celebrate-pi-recite-its-continued-fraction-instead/

I’ve also been obsessed with creating visualizations of concepts I’ve seen before but never really understood, like here: https://www.desmos.com/calculator/sfksyqhrxa. I find that these visualizations help me understand and remember a concept so much better than traditional algorithms and proofs ever do.

After doing a lot of new math over the break, I seem to have more empathy for my students as they struggle with material which is new to them but not new to me. So far today, the pace of my classroom has been much slower but at the same time much more productive.

11. #### Pavlo Fesenko

January 9, 2017 - 2:01 am -