I thought that talking through these problems would make a nice project for the boys today, so we started in on Alex’s problem. The nice thing right from the start is that the boys had different guesses at what the expected value of one die would be when the combined roll was 8:
Now that we had a good discussion of the case where the sum was 8, we looked at a few other cases to get a sense of whether or not the intuition we developed from that discussion was correct:
Next I introduced the problem on Gil Kalai’s blog – again the boys had different guesses for the answer:
I had the boys write computer programs off screen to see if we could find the answer to the problem on Kalai’s blog via simulation. The interesting thing was that the boys approached the problem in two different ways.
First, my younger son started looking at dice roll sequences and he stopped when he found a 6 and always started over when he saw an odd number. He found the expected length of the sequence of rolls was roughly 1.5:
My older son looked at dice roll sequences and he stopped when he found a 6 but instead of starting over when he found an odd number, he just ignored the odd number. He found the expected length of the sequences looking at them this way was 3:
This turned out to be a great project. I’m glad that the boys had different ideas that we got to talk through. These conditional expectation puzzles can be tricky and subtle, but they are always fun!
The game is easy to explain. You roll a fair 6-side die N times, where N is any number you pick. You also choose an amount of money to bet – say X. If you never roll a 6 in your N rolls, you win 2^N times your money back. If any of your rolls are a 6, you get $0 back. Salmon’s questions are -> (1) How much money would you bet if you could play this game once, and (2) how many rolls would you select?
I thought this game would be fun to talk through with my older son. Here I explain the game and he talks about a few of the ideas he thinks will be important for answering Salmon’s questions. He has some interesting ideas about “high risk” and “low risk” strategies. We also talk through a few simple cases:
In the last video my son was calculating the probability of winning the game in N rolls by calculating the probability of not losing. That’s, unfortunately, a fairly complicated way to approach the problem so I wanted to talk a little more so he could see that a direct calculation of the probability of winning wasn’t actually too hard. We talked through that calculation here. We also find that if you roll 4 times you have roughly a 50/50 chance of winning the game.
Before we played the game he wanted to calculate the expected value for your winning in this game. Here we do that calculation and find the surprising answer. We then play the game. He decided to bet $100 and roll three times, and . . .
This was a fun problem to talk through with my son, and I’m excited to talk through it with my younger son tomorrow to see if he reaches a different conclusion. It had never occurred to me to talk through this or any version of the St. Petersburg Paradox with the boys before, so thanks to Felix Salmon for sharing this problem.
20 or so projects is only scratching the surface, though, since she comes out with fantastic geometry puzzles all the time! The one from yesterday is fantastic and I thought it would be great for another project for my son:
His solution to the problem was computational. He explains the main ideas here without going into all of the computational details:
In all of our projects we return to the problem’s twitter thread and my son picks out a solution that he thinks is interesting. Today he picked the solution from @lucythepoet
Consider red-yellow as a unit, we have 3 units on top and 1 unit on the bottom. Everything's similar, so scaling the length of a unit by 3 also scales the height by 3. So green:blue is a 1:3 ratio Similarly, each red-yellow block is also a 1:3 ratio
Here’s his explanation of this solution and a bit about why he liked it:
I think – and have thought for a long time! – that Catriona’s puzzles are great to use with kids. The process of attempting to solve the puzzle (sometimes getting it, sometimes not) and then going to the twitter thread to see all of the neat solutions has been a great way for my younger son to review geometry.
Yesterday we looked at a very simple model of how a virus spreads through a network – the assumption was that everyone infects everyone they are connected to. In that (obviously simplified model) the structure of the network affects the structure of the spread:
In Christopher Wolfram’s model, we use Mathematica to make a network and then study how the virus spread through the network by varying the average number of connections per day that people in the network have. The surprise (that we discuss mostly in the last video) is here the different network structures seem to behave in nearly identical ways. So the result today is very different than yesterday’s result.
I introduced today’s idea by asking the boys to think about how to build a more realistic model of how a virus spreads. The first network we looked at was a simple 2d grid:
Now we looked at a 3d grid:
Next up was a Delaunay triangulation:
Now we looked at a pure random graph network:
For the last two we looked at two graph networks that look a lot like connections in the “real world.” First up was a Watts-Strogatz graph:
Finally we looked at a Barabasi Albert Graph. This graph looks like the pure random graph we looked at, but you can see in the video that the degree distribution is really different. At the end of this video the boys talk about some of the surprises in this project and what they learned.
I think Christopher Wolfram’s program is one of the best I’ve seen for helping students understand some of the difficulties in modeling how a virus spreads. It seems like a big surprise that all of these networks seem to behave the same way, but understanding why it maybe isn’t a huge surprise helps kids see some of the key ideas in these simple models.
This week I watch an interesting live coding video from Stephen Wolfram:
Right at the beginning of this video Wolfram shows how to use some simple Mathematica commands to make a simple model of how a virus spreads through a network. I thought it would be fun to share this idea with the boys for several common networks.
I introduced the idea on a 2d grid:
Then we moved to a 3d grid:
Then we moved to a type of network called a Delaunay triangulation:
Now we moved away from these relatively simple graph networks and looked at a completely random one:
With these examples out of the way, we moved to two types of networks that more more commonly used to model a network of human interactions. The first was a Watts-Strogatz network:
Finally we looked at a Barabasi-Albert graph:
This was a really fun project and I was really excited to hear how the boys thought about the different types of networks. The math to properly describe what’s going on in these networks is over my head but I am really happy that Mathematica makes it so easy to explore.
Finally, the idea for looking at these 6 different graphs comes from Christopher Wolfram’s fantastic agent based modeling example. In that program he dives into these different networks much more deeply than we do here – this program is definitely worth checking out if you’ve not see it already:
Since the boys have been learning more about programming in Mathematica this summer, I thought it would be fun to review Wolfram’s program again. My older son spent the week looking through the notebook. Tonight we talked about some of the things he thought were interesting.
The first thing that caught his eye was how the average number of interactions per time step affects the spread:
The second thing that caught his attention was how Wolfram was able to model how the virus spread across different kinds of graph networks:
Finally, he thought the “network of networks” model was really interesting and Wolfram’s graph of how the number of connections between the individual networks changed how the infection spread, in particular, caught his eye.
I think that Wolfram’s work here is one of the best examples I’ve seen that makes virus modeling accessible to students. I also really love that there are many different areas to explore further in Wolfram’s work. Definitely interesting for my son to play around with this program a bit more.
Today I wanted to extend that conversation to an example I first learned from a talk by Ole Peters.
I introduced the additive version of the game first and asked the boys what they thought would happen:
After the boys thought about the coin flipping game where you bet $100 each time we moved to the same game where you bet your entire net worth each time/. This one is a little harder to think through, but they boys has some good intuition:
Now we moved to looking at the two games in Mathematica. Here’s how the additive game plays out:
Finally the boys got to see the surprise in the multiplicative game – average wealth increases, but eventually any individual players ends up losing all of their money:
This game is really fun to think through and also a nice example to share with kids to illustrate additive and multiplicative games. Happy that yesterday’s detour into multiplicative processes led us to this conversation today!
I saw an interesting tweet earlier in the week from Atrin Assa:
@nntaleb demonstrating central limit theory by adding together increasing numbers of individual uniform distributions. By the time you add 3 uniform distributions, the resulting distribution looks like a normal distribution. Beautiful. #RWRIpic.twitter.com/7guT0c4dig
I thought the idea would make a neat project for the boys today, and it turned out to be even more interesting that I expected.
I started by just having the boys look at draws from a uniform distribution. You never really get a chance to go back and see ideas like this for the first time, so I’m always really interested to hear how kids describe what they are seeing:
Next we looked at the average of two draws from a uniform distribution. The boys had different thoughts about what this would look like before we saw the outcome – that was fun to hear. My younger son had some interesting intuition based on a dice game he’d played previously:
At the end of the last video my older son wanted to check what would happen if we looked at the geometric mean rather than the arithmetic mean. This idea wasn’t what I was planning to study, but it seemed like a great idea so we tried it out. They both had interesting guesses at what this new distribution would look like:
We wrapped up today looking at the arithmetic mean and geometric mean of three draws. With the arithmetic mean they started to see the normal distribution appearing. With the geometric mean they didn’t recognize the shape – my guess is that they’d never seen a distribution like this before:
This was a really fun project. Nassim’s intuition about how to explain important ideas from probably and statistics is incredible. I love working through his ideas with my kids.
Yesterday I learned about a fantastic video that John Urschel made:
Right now is obviously a tough time for education. So when @NMSI asked me to make a free math lesson, I was more than happy to. They let me pick whatever hs topic I wanted. I chose rational and irrational numbers. Check out the video here:https://t.co/vInfaUjK4Z
This morning I asked my younger son (going into 9th grade) to watch the video so we could talk about it. Here are his initial thoughts and what he thought was interesting:
Now we talked through three of the ideas he thought were interesting. The first was how to find the rational representation of a number like 0.64646464…..
Next he talked about the proof that is irrational:
Finally, we talked about a really neat proof in Urschel’s video -> why log base 2 of 3 is irrational:
I love Urschel’s video and think it is an absolutely terrific one to share with kids. It is a great way for kids to see some advanced mathematical proof ideas, but also a great way to review some important ideas in math. We had a really fun morning going through it.
Nassim’s problem really isn’t accessible to kids, but a slight variant is -> how many sequences of 250 coin flips are there where no run of heads or tails is longer than 2 flips?
I decided to go through that problem with the boys this morning. It was just at the right level to really challenge them, but still fit inside of a 30 min project.
We started by looking at Xi’s problem and they both had pretty good intuition for which sequence was which:
After the short introduction we started trying to figure out how to tackle the problem about sequences where the longest run was at most 2. After thinking of a few other ideas first, they decided to take a look at some shorter sequences to see if that would help us get some intuition for how many had runs that were no longer than 2:
Now we took a look at the sequences of length 4. Luckily there are only 16 different coin flip sequences of length four, so we could write them all out. The boys found that there were 10 sequences with runs no longer than 2. That led to an idea about how many would work in general:
Now we had a conjecture – there would be 16 sequences of length 5 that had runs no longer than 2 – so we tried to count those sequences directly to see if the conjecture was right:
Finally, we sketched a general proof of the conjecture (I’m intentionally being vague on what it is to not give it away). This part was also a little difficult for the boys, but they eventually saw the right pattern and that pattern led to the general proof:
This problem made for a really fun project this morning and Nassim’s problem led to some great twitter discussions that lasted all week. I was happy to be able to find a piece of Nassim’s problem that the boys could tackle.