Tag probability

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.

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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 an advanced expected value problem from Nassim Taleb with kids

Earlier in the week I saw this interesting problem posted by Nassim Taleb:

Solving this problem requires calculus, and trig to even begin to understand how to approach it, but it still seemed like one that would be interesting to talk through with kids. Especially since a Monte Carlo-like approach is going to lead you down a surprising path.

So, I presented this problem to the boys this morning. It took a few minutes for them to get their arms around the problem, but they were able to understand the main ideas behind the question. That made me happy.

Here’s the introduction to the problem:

Next I asked the boys what they thought the answer to this question would be. It was fascinating to hear their reasoning. Both kids had the same guess -> the expected average distance was 1.

Now we went to the computer to see what the average was when we did a few trials. We started by doing 100 trials to estimate the average and then moved up to 10,000 trials.

Next we went to 1 million trials and found a few big surprises including this amazing average:

Screen Shot 2018-04-08 at 9.20.06 AM.png

We wrapped up by discussing how you might get an infinite expected value by looking at the values of Tan(89), Tan(89.9), Tan(89.99), and so on. It was interesting for them to see how individual trials could have large weights, even with large numbers of trials.

Definitely a fun project to show kids, and a nice (though advanced) statistics lessonm too -> What happens when the mean you are looking for is infinite?

15 (+1 bonus) Math ideas for a 6th grade math camp

Saw an interesting tweet last week and I’ve been thinking about pretty much constantly for the last few days:

I had a few thoughts initially – which I’ll repeat in this post – but I’ve had a bunch of others since. Below I’ll share 10 ideas that require very few materials – say scissors, paper, and maybe snap cubes – and then 5 more that require a but more – things like a computer or a Zometool set.

The first 4 are the ones I shared in response to the original tweet:

(1) Fawn Nguyen’s take on the picture frame problem

This is one of the most absolutely brilliant math projects for kids that I’ve ever seen:

When I got them to beg

Here’s how I went through it with my younger son a few years ago:

(2) James Tanton’s Mobius strip cutting exerciese

This is a really fun take on this famous scissors and paper cutting exercise:

You will honestly not believe what you are seeing when you go through Tanton’s version:

Here’s the link to our project:

James Tanton’s incredible mobius strop cutting project

(3) Martin Gardner’s hexapawn “machine learning” exercise

Screen Shot 2017-09-03 at 10.07.08 AM

For this exercise the students will play a simple game called “hexapawn” and a machine consisting of beads in boxes will “learn” to beat them. It is a super fun game and somewhat amazing that an introductory machine learning exercise could have been designed so long ago!

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

(4) Katie Steckles’ “Fold and Cut” video

This video is a must see and it was a big hit with elementary school kids when I used it for “Family Math” night:

Here are our projects – all you need is scissors and paper.

Our One Cut Project

Fold and cut project #2

Fold and cut part 3

(5) Along the same lines – Joel David Hamkins’s version of “Fold and Punch”

I found this activity in one of the old “Family Math” night boxes:

Joel David Hamkins saw my tweet and created an incredible activity for kids.ย  Here’s a link to that project on his blog:

Joel David Hamkins’s fold, punch and cut for symmetry!

(6) Kelsey Houston-Edwards’s “5 Unusual Proofs” video

Just one of many amazing math outreach videos that Kelsey Houston-Edwards put together during her time at PBS Infinite Series:

Here is how I used the project with my kids:

Kelsey Houston-Edwards’s “Proof” video is incredible

(7) Sharing the surreal numbers with kids via Jim Propp’s checker stacks game

Screen Shot 2018-03-24 at 7.52.55 PM

Jim Propp published a terrific essay on the surreal numbers in 2015:

Jim Propp’s “Life of Games”

In the essay he uses the game “checker stacks” to help explain / illustrate the surreal numbers. That essay got me thinking about how to share the surreal numbers with kids. We explored the surreal numbers in 4 different projects and I used the game for an hour long activity with 4th and 5th graders at Family Math night at my son’s elementary school.

This project takes a little bit of prep work just to make sure you understand the game, but it is all worth it when you see the kids arguing about checker stacks with value “infinity” and “infinity plus 1” ๐Ÿ™‚

Here is a summary blog post linking to all of our surreal number projects:

Sharing the Surreal Numbers with kids

(8) Larry Guth’s “No Rectangle” problem

I learned about this problem when I attended a public lecture Larry Guth gave at MIT.ย  Here’s my initial introduction of the problem to my kids:

I’ve used this project with a large group of kids a few times (once with 2nd and 3rd graders and it caused us to run 10 min long because they wouldn’t stop arguing about the problem!). It is really fun to watch them learn about the problem on a 3×3 grid and then see if they can prove the result. Then you move to a 4×4 grid, and then a 5×5 and, well, that’s probably enough for 80 min ๐Ÿ™‚

Larry Guth’s “No Rectangles” problem

(9) The “Monty Hall Problem”

This is a famous problem, that equally famously generates incredibly strong opinions from anyone thinking about it. These days I only discuss the problem in larger group settings to try to avoid arguments.

Here’s the problem:

There are prizes behind each of 3 doors. 1 door hides a good prize and 2 of the doors hide consolation prizes. You select a door at random. After that selection one of the doors that you didn’t select will be opened to reveal a consolation prize. At that point you will be given the opportunity to switch your initial selection to the door that was not opened. The question isย  -> does switching increase, decrease, or leave your chance of winning unchanged?

One fun idea I tried with the boys was exploring the problem using clear glasses to “hide” the prizes, so that they could see the difference between the switching strategy and the non-switching strategy:

Here’s our full project:

Exploring the Monty Hall problem with kids

(10) Using the educational material from Moon Duchin’s math and gerrymandering conference with kids

Moon Duchin has spent the last few years working to educate large groups of people – mathematicians, politicians, lawyers, and more – about math and gerrymandering.ย  . Some of the ideas in the educational materials the math and gerrymandering group has created are accessible to 6th graders.

Screen Shot 2018-01-14 at 9.08.06 AM

Here’s our project using these math and gerrymandering educational materials:

Sharing some ideas about math and gerrymandering with kids

(11) This is a computer activity -> Intro machine learning with Google’s Tensorflow playground.

This might be a nice companion project to go along with the Martin Gardner project above. This is how I introduced the boys to the Tensorflow Playground site (other important ideas came ahead of this video, so it doesn’t stand alone):

Our complete project is here:

Sharing basic machine learning ideas with kids

(12) Computer math and the Chaos game

The 90 seconds starting at 2:00 is one of my all time favorite moments sharing math with my kids:

The whole project is here, but the essence of it is in the above video:

Computer math and the chaos game

(13) Another computer project -> Finding e by throwing darts at a chess board

This is a neat introductory probability project for kids. I learned about it from this tweet:

You don’t need a computer to do this project, but you do need a way to pick 64 random numbers. Having a little computer help will make it easier to repeat the project a few times (or have more than one group work with different numbers).

Here’s how I introduced the project to my kids:

Here’s the full project:

Finding e by throwing darts

(14) Looking at shapes you can make with bubbles

For this project you need bubble solution and some way to make wire frames. We’ve had a lot of success making the frames from our Zometool set, but if you click through the bubble projects we’ve done, you’ll see some wire frames with actual wires.

Here’s an example of how one of these bubble projects goes:

And here’s a listing of a bunch of bubble projects we’ve done:

Our bubble projects

(15) Our project inspired by Ann-Marie Ison’s math art:

This tweet from Ann-Marie Ison caught my eye:

Then Martin Holtham created a fantastic Desmos activity to help explore the ideas:

It is fun to just play with, but if you want to see how I approached the ideas with my kids, here are our projects:

Using Ann-Marie Ison’s incredible math art with kids

Extending our project with Ann-Marie Ison’s art

(16) Bonus project!!A dodecahedron folding into a cube

This is a an incredible idea from 3d geometry.

We studied it using our Zometool set – that’s not the only way to go, but it might be the easiest:

dodecahedron fold

Here’s the full project:

Can you believe that a dodecahedron folds into a cube?

A terrific probability problem for kids shared by Alexander Bogomolny

Saw this tweet from Alexander Bogomolny yesterday and knew immediately what today’s project was going to be ๐Ÿ™‚

The problem is, I think, accessible to kids without much need for additional explanation, so I just dove right in this morning to see how things would go.

My first question to them was to come up with a few thoughts about the problem and some possible strategies that you might need to solve it. They had some good intuition:

Next we attempted to use some of the ideas from the last video to begin to study the problem. Pretty quickly they saw that the initial strategy they chose got complicated, and a more direct approach wasn’t actually all that complicated:

I intended to have them solve the 4x4x4 problem with one of our Rubik’s cubes as a prop, but we could only find our 5x5x5 cube. So, we skipped the 4x4x4 case, solved the 5x5x5 case and then jumped to the NxNxN case:

Finally, I wanted the boys to see the “slick” solution to this problem – which is really cool. You’ll hear my younger son say “that’s neat” if you listen carefully ๐Ÿ™‚

Definitely a fun problem – would be really neat to share this one with a room foll of kids to see all of the different strategies they might try.

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.

A nice probability coincidence today

This morning my older son and I talked through the following problem from the 2003 AMC 10b:

Screen Shot 2018-02-15 at 7.11.31 PM.png

It turned out that an arithmetic problem is what led to his confusion on this problem, but discussing this problem was a nice opportunity to talk about discrete probability. At the end of our conversation I told him to always remember the underlying idea in discrete probability is simple -> count the cases that work and then count the total cases. It may not always be so easy to do, but it really is all that you have to do to solve the problem.

After he went off to school I this problem posted on Twitter:

This is a terrific problem, and it is really tempting to try to break the problem into pieces and essentially to try to solve it with recursion.

But remember the simple little idea I told my son -> count the cases that work and count the total cases. You’ll find a delightful solution to the problem!

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:

Sharing the ABRACADABRA problem with kids

Yesterday we did a fun project on Markov chains and sharing the “COVFEFE” problem with kids:

Sharing Markov chains and the “covfefe” problem with kids

For me math behind this problem was the most interesting math I learned in 2017:

The most interesting piece of math I learned in 2017 -> the “covfefe” problem

Today we moved on to a really neat surprise, and what makes the math behind this problem incredibly fun -> the “ABRACADABRA” problem.

First, we reviewed the ideas from yesterday:

After that review, we though through a few of the states and the transition probabilities in the new word. The transition probabilities are subtly different than in the “COVFEFE” problem:

Now we went to Mathematica to code in the ideas we discussed in part 2. We did about half of the coding on camera and did the other half off camera:

Finally, having finished the code we discussed what results we expected. I don’t see how anyone could get the right intuition here seeing the problem for the first time, so what do you expect here is almost an unfair question. Still, the boys had some nice ideas and then we checked out the results:

There are other approaches to these problems – the approach via Martingales, for example:

What that approach is also interesting (and incredible – you can solve the stopping time in your head!) I think the Markov chain approach is a bit more accessible to kidsd. Well . . . maybe because the math is buried in the background.

Anyway – super fun project, and an great piece of math to share with kids.