Tag counting

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.

Advertisements

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.

A challenging but worthwhile probability problem for kids

Alexander Bogomolny shared a great problem from the 1982 AHSME yesterday:

I remember this problem from way back when I was studying for the AHSME back in the mid 1980s. I thought it would be fun to talk through this problem with my older son – it has some great lessons. One lesson in particular is that there is a difference between counting paths and calculating probabilities. It was most likely this problem that taught me that lesson 30+ years ago!

So, here’s my son’s initial reaction to the problem:

Next we talked through how to calculate the probabilities. This calculation gave him more trouble than I was expecting. He really was searching for a rule for the probabilities that would work in all situations – but the situations are different depending on where you are in the grid!

Despite the difficulty, I’m glad we talked through the problem.

(also, sorry about the phone ringing in the middle of the video!)

So, definitely a challenging problem, but also a good one to help kids begin to understand some ideas about probability.

Sharing a card shuffling idea from Jim Propp’s “Who knows two?” essay with kids

Jim Propp published a terrific essay last week:

Here’s a direct link in case the Twitter link has problems:

Who knows two? by Jim Propp

One of the topics covered in the essay is a special type of card shuffle called the Faro shuffle. We have done a few projects on card shuffling projects previously, so I thought the kids would be interested in learning about the Faro shuffle. Here are our prior card shuffling projects:

Card Shuffling and Shannon Entropy

Chard Shuffling and Shannon Entropy part 2

Revisiting card shuffling after seeing a talk by Persi Diaconis

I started the project by asking the kids what they knew about cards. They remembered some of the shuffling projects and then introducing the idea of the Faro shuffle.

My younger son thought he saw a connection with pi, which was a fun surprise.

We continued studying the Faro shuffle with 8 cards and looked for patterns in the card numbers and positions. The boys noticed some neat patterns and were able to predict when we’d return to the original order of cards!

Next we looked at the paths taken by individual cards. My older son thought that there might be a connection with modular arithmetic (!!!) and the boys were able to find the pattern. I’d hoped that finding the pattern here would be within their reach, so it was a really nice moment when he brought up modular arithmetic.

Finally, we wrapped up by talking about how to extend the ideas to a 52 card deck and calculated how many Faro shuffles we’d need to get back to where we started.

I think that kids will find the idea of the Faro shuffle to be fascinating. Simply exploring the number patterns is a really interesting project, and there’s lots of really interesting math connected to the idea. I’m really thankful that Jim Propp takes the time to produce these incredible essays each month. They are a fantastic (and accessible) way to explore lots of fun mathematical ideas.

Sharing a great Alexander Bogomolny probability problem with kids

[note: I’m trying up this post at my son’s karate class. It is loud and unfortunately I forgot my headphones. I’m left having to describe the videos without being able to listen to them . . . . ]

I saw a really great problem today from Alexander Bogomolny:

By coincidence I heard the recent Ben Ben Blue podcast yesterday which had a brief mention / lament that it was hard to share mistakes in videos.

This problem is probably a good challenge problem for my older son and definitely above the level of my younger son. But listening to both of them try to work through the problem was really interesting.

I started with my older son – he initially approached the problem by comparing the individual probabilities:

After his initial work, I talked with him about comparing the probabilities of the complete events described in the problem. Initially there was a little confusion on his part, but eventually he understood the idea:

Next up was my younger son – not surprisingly, he had a hard time getting started with the problem. His initial approach was similar to what my older son had done – he looked at the one head and two heads events separately to see which one was more likely for each coin:

As I did with my older son, I asked him to look at the two events as a single event and see which one was more likely when each coin went first:

So, a nice project and an opportunity to see a few mistakes and as well as how kids approach a challenging probability problem.

Comparing double stuffed Oreos to thin Oreos

In 2015 we did a project comparing double stuffed oreos to regular ones:

Do double stuffed oreos have double the stuffing?

As I said in that blog post, I’d seen a few teachers discussing the idea, but I don’t remember who originally shared the project. So, to be clear again, the idea for this line of study isn’t mine, but I’m happy to have some fun with it.

Instead of revisiting the prior project today, I tried something slightly differnt -> comparing double stuffed oreos with thin ones. The prep work for this project proved to be a little harder than I was expecting because the double stuffed oreo shells were really fragile. So, if you want to repeat this project, be prepared for lots of broken oreo shells!

To start I introduced my son to the problem we were going to try to solve today and asked for his thoughts. The problem was to try to find the ratio of the volume of stuffing in the double stuffed oreos to the volume of stuffing in the thin ones.

Our original intention was to weigh 10 of the crackers from each of the 2 types of cookies. We were able to get only 8, though. The 8 thin crackers weighted 22 grams and the 8 double stuffed crackers weighed 33 grams.

Sorry the writing was off screen.

Next we moved to weighing the full cookies. I didn’t communicate really well at the start of this video, and confused my son a bit. Eventually we got back on the same page weighed 4 cookies of each type.

The 4 double stuffed cookies together weighed 60 grams. That led us to conclude that the stuffing weight was approximately 27 grams for 4 cookies.

Finally, we repeated the process in step 3 with 4 thin oreos. We found that the 4 cookies together weighed roughly 30 grams, meaning the total weight of the filling was 8 grams.

So, my son’s guess of 4 to 1 for the ratio of the filling weight was pretty close. Turned out be 27 grams to 8 grams for 4 cookies, or about 3.5 to 1.

Definitely a fun project. I haven’t done much in the way of introductory statistics for kids – this project definitely gets kids engaged!

Revisiting Joel David Hamkins’s “Graph Theory for Kids”

A few years ago we did a fun project with Joel David Hamkins’s “Graph Theory for Kids”:

Going through Joel David Hamkins’s “Graph Theory for Kids”

Here’s the link to Hamkins’s notes for the project:

Graph Theory for Kids

This project was also inspired by the project we did yesterday on the graph isomorphism theorem:

Sharing Lazlo Babi’s graph isomorphis talk with kids

For today’s project I printed two copies of Hamkins’s booklet and had the boys work through it on their own. After they were finished, we talked through the project after they were finished. Here’s the conversation broken into 4 parts – as you’ll see, Hamkins has made an absolutely fantastic project for kids:

Part 1: An introduction to graphs and one surprising property

Part 2: Looking at some more complicated or “extreme” examples and also illustrating how some of the more complicated graphs make for nice counting exercises for kids

Part 3: Now a few examples that the kids made on their own – this part led to a nice discussion about crossings

Part 4: Some 3d shapes and a really fun observation from my older son about the sphere

Talking through two problems from the 2005 AMC 10

I really enjoy using old AMC problems to talk about math with the boys.

These two problems gave my older son some trouble today:

Tonight I had a chance to talk through these problems with them.

Here’s the probability problem:

Here’s the GCD problem:

Talking though Richard Evan Schwartz’s Gallery of the Infinite with kids

We received Richard Evan Schwartz’s Gallery of the Infinite in the mail this week:

I thought that the boys would love reading the book and asked them to each read it twice prior to today’s project. Here are some of the things that they thought were interesting (ugh, sorry for the focus problems . . . .) :

The first thing the boys wanted to talk about was the “smallest” infinity -> \aleph_0. Here we talked about that infinity and other sets of integers that were the same size.

Next we moved on to talk about the rational numbers – we had a good time talking through the argument that the “size” of the rational numbers was the same as the positive integers.

This argument is represented in the book by a painting of a shark!

Now my older son wanted to talk about Cantor’s diagonal argument. He was a little confused about the arguments presented in the book, but we (hopefully) got things straightened out. I think this shows kids can find ideas about infinity to be really interesting.

Finally, we wrapped up by talking about the implications of the infinity of binary strings being larger than the infinity of counting numbers.

Definitely a fun project. I love the content of the book and so do the kids. The only problem is that the quality of the binding is awful and although we’ve only had the book for a few days, it is falling to pieces. Boo 😦

Working through a challenging AMC 10 problem

My son was working on a few old AMC 10 problems yesterday and problem 17 from the 2016 AMC 10a gave him some trouble:

Screen Shot 2018-01-07 at 5.14.40 PM

I thought this would be a nice problem to go through with him. We started by talking through the problem to make sure that he understood it:

In the last video he had the idea to check the cases with 10 and 15 balls in the bucket, so we went through those cases:

Now we tried to figure out what was happening. He was having some difficulty seeing the pattern, so I spent this video trying to help him see the pattern. The trouble for me was that the pattern was 0, 1, 2, . . ., so it was hard to find a good hint.

Finally he worked through the algebraic expression he found in the last video:

This isn’t one of the “wow, this is a great problem” AMC problems, but I still like it. To solve it you have to bring in a few different ideas, and combining those different ideas is what seemed to give my son some trouble. Hopefully going through this problem was valuable for him.