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Sharing Tim Gowers’s nontransitive dice talk with kids

During the week I attending a neat talk at Harvard given by Tim Gowers. The talk was about a intransitive dice. Not all of the details in the talk are accessible to kids, but many of the ideas are. After the talk I wrote down some ideas to share and sort of a sketch of a project:

Thinking about how to share Tim Gowers’s talk on intransitive dice with kids

One of the Gowers’s blog posts about intransitive dice is here if you want to see some of the original discussion of the problem:

One of Tim Gowers’s blog posts on intransitive dice

We started the project today by reviewing some basic ideas about intransitive dice. After that I explaine some of the conditions that Gowers imposed on the dice to make the ideas about intransitive dice a little easier to study:

The next thing we talked about was 4-sided dice. There are five 4-sided dice meeting Gowers’s criteria. I thought that a good initial project for kids would be finding these 5 dice.

Now that we had the five 4-sided dice, I had the kids choose some of the dice and see which one would win against the other one. This was an accessible exercise, too. Slightly unluckily they chose dice that tied each other, but it was still good to go through the task.

Now we moved to the computer. I wrote some simple code to study 4-sided through 9-sided dice. Here we looked at the 4-sided dice. Although it took a moment for the kids to understand the output of the code, once they did they began to notice a few patterns and had some new ideas about what was going on.

Having understood more what was going on with 4-sided dice, we moved on to looking at 6-sided dice. Here we began to see that it is actually pretty hard to guess ahead of time which dice are going to perform well.

Finally we looked at the output of the program for the 9-sided dice. It is pretty neat to see the distribution of outcomes.

There are definitely ideas about nontransitive dice that are accessible to kids. I would love to spend more time thinking through some of the ideas here and find more ways for kids to explore them.


Thinking about how to share Tim Gowers’s talk on intransitive dice with kids

Yesterday Tim Gowers gave a really nice talk at Harvard about intransitive dice. The talk  was both interesting to math faculty and also accessible to undergraduates (and also to math enthusiasts like me).

The subject of the talk – properties of intransitive dice – was based on a problem discussed on Gowers’s blog earlier in 2017.  Here’s one of the blog posts:

One of Tim Gowers’s blog posts on intransitive dice

Part of the reason that I wanted to attend the talk is that we have played with non-transitive dice previously and the kids seemed to have a lot of fun:

Non-Transitive Grime Dice

Here’s the punch line from that project:

The starting example in Gowers’s talk was due to Bradley Efron. Consider the four 6-sided dice with numbers:

A: 0,0,4,4,4,4
B: 3,3,3,3,3,3
C: 2,2,2,2,6,6
D: 1,1,1,5,5,5

If you have little dice rolling competition in which the winner of each turn is the die with the highest number, you’ll run across the following somewhat surprising expected outcome:

A beats B, B beats C, C beats D, and D beats A.

The question that interested Gowers was essentially this -> Is the situation above unusual, or is it reasonably easy to create intransitive dice?

Although the answering this question probably doesn’t create any groundbreaking math, it does involve some fairly heavy lifting, and I think the details in the talk are not accessible (or interesting) to kids.  Still, though, the general topic I think does have questions that could be both fun and interesting for kids to explore.

In discussing a few of the ideas that I think might be interesting to kids, I’ll use a constraints that Gowers imposed on the dice he was studying. Those are:

(i) The numbers on each side of an n-sided can be any integer from 1 to n

(ii) The sum of the numbers must be (n)(n+1)/2

I’ll focus just on 6-sided dice for now. One question that kids might find interesting is simply how many different 6-sided dice are there that meet these two criteria above? Assuming I’ve done my own math right, the answer is that there are 32 of them:


Next, it might be interesting for kids to play around with these dice and see which ones have lots of wins or lots of losses or lots of draws against the other 31 dice.  Here’s the win / draw / loss totals (in the same order as the dice are listed above):


So, for clarity, the 4th die on the list – the one with numbers 1,1,3,5,5,6  – wins against 16 other dice, draws with 7 (including itself), and loses to 9.  Not bad!

The die three down from that – the one with numbers 1, 2, 2, 4, 6, 6 – has the opposite results.  Such poor form 😦

It certainly wasn’t obvious to me prior to running the competition that one of these two dies would be so much better than the other one.  Perhaps it would be interesting for kids to try to guess ahead of time which dice will be great performers and which will perform poorly.

Also, what about that one that draws against all the others – I bet kids would enjoy figuring out what’s going on there.

Once I had the list, it wasn’t too hard for me to find a set of three intransitive dice.  Choosing

A ->  2, 2, 3, 3, 5, 6

B -> 1, 1, 3, 5, 5, 6, and

C -> 1, 2, 4, 4, 4, 6

You’ll see that A beats B on average, B beats C, and C beats A.

It is always fun to find problems that are interesting to professional mathematicians and that are also accessible to kids.   A few ideas I’ve found from other mathematicians can be found in these blog posts:

Amazing math from mathematicans to share with kids

10 more math ideas from mathematicians to share with kids

I think exploring intransitive dice will allow kids to play with several fun and fascinating mathematical ideas.   I’m going to try a project (a computer assisted project, to be clear) with the kids this weekend to see how it goes.


An introductory stars and bars problem

Yesterday a counting problem from my son’s math team homework gave him a little trouble. The problem went something like this:

There are 5 different types of fasteners and you need to buy 10 total. If you need to buy at least one of each, how many different ways can you do it?

First we talked about the problem and got their initial thoughts. Then I introduced the stars and bars counting idea:

Next I tried to go through a few more examples by changing the numbers a little. The main ideas seemed a little confusing to the boys and I’d hoped a few extra examples would help. Unfortunately things weren’t going so well.

The last example in the prior video confused my younger son, so I moved on to the next video to talk about that example in more detail. By the end of this example I hoped that the general idea had sunk in, but there was still a little confusion.

So, we talked through the problem a few more times. Now the ideas seemed to be sinking in. IF you have N groups of objects (in the original problem 5 fasteners) and you have to pick M total objects (in the original problem we were trying to pick 5 fasteners) you can represent the problem with M stars and N – 1 bars. So the total number of different ways to make the selections are (M + N – 1) “choose” M or, alternately, (M + N – 1) choose (N – 1).

Not all of our projects go super well. Here my mistake was thinking that I could introduce an advanced concept and the boys would immediately understand it. I feel like the ideas here are definitely within their grasp and will probably spend a bit more time this weekend covering the concept. Hopefully a few more examples will do the trick. Stars and Bars.jpg

Working through two old contest problems

I’ve been sort of on pause doing new math with the boys for the last couple of weeks. I want them to find their stride with the new school year before seeing what additional enrichment math we can do at home.

So, while on pause they’ve just been working through problems from old AMC tests in the morning. When they finish we talk through some of the problems that gave them trouble. Both problems were pretty interesting lessons (for them and me, I mean) today.

Here’s what my older son had to say about problem 18 from the 2013 AMC 10b. It was fascinating to me how he counted the numbers in this problem.

and here’s what my younger son had to say about problem #17 from the 1985 AMC 8 (which then was the American Junior High School Math Exam). It was fascinating to me to see both how he played with the averages and how he found his arithmetic mistake.

I love using these old AMC problems to keep the kids engaged with math. It is always fun to see what sorts of ideas give them problems and just as fun to see their problem solving strategies.

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:


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

Counting in 4 dimensions

Yesterday we did a neat project based on problem #12 from the 2015 AMC 8:


That project is here:

A great counting problem for kids from the 2015 AMC 8

Then I got a nice comment on the project from Alison Hansel:

So, for today’s project we extended the problem from yesterday to 4 dimensions.

Here’s the introduction and a quick reminder of yesterday’s problem. I had both boys review their solutions and then we began to discuss how to approach the same problem in 4 dimensions.

Next we dove a bit deeper into how to approach the 4 dimensional problem. They boys thought a bit about the symmetry that a 4d cube would have and at the end (after a long and quiet pause) my younger son thought that looking at how a square turns into a cube might help us.

In studying how a square transforms into a cube, the critical idea is how 4 edges turn into 12 edges. This video is a little on the long side, but I think the discussion is really interesting. By the end the boys have found the main idea for how to count edges as you move up in dimension.

Next I brought out Henry Segerman’s 4-d cube model and compared the model to the ideas we’d developed up to this point.

An important idea from earlier in this project was that my older son thought that each edge of the 4d-cube would be part of two 3d cube “faces”. Using the model we were able to see that, in fact, each edge is part of 3 cubes.

Finally – with the 4 pieces of prep work behind us! – we were able to answer the AMC 8 question about a 3d cube in 4 dimensions. So . . . how many pairs of parallel edges does a 4d cube have? The answer is 112 🙂

Thanks to Alison Hansel for the great suggestion for how to extend yesterday’s project. I think her idea makes a great way to introduce kids to some simple ideas in 4d geometry.

A great counting problem for kids from the AMC 8

This problem from the 2015 AMC 8 gave my son some trouble today. Actually quite a bit of trouble:



I’m not 100% sure what caused the difficulty. It might be that once you start thinking about this problem one way that it is hard to switch. Whatever the cause, though, we had a really good conversation about the problem.

Here’s his original approach that is incorrect:

So, after finding out that the answer in the last video was incorrect, we went to try to find the error. He found it pretty quickly.

After that I tried to explain an alternate approach to the problem. Unfortunately my explanation ended up causing quite a bit of confusion:

In the last part of our discussion I tried to dig my way out of the hole I created in the last video.

Even watching this video after the fact, I’m not sure what was the original source of his confusion. There was definitely some difficulty going from 4 parallel edges to 6 pairs of parallel edges.

By the end of our conversation he was able to walk through the argument, but I think that I’ll revisit some similar problems with him just to be sure the main ideas have sunk in.

I think this short project is a nice example of how old contest problem can help kids learn math. For me anyway, it is really challenging to come up with good problems and the fact that all of the old AMC problems are available for kids to work through is an incredibly helpful resource. Hopefully I can find some similar counting problems on other old AMC contests.

Going through three AMC 8 problems

My younger son has been doing a little practice for the AMC 8. Yesterday three problems from the 2013 exam gave him a little trouble. We went over them together.

The first was a tricky geometry problem. Both the words and ideas needed to solve this problem were new to him.

Next up was a challenging counting problem – we broke this into two pieces. This is a great counting problem for kids. In the first part we found out how to calculate the answer, but didn’t finish the calculation:

In the second part we talked about strategies to finish the calculation:

Finally – a fantastic geometry problem. It has a few little traps in it, but my son found a nice solution.

I love using the old AMC contest problems to help the boys see math that is both fun and challenging. These problems were really fun to talk through.

Playing with Jim Propp’s essay on Arthur Engel

Jim Propp’s August 2017 blog post is absolutely terrific:

Prof. Engel’s Marvelously Improbably Machines

Even though we are visiting my parents in Omaha, I couldn’t resist having the boys watch the video in the essay and then play with the challenge problem.

Here’s what they thought after watching the video – the nice thing is that they were able to understand the problem Propp was discussing [also, I shot these videos with my phone, so they probably don’t have quote the quality or stability of our usual math videos]:

Next I had them play the game that Propp explained in his video. The idea here was to make sure that they understood how the [amazing!] solution to the problem shown in the video:

Next we tried the challenge problem from the essay. I almost didn’t do this part of the project, but I’m glad I did. It turned out that there were a few ideas in the Propp’s video that the boys thought they understood but there was a bit more explanation required. Once they got past those small stumbling blocks, they were able to solve the problem.

I’m really excited to dive a little deeper into the method of solving probability problems that Propp explains in his essay. What makes me the most excited is that the method came from someone thinking about how to explain probability to kids.

The last video shows that understanding Engel’s method does take a little time. Once kids get the general idea, though, I think they’ll find that applying to a wide variety of problems is pretty easy. It is amazing how such a simple method can made fairly complex probability problems accessible to kids.

The “Dungeons and Dragons” problem

My kids have suddenly been drawn into D&D. They are having a ton of fun with it and I thought that there was at least one fun little math problem we could talk through that related to the game.

At the start of the game you create a character with various properties. To determine the value of some (maybe all) of those properties you roll four 6-sided dice and add up the three highest numbers.

The question we looked at today was what is the expected value of that sum?

First we introduced the problem and come up with a few ideas about what the answer might be.

Next we did 10 trials to see what average we’d find:

Now we had a longish talk about how you might solve the problem. The boys jumped to a computer simulation pretty quickly. After talking about how that simulation would work we talked about how to solve a similar problem with two dice.

Finally, we did go the computer to see what the answer would be. Talking about how to write the program was pretty fun.

Nice project – I might revisit this one to talk through the geometry of the solution of the 2 and 3 dice problem and see if the boys can figure out how to generalize to the 4 dice case.