We’ve looked carefully at the Elchanan Mossel’s dice probability problem a few times – including last week. The problem goes like this:
If you have a fair 6-sided die with sides marked 1, 2, 3, 4, 5, and 6, how many rolls on average will it take to roll a 6 if any sequence of rolls containing an odd number prior to seeing a 6 doesn’t count. So, 2, 4, 4, 6 would count, for example, and 2, 4, 5, 6 would not count.
I was looking for additional ways to discuss the problem in a slightly different way and found a fascinating idea in the comments (attributed to Paul Cuff) on Gil Kalai’s blog. We talked through that idea today and then looked at the problem on a 120-sided die.
Here’s now I introduced Paul Cuff’s idea:
Now that we had a series to sum, we had to talk about how to sum it! My younger son had a pretty clever idea and was able to find the sum with a little help. My older son found the sum using an idea from Calculus:
With the series summed, we could now calculate the answer to the original problem with a 6-sided die. We also looked at the same problem with a 120-sided die.
For the last part of our project we went to Mathematica to check the answer for the 120-sided die. There was one mysterious result in the histogram, but I think the simulation showed that our answer in the last video was correct.
Also, whoops . . . , sorry for forgetting to fix the focus on this one.
I had the boys watch the full video and come up with two things that they thought were interesting. Here they explained their choices and gave a few thoughts about the video:
My younger thought the approximation of a circle by 1×1 boxes was interesting. Here we talked about that idea and sort of hand waved why the approximation gets good:
My older son thought that the concepts of the “good” and “bad” numbers was interesting. I let my younger son do a lot of the talking when we were talking about sums of squares. It was also interesting talk about why the proof that none of the integers of the form 4n + 3 can be written as the sum of two squares was easy, but the proof that all integers of the form 4n + 1 can be is hard.
I hope to return to the more complex questions the boys found interesting in a different project. Maybe next week!
A lot of people have been talking about recent observations of the star Betelgeuse this week. Here’s one great thread I happened to see:
Regarding #Betelgeuse's "historic" dimming, here's V-band & photovis magnitude estimates from @AAVSO database over past century. *You* try maintaining constant luminosity w/~20 solar masses spread out within the size of Jupiter's orbit, w/convection & nucleosynthesis is 3D! pic.twitter.com/e2kojEZXAC
After seeing this thread I thought it would be fun to share some of the ideas about the recent observations of Betelgeuse with the boys. Although I’m way out of my league here, there were some great resources I found that I thought would help the boys understand what was going on. Two of those resources were:
I started today’s project by showing the boys the article on Astroblog and then the graph in Eric Mamajek’s tweet:
Next we looked at a graph from the Light Curve generator showing how the brightness of Betelgeuse has varied going back about 6 months. Sorry for the glare on the computer screen 😦
The boys had different ideas about how to interpret the data – which was fun to hear:
Next I had each on my son’s create a new graph. My younger son went first and he wanted to look at the observations from a single astronomer. We did this by using the green dots since there were only to people who collected that data. The astronomer whose data we looked at was Wolfgang Volmann:
My older son went second – he wanted to look at the observations of Betelgeuse going back a long time. We were able to zoom in on a time period in the 70s and 80s in which many observations showed that Betelgeuse was pretty dim.
This was a really fun project to work through with the kids. It really highlights the difficulty of collecting data in astronomy, and in the real world in general! It was fun to hear their ideas about how to think through the
For today’s project we looked at two problems inspired by these two projects. The problems seem pretty similar:
(1) If you have a fair 6-sided die with sides marked 2, 2, 4, 4, 6, and 6, how many rolls on average will it take for you to roll a 6.
(2) If you have a fair 6-sided die with sides marked 1, 2, 3, 4, 5, and 6, how many rolls on average will it take to roll a 6 if any sequence of rolls containing an odd number prior to seeing a 6 doesn’t count. So, 2, 4, 4, 6 would count, for example, and 2, 4, 5, 6 would not count.
I started the project today looking at the first problem, which is inspired by the Ox Blocks project:
Now we moved to the 2nd problem. To introduce the problem I had the boys play the game a few times and we found that lots of sequences of rolls were thrown out:
To help the boys understand this second game a bit more I moved to a slightly different question -> for valid sequences of rolls in the 2nd game, how often do you see a 6 on the first roll.
This question was slightly difficult for the kids to understand, but we made pretty good progress:
Finally, we went to the computer to run a simulation for the 2nd game. This video runs a little long as I asked my younger son to explain the program. But once we get through the explanation we see that their guesses for the expected number of rolls and also the percentage of 6’s on the first roll were roughly right!
There was yet another absolutely fantastic episode of the Mathematical Objects podcast published last week. I just can’t say enough about the great work that Katie Steckles and Peter Rowlett are doing with this blog:
After listening to the post yesterday I did a little google searching for the game online and thought a project based on the Ox Blocks would be really fun.
First we talked about the game:
After the short introduction the boys played one round of Ox Blocks. Since we didn’t have the specific blocks the game uses, we used a 12 sided die and looked at the rolls mod 3. I think the video will show you (i) how easy the game is to play, and (ii) what a fantastic game it is to play with kids:
Next we rounded up to discuss some of the surprises and strategy from the game:
Finally, we went to Matheamatica to explore some of the statistics from the podcast. Unfortunately iMovie didn’t like something about how the video got recorded, and an hour of trying to fix the problem produced essentially no results.
What we were exploring was the situation that Peter Rowlett described in the podcast. He rolled the cubes 501 times and found roughly 180 “take away” rolls, 160 X’s and 160 O’s. So, how likely was it that he’d have seen one of the rolls come up 180 times in 500 rolls. We found that one of the sides coming up 180 times would happen about 30% of the time if the cubes were fair:
My old son wondered what would have happened had Peter found one of the sides coming up 200 times. Turned out that was far less likely – about 0.25%
I’m sorry that the video for this part of the project broke – it was a fun discussion.
A couple of years ago Jim Propp suggested a neat counting exercise for the boys – counting tilings of 2xN rectangles by 2×1 dominos. We’ve played with this idea twice before, but thought it would be fun to revisit it today.
I saw a really neat problem in Strang’s Linear Algebra book earlier this week:
Tonight I had my son work through them on camera. These problems bring together ideas not just from linear algebra, but also from a high school algebra class.
Here’s his work on the first problem:
Here’s his work on the 2nd problem – this one is a fun surprise. The numbers don’t get big at all. In fact, this matrix has powers that are the identity matrix:
Here’s the third problem – a lot of the work in this problem is him remembering how to multiply complex numbers. I really like this problem because it brings in quite a bit of math from outside of linear algebra:
Here’s the last problem which is another fun surprise. We change one entry of the matrix in the previous problem by a tiny amount, and the powers of the matrix behave in a completely different way:
Yesterday we did a fun project exploring how long it takes, on average, to create certain words like COVFEFE or ABRACADABRA when selecting letters at random. We also simplified the problem a bit by looking at sequences of H’s and T’s for coin flips. That project is here:
Today’s project was writing a computer program to simulate flipping a coin until we saw HHHH. In yesterday’s project we found that it would take 30 coin flips on average. We started today’s project by talking about how to write a program to do this simulation. Following this discussion the boys wrote their program off camera:
When the boys finished their program we talked through it and looked at the shape of the distribution of the number of flips it took to get to HHHH. They were pretty surprised by this shape:
To wrap up the project we spent 5 min talking about how the program would need to change to look at a general sequence of 4 flips – HTHT, for example. We didn’t actually make the changes, though, as we’d already spent enough time working through the ideas this morning:
This was a fun statistics / programming project that has a pretty surprising result. We’ll definitely have to follow up with the program for a generic sequence of flips soon!