Tag games

Exploring Elchanan Mossel’s fantastic probability problem with kids

Saw a really great problem via a Lior Patcher tweet:

Here’s the problem:

You throw a dice until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that
all throws gave even numbers.

Here are direct links to Kalai’s two blog posts on the problem:

Gil Kalai’s “TYI 30: Expected number of dice rolls

Gil Kalai’s follow up post: Elchanan Mossel’s Amazing Dice Paradox (Your Answers to TYI 30)

It is fun to click through to the first Kalai blog post linked above to cast your vote for the answer if you haven’t seen the problem before.

We actually started the project today by doing that:

Next we rolled some 6-sided dice to see how this game worked. I note seeing the video that a few of the rolls went off camera, sorry about that 🙂

At the end we discussed what we saw and why what we found was a little surprising.

The next part of the project was having the boys play the game off camera until they found 5 rolls meeting the criteria.

After this exercise the boys started to gain some confidence that the answer to the problem was 3/2.

Now I walked them through what I think is the easiest solution to understand. It comes from a comment on the first Gil Kalai’s blog post linked above:

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Listening to this discussion now, I wish I would have done a better job explaining this particular solution. Still, I hope the discussion is instructive.

Finally, we went to Mathematica to evaluation the sum from the last video and then to explore the problem via a short program I wrote.

At the end of this video the boys some up their thoughts on the problem.

I love this problem. It isn’t that often I run across a clever problem that is interesting for both professional mathematicians and kids. Those problems are absolute

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Intro “machine learning” for kids via Martin Gardner’s article on hexapawn

Last month I had the nice surprise of finding Martin Gardner’s book The Colossal Book of Mathematics at the Omaha Public Library’s book sale:

I’ve been flipping through the book and thinking about how to share some of the ideas with the boys.  Chapter 35 – “A Matchbox Game-Learning Machine” really caught my attention.  In particular, Gardner’s discussion of the game “hexapawn” inspired me to try this introductory “machine learning” idea with kids.

I had the boys read the (approximately) 2 pages on the game and the approach and then we talked through the game to make sure they understood it:

Next we started playing. We were very lucky to have a coffee table that allowed us to easily show the 24 cases and their snap cubes. This video shows the first two times through the game. I hope that it shows that playing through the game is something that is accessible to kids:

The next part shows 3 more turns of the game. My main reason for showing these three turns is so you can see some of the parts that kids find challenging. I think these parts are a big part of what makes sharing Gardner’s idea with kids so fun. The pattern matching and the general walk through the game keeps their attention while they learning about machine learning.

Next we played for a while with the camera off. After a while the kids (and the computer) learned something:

Next we played a bit more with the camera off and the before long the “computer” learned to win the game every time. Amazing!

In the last 3 min of this video the boys talk about some of the things that they learned in this project.

This is one of the most fascinating projects that we’ve ever done. It does require a bit of set up and probably a bit more careful supervision than usual to make sure that the kids don’t go down the wrong path, but wow is there a lot to learn here. I think that opening the door for kids to see how computers / machines might “learn” is an amazingly valuable lesson.

Talking a bit more about my son’s probability problem

Yesterday we did a fun project on a probability problem / game my son was working on. The game involves rolling three 10-sided dice and adding up the numbers. Repeat the process until you’ve seen one of the sums 40 times. Yesterday 15 was the winner:

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Here’s yesterday’s project:

A probability and stats problem with dice my younger son was working on today

I wrote a short program on Mathematica to play my son’s game 1,000,000 times. I was interested to see how each of the boys would interpret the results.

Here’s what my older son had to say:

Here’s what my younger son had to say:

It is interesting to hear what kids have to say about the various probabilities and distributions. The results of the 1,000,000 simulations are probably pretty surprising. This problem that my son made up is actually a pretty fun problem to explore with kids.

A probability and stats problem with dice my younger son was working on today

When I got up this morning my younger son was playing some sort of dice game in the kitchen. An hour later he was still rolling dice so I finally asked him what he was doing:

It turns out what he was actually trying to was find the first sum that would appear 40 times, but I only understood that later.

This seemed like an easy activity to turn into a project, so we got started by having him explain what he was doing:

Next we turned to Mathematica to play around a little bit with the problem. I had to explain some terms first (and sorry I had part of the screen out of view for a bit). After explaining those terms we looked at the distribution of the sums:

Finally we wrapped up by taking a very deep dive into the distribution of the sum of three 10 sided dice. The kids were able to understand the probability of getting a 3 or a 30, and then we talked about a few of the other probabilities that Mathematica was showing us.

Later in the morning my son finished his game. 15 was the first roll to appear 40 times.

It was really fun to base a project on a math problem that my son came up with on his own.

Playing with Three Sticks

I saw this tweet from Justin Aion at the end of July and immediately ordered the game:

When I returned from a trip to Scotland with some college friends the game was on the dining room table – yes!! Today we played.

In this blog post I’ll show how the game ships and two rounds of play (and we might not be playing exactly right) to show how fun and accessible this game is for kids.

First, the unboxing. The game comes out of the box nearly ready to play.

Here’s our first round of game play. I think we misunderstood one of the rules here, but you’ll still see that the game is pretty easy to play:

Here’s the 2nd round of play. I think we understood the rules better this time, which is good. You’ll also see how this game gets kids talking about both numbers and geometry:

Finally, here’s what the boys thought about the game:

I’m really happy that I saw Justin Aion’s tweet and now have this game in our collection. It is a great game for kids!

The coupon collection problem with kids

Yesterday my younger son was playing a dice game (explained in the first video) that reminded me a bit of the coupon collection problem. I thought it would be fun to try out that problem with the boys this morning. We were a little low energy, but I think it was still a good project. I’ll have to figure out how to revisit it to make sure the points stuck.

Here’s the introduction, including the game my son was playing:

Next we worked through one case of the problem – rolling dice trying to collect 6 “coupons”. My older son thought it would take 15 rolls and my younger son thought it was take 20.

Now I tried to help the kids dive into the math. We ended up going down a path that was much more complicated than I intended. I’m not sure why I made the choice that I did here, but . . . it happens sometimes 🙂

So, at the end of the last video we were caught in a seemingly complicated infinite series. I tried to explain why the expression we had on the board had to be equal to one. Then I tried to explain why the expected number of rolls had to be greater than one. The explanation here is a disaster, though.

Now that things had gone totally off the rails, I tried to pull it back. Luckily things did go better, and it was easier for the boys to see the expected number of rolls when there were fewer open slots.

Finally I wanted to show the kids how the ideas we talked about here would apply to a more difficult problem – say 100 coupons. We got off on the wrong foot here, but we eventually saw how the ideas we’d talked about previously applied.

Despite the low energy and going doing a path that was a bit too complicated, I think this is a fun problem for kids to study. It looks very difficult initially, but through a bit of calculation (and maybe a bit of hand waving) we can break it down into some smaller problems that we are able to solve. Putting the solutions of those smaller problems together, we can show that the solution to the original coupon collection problem isn’t too hard to understand.

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Jim Propp’s “Swine in a line” game part 2

Last week I saw a really fun new question from Jim Propp:

Here’s the first project that we did on the game:

Jim Propp’s Swine in a Line game

Today we returned to the game to see if we could make any more progress understanding how it worked.

First we reviewed the rules and decided on an initial approach to studying the game for today:

Their first idea was to try to keep the two ends open since they knew the result when you reached the position with only 1 and 9 open.

Now we tried to study a bit more. The kids were having trouble seeing a path forward, so I just let them play.

At the end of the last video we were studying a position with all of the pens filled except for 7 and 9. In this video we searched for a winning move.

Finally, we took one more crack at solving the game. They boys got very close to the main idea, about an inch away(!), but didn’t quite get over the line.

The boys were really interested in the game and we kept talking for about 30 min after the end of the project. During that talk they did uncover the main idea. After that we played several games where they followed the strategy and they were able win against me every time following that strategy. It was a fun way to end the morning.

Jim Propp’s “swine in a line game”

Saw this great new video from Jim Propp yesterday:

This morning I had the boys watch the video and then we spent maybe 15 min talking through the game and seeing what we could learn.

First I asked them what they thought after seeing the video:

Now we played the game and the boys made a couple of initial discoveries. You can see quickly why this is a fun game for kids to play around with:

Next we played the game one more time. We aren’t trying to solve the game in this project, just to try to learn a few things about it.

Finally, we wrapped up by talking about a few of the things they learned playing the game. This part didn’t quite go how I wanted, but it was still interesting to hear what they had to say.

I’m excited to play around with this game a bit more later in the week. It’ll be interesting to see if the boys can continue to make project towards the solution.

Playing with 4d Toys

Quick post tonight because I’m running out to dinner . . . .

I learned about the new iPad app “4D Toys” last week:

Here’s a link to their site:

The 4D Toys site

It is a nice compliment to some of the 4th dimensional projects we’ve been doing. Here’s what my younger son thought after playing around with it for a bit:

Here’s what my older son thought after playing with it for 10 min:

Excited to use this app a bit more!

Studying shuffling and Shannon entropy part 2

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We did a fun project about Shannon Entropy and Shuffling yesterday:

Chard Shuffling and Shannon Entropy

That project was based largely on an old Stackexchange post (well, comment) here:

See the first comment on this Stackexchange post

Today I wanted to extend that project a little bit and thought it would be fun to look at a different kind of shuffle to see if there was a difference in entropy.

Here’s the shuffle for today as well as what the boys think will happen with this shuffle:

Next we took some time off camera to enter the card numbers in our spreadsheet. Here’s what we found for the new entropy after one of these new shuffles:

Finally, we took even more time off camera to do 4 more shuffles and write down the order of all the cards. After that we did 5 successive shuffles and wrote down the numbers after the 10th shuffle.

The kids didn’t think the cards were as mixed as they were in yesterday’s project, and here’s what the entropy calculation said:

I really enjoyed these two projects. It was especially fun to see how kids could get their arms around the idea of entropy even though the math itself is pretty advanced.