Playing with Annie Perkins’s counting problems

I saw a neat tweet from Annie Perkins last week:

Today I thought it would be fun to play around with this idea with my younger son. First I introduced the 4-person problem and let him think through it. His thought process is a great example of what a kid learning math can look like:

At the end of the last video he’d determined that there were 3 different arrangements of the 4 people sitting around the table. In this video I asked him to find those arrangements:

Next we moved to the 5 person problem:

Finally, having decided that there were 12 different arrangements with the 5 person problem, I asked him to try to write down all 12. This is a good exercise in using counting techniques to make an organized list:

Definitely a fun problem for students, and also a really nice introduction to counting and symmetry. Thanks to Annie for sharing!

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Bob Lochel’s random “Matt” problem

Saw a fun tweet from Bob Lochel last week:

For today’s Family Math problem I thought I’d talk through the math behind this situation with my younger son.

First I showed him what had happened and asked for a few ideas about how to approach finding the probability that all of the Matt’s would end up in group d. It has been a while since we’ve done a probability and counting project, but the ideas came back to him as we talked:

Next we talked about how to calculation the number of different ways to put the students into the the groups of 5 and 4. Then I asked my son to estimate what the number was without calculating. His estimate was off and we returned to why in part 4 of the project:

Now we calculated the number of ways to arrange the students with all 3 Matt’s landing in group D. Once we had that number we found the probability of the outcome that happened. I also asked my son if he thought the outcome happened by random chance, or was Bob possibly playing a little joke:

Finally, we revisited the calculation from part 2 of the project. It turned out that his estimate actually was nearly right – investigating where it went wrong was a good use of time. Once we had the exact value by hand, we also computed the exact probability for the original problem by hand.

Thanks to Bob Lochel for sharing this fun outcome. If you’d like to see a similar probability / counting problem check out our exploration of the “Snapchat problem” from a few years ago:

Exploring the “Shapchat Problem” with my kids

Using the Pringles Super Bowl commercial for a fun counting project for kids.

During the Super Bowl Pringles ran a commercial that caught the attention of math twitter:

The twitter team from Pringles was even helpful in clarifying what they meant in the commercial:

I thought that showing the kids where the 318,000 number came from would make a fun math project and I also thought it would be fun to see whether or not you could actually taste the difference when stacks of the same chips were arranged differently.

We started with the counting problem:

Now that we knew there were two different ways to count the stacks, we had to figure out which one was right. Could you tell the difference if the same chips were stacked in a different order?

We started with the simple case using a stack of two chips:

Next we went all in and tried to identify a stack of 4 chips. Amazingly, my younger son was able to identify the correct order of the chips!

Definitely a fun counting project (our dog also gave this project an A+). AND after you finish this project you can try another fun math-related Pringles project -> making the Pringle Ringle!

Part 2 of studying Futility Closet’s “Paradox of the Second Ace” with kids

Yesterday we did an introductory project for kids on Futility Closet’s Paradox of the Second Ace:

Here’s that project:

Introducing the boys to Futility Closet’s Paradox of the Second Ace

Today we continued the project and calculated the two probabilities in the “paradox.” These calculations are pretty challenging ones for kids, but even with the counting challenges, this was a really fun project.

I started by reminding them of the problem and getting their thoughts from yesterday:

Now we calculated the probability of having a second ace given that you have at least one ace. It took a while to find the right counting ideas, but once they did the calculation went pretty quickly. The counting technique that we used here was case by case counting:

Next we moved to the 2nd problem -> If you have the Ace of Spades, what is the probability that you have more than one ace? The counting technique that we used here was complimentary counting:

Finally, I asked the boys to reflect on the problem – was it still a “paradox” in their minds or did it make a bit more sense now that we worked through it?

I really loved talking through this problem with the boys – thanks to John Cook for sharing it and to Futility Closet for writing about it originally!

Introducing the boys to Futility Closet’s “Paradox of the Second Ace”

I saw this tweet from John Cook’s Probability Fact earlier this week:

My reaction was that it would be fun to turn this into a project for kids, but this one would need a little introduction since conditional probability can be incredibly non-intuitive. During the week I came up with a plan, and we began to look at the problem this morning.

Here’s the introduction – I asked the boys to give their initial reaction to the seeming paradox:

Next we looked at an example that is slightly easier to digest -> rolling two dice and asking “do you have at least one 6?”

My younger son had a little trouble with the conditional probability, so I’m happy that we took this introductory path:

Next we moved to a slightly more difficult problem -> rolling 3 distinct dice. I used a 6x6x6 Rubik’s cube to represent the 216 states. To start, I asked the boys to count the number of states that had at least one six. Their approach to counting those 91 states was really fascinating:

Finally, we looked at the analogy to the 2nd ace paradox in our setting. So, if you have “at least one 6” what is the chance that you have more than one six, and if you have “a six on a specific die” what is the chance that you have more than one six?

Again, my younger son had a little trouble understanding how the cube represented the various rolls, but being able to hold the cube and see the states helped him get past that trouble:

Tomorrow we’ll move on to studying the paradox with the playing cards. Hopefully today’s introduction helped the boys understand

Playing with Colin Wright’s card puzzle

Aperiodical is hosting an “internet math off” right now and lots of interesting math ideas are being shared:

The Big Internet Math Off

The shared by Colin Wright caught my attention yesterday and I wanted to share it with the boys today:

The page for the Edmund Harriss v. Colin Wright Math Off

The idea is easy to play with on your own -> deal out a standard deck of cards (arranged in any order you like) into 13 piles of 4 cards. By picking any card you like (but exactly one card) from each of the 4 piles, can you get a complete 13-card sequence Ace, 2, 3, . . . , Queen, King?

Here’s how I introduced Wright’s puzzle. I started the way he started – when you deal the 13 piles, is it likely that the top card in each pile will form the Ace through King sequence:

Now we moved on to the main problem – can you choose 1 card from each of the 13 piles to get the Ace through King sequence?

As always, it is fascinating to hear how kids think through advanced mathematical ideas. By the end of the discussion here both kids thought that you’d always be able to rearrange the cards to get the right sequence.

Now I had the boys try to find the sequence. Their approach was essentially the so-called “greedy algorithm”. And it worked just fine.

To wrap up, we shuffled the cards again and tried the puzzle a second time. This time it was significantly more difficult to find the Ace through King sequences, but they got there eventually.

They had a few ideas about why their procedure worked, but they both thought that it would be pretty hard to prove that it worked all the time.

I’m always happy to learn about advanced math ideas that are relatively easy to share with kids. Wright’s card puzzle is one that I hope many people see and play around with – it is an amazing idea for kids (and everyone!) to see.

How a kid approaches a challenging problem

We stumbled on this problem in the book my older son is studying over the summer:

A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.

(i) What is the probability of finishing the game with a positive score?

(ii) What is the expected win when you play this game?

The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.

Here’s how the conversation went:

(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.

(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.

(3) Now that we made progress on one of the tough cases, we tackled the other two:

(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.

(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.

I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.