We started off this week going back to some of the Eulerian circuits we worked on last week and tried out Euler’s Formula on them.
After defining what vertices, edges, and regions are, as defined in graph theory, we counted up those elements in each of the shapes and found that the sum of the number of vertices and edges minus the number of regions always equaled two.
We found, though, that this only worked if we followed a couple of strict rules.
This star, for example, doesn’t work:
But if we make it planar (edges can’t cross without vertices), it works out:
Similarly, every edge needed at least one vertex:
And only one figure at a time. Not this:
But this is okay:
Most of these exercises were taken from Math Lab for Kids by Rebecca Rapoport and were designed by Fan Chung.
I mentioned this is all a branch of mathematics called topology. Most people, when they think of topology, think of Mobius strips. Much to my excitement, most of our Mad Math-ers hadn’t heard of them!
We first saw this clip of Avengers: Endgame that featured a Mobius strip:
And then we made a regular loop from a strip of paper. Then a loop with a half-twist (Mobius strip). Then a loop with a full twist. We saw the former and latter loops had two sides. The Mobius strip only had one. (!!!)
We then cut each of the loops in half. The first loop divided into two separate loops. The loop with the full twist separated into two intertwined loops (of the same size). The Mobius strip just divided into one loop of double the size.
We then watched the following video of Vi Hart and emulated what she was doing:
She mentions a lot of interesting things, but I was able to talk a little bit more about rotational symmetries, frieze patterns, and glide symmetries so that we could follow what she was doing.
I strongly encourage anyone to experiment with Mobius strips, either with paper strips or with fruit leathers! They have such counter-intuitive properties that don’t really make sense until you think of them mathematically.
A little bit of extra credit. Before Avengers: Endgame was released in theaters, this gentleman had a theory about how the movie could utilize time travel in interesting ways using Mobius strips. If the dimension of time was a Mobius strip, you could resolve time-traveling paradoxes with something like quantum superpositioning! This ultimately wasn’t the direction the movie went in, but mind-blowing all the same:
This week we picked up from last week’s last activity. If you recall, we left with some diagrams that we tried to trace using a continuous path without backtracking.
We tried a number more examples of this activity:
Remember the rules:
Try to get to every point (also called a node)
Use one continuous path
Do not retrace any edge already traversed
Can all of them be done? Does it matter which point you start from? How, in some ways, are some of these shapes similar to others?
After some practice with these shapes, I played this video from Numberphile about Euler and the bridges of Konigsberg:
As it turns out, what we’ve been doing is a fundamental part of graph theory called Eulerian circuits. The video gave us some key insights Euler came to about these circuits. They should help with the challenge puzzle I left at the end:
This puzzle actually has a neat story behind it. According to Daniel Finkel:
A less obvious version comes from the Kuba children in the Congo, who took this puzzle much, much farther. The story, related by anthropologist Emil Torday, goes like this: the children were drawing complicated networks in the sand. When they saw Torday, they challenged him to draw the pictures in the sand without picking up his finger or drawing the same stretch more than once. “The children were drawing,” said Torday, “and I was at once asked to perform certain impossible tasks; great was their joy when the white man failed to accomplish them.”
Goes to show you that mathematical insight comes from all sorts of places!
We have a new name! Mina suggested Multiplication Madness — I think she was a little occupied with memorizing her multiplication tables last week — and we agreed that Math Madness was a little more generalizable and retained the nice alliteration.
I recently heard a Planet Money podcast episode that reminded me of last week’s session, and so I decided we should discuss it in today’s session.
I first presented this graph and talked about what information it tells us:
What does the horizontal axis explain? What are the numbers on the vertical axis? 3PA means “3-Point Attempts.”
1979 is when the NBA implemented the 3-point shot rule. What do you think accounts for the steady and steep rise of 3-point shot attempts? Why did it start so small, and why is it so large now? Do you think it will continue to get larger? Forever?
What new information does this graph tell us? What do you think are “mid-range” shots? Why do you think they went down over time while 3-point attempts rose? Do mid-range and 3-point attempts account for all the shot attempts? Why is the vertical axis in decimals less than 1?
“FGA” stands for Field Goal Attempts. Field goals are any shots but free throws. (Free throws, it turns out, have their own interesting math.)
Here’s a variation on a game we played last week. Again, vowels (including Y) get 5 points, and consonants get 1 point. In this game, everyone takes turns and offers a word. If that word does not have a higher point value than the previous word, you get eliminated.
What words would survive in later rounds in this game? In other words, what kind of words have a really high point value? Is it just words with a lot of letters? Can you think of a word that has a lot of letters that actually has a pretty underwhelming point value?
We talked about how, just like vowels in my made-up game, 3-point shots have an artificially high value.
By the way — in the first graph, why do you think there is a bump in the middle of it?
It turns out, in 1994 the NBA shortened the distance of the 3-point line from about 24 feet to about 22 feet. It then moved it back to the original distance after a few years.
Why would that make such a significant difference in 3-point-shot attempts? What does it tell us about the rise of those attempts over the years?
What does this graph tell us? Shooting percentage is the percentage of shots you successfully make. Why do you think this percentage is so high at first? After a precipitous drop-off, it seems to level out — what does that tell us? At the end, it drops off again — at what distance does that happen?
Remember, the 3-point line is about 24 feet away from the basket.
How is this graph different from the previous one? Notice the vertical axis — it’s not in percentages here; it’s in points. The plot looks pretty much identical to the previous graph up until around the 3-point line. Why is that? What does that tell us about the value of the 3-point shot?
Here’s another visualization of data. What do the colors mean? What would happen if we took out the colors?
What information is removed when we take out the colors? Is any information perhaps clearer to see without the colors? Which do you prefer — why?
Each dot represents a shot taken in the 2014-15 season. Where are the most shots taken? Where are less shots taken? Notice there is a white band around the 3-point line (and behind the basket). Why is that?
Notice how this new visualization tells us similar information as a previous graph but actually adds another dimension of insight. What areas of the court seem like particularly good opportunities to take shots? Lots and lots of teams in the NBA are specifically training players hard on the corner 3’s.
Do you notice any discrepancies in the data between the line plot and court map? Why would that be?
Does this new visualization tell us anything more? Does it tell previous insights better? What is it about how the court map is designed that enhances its clarity?
We ended this discussion with a consideration of the implications of all of this. Does the value of the 3-point-shot improve basketball? Is it now to the point that it’s detracting from the sport? Some people think it is. What are some ways that the NBA can fix this problem (if it is one)?
One of our attendees shared how they played a version of basketball that allowed another team to play very physically and dominate games that way. That was a good example of how structural allowances incentivizes certain behaviors. I had to restrain myself from talking on and on about the Detroit Pistons in the 80s.
We then transitioned into a more interactive activity. Here’s a simplified diagram of a basketball court:
Can you trace the entire diagram without backtracking or lifting your finger? What’s the minimum number of times you have to lift your finger?
What about this tennis court diagram? Is it harder or easier to do? Do you need lift your finger more or less? Does it matter where you start? Are there any strategies you can develop to minimize getting stuck?
I mentioned the artist Tyler Foust, who specializes in making intricate drawings using only one continuous line. Here’s a video of him at work that I wasn’t able to share because of technical difficulties:
Finally, we ended with this basketball poem by Edward Hirsch. Notice that it’s all one sentence, which gives its this breathless momentum.
Some optional follow-up activities:
What sports are you into? Have you thought about that sport’s rules? What are the weirdest ones? Why do they exist? How do they shape players’ behavior?
Look for (or make up) more drawings that you can trace. Which ones are impossible to do in a continuous line? Why?
Challenge yourself to draw something using only one continuous line. Can you make a happy face? What about an emoji?
Here are a list of links and resources I used for the session:
At our first Math Club session we started off with an activity around our names. Everyone shared the name they wanted to be referred to, spelled as they wanted, and then transformed that name into a number using the following rules: vowels get 5 points, consonants get 1 point, and add up the sum.
We then tried to guess which name corresponded to which number. I suggested that a robot would probably solve this puzzle by brute force, but a lazy mathematician would first want to use logic to make some educated guesses.
How does knowing whether the number is odd or even help?
Would this challenge be more or less difficult if, instead of adding the points, you multiplied them?
We then did a variation of this challenge, where each letter of the alphabet gets a unique number:
Do our previous strategies translate to this variation?
The kids noticed that certain repetitions and patterns in the words made it easier to guess their corresponding numbers.
Tom Kim 