A scary approach and a not-so-scary approach to a challenging math problem

Today was the last day of math projects for at least a week because of some vacation and work travel. Rather than jumping ahead in our book I thought it would be neat to show them an interesting identity in Pascal’s triangle.

Here’s a scary way to ask about this identity:

Prove that for any integer n greater than 0:

{n \choose 0}^2 + {n \choose 1}^2 + \ldots {n \choose n}^2 = {2n \choose n}

Instead of taking the scary approach, we started today’s problem by talking about counting paths in a square grid. We explored this type of question for the first time in yesterday’s project:

What learning math sometimes looks like: Counting paths in a grid

Here’s what the boys thought about counting these paths today followed by their approach to counting a few subsets of the total paths:


In the last video we started counting paths that passed through specific points in the diagonal of the grid. Here we finish off that calculation to find an interesting, though fairly complicated-looking expression:


For the last part of the project we take a look at what this identity means in terms of Pascal’s triangle. The fairly easy to see relationship here is a nice surprise! Once the boys see the surprise, they are able to find other cases of this identity in Pascal’s triangle


So, a fun project before we take a break for a week. The connections with counting and Pascal’s triangle really are amazing 🙂

What learning math sometimes looks like: Counting paths in a grid

We started a new chapter in our Introduction to Counting and Probability book today. This chapter covers more applications of choosing numbers and the first example was counting paths in a grid.

The boys and I spent 20 to 30 minutes talking through this example and found an answer to the specific problem. The first video below is their explanation of that solution. In the following videos we talk about an alternative way to solve the problem that involves choosing numbers.


The solution that the boys described in the last video is a fairly standard solution to the problem, though the connection to choosing numbers didn’t really emerge. I wanted to help them see that connection, but didn’t want to give it away. So, talking through that connection takes up most of the next three videos. It was harder to get to that connection that I was expecting, but I think the roughly 15 minute conversation we had about that connection was really productive.

Here is that conversation:




So, definitely challenging, but a really fun project. The connection with the choosing numbers here is one of those ideas that is “obvious” after you learn it, but takes some work to see when you are encountering it for the first time. I think this is a great example of why learning math isn’t always a nice, easy straight line.

How many different “three of a kinds” are there?

We came to the end of Chapter 4 of our Introduction to Counting and Probability book today. While the boys were working through some of the review problems I was searching for a nice challenge problem for our project today. A question about counting the number of different “three of a kind” poker hands seemed perfect for today’s project.

Well, almost perfect – we don’t really play cards that much so this one required a longer than usual introduction. One important thing in counting “three of a kinds” is that the other two cards cannot match each other – that’s a full house.

Once we find exactly what we want to count, we start by figuring out how to count the number of ways for us to get a group of three matching cards.

For the next part we have to try to find the number of ways to select the remaining two cards. The boys find the right ideas here, but miss that they’ve over counted by a factor of 2. We’ll go back and find the over counting in the next video, here we spent a little extra time making sure the arithmetic was right:

In the last section we compute the value of the expression from the last video. I tell them that the number isn’t right, and they then search through their solution to find where they over counted. Fortunately they found the place where they accidentally double counted.

I end the video telling them about an old work project in we insured a $100 million prize at a poker tournament – the cards we used in this project were from that tournament 🙂

So, a fun project for kids learning about counting and choosing numbers. It is a fun exercise to go through the same calculation for all of the different poker hands, actually. In fact, working through those calculations was the final project for a former student of mine who – what a surprise – did indeed go on to become a professional poker player 🙂

A challenging – but illuminating – counting problem

Yesterday “choosing numbers” came up in our summer counting project. The boys seemed to think of them in a fairly formulaic way yesterday. While they were working through a few simple exercises this morning I was trying to think of a question that involved a bit more intuition.

Don’t know what made me think of it, but this pretty well-known problem came to mind:

How many three digit numbers have their digits in strictly increasing order?

When you see this problem for the first time, the connection to choosing numbers is not even remotely obvious. It was fun to hear the boys thinking about this problem and by the end of the project they were able to see the connection.

We started off by talking through some of their initial ideas about how to approach this problem. Those initial thought mainly involve case by case counting.


In the next part I asked them to try to think about the problem in terms of choosing numbers. By the end of this discussion the boys are able to understand that to pick a number with increasing digits we need to pick three digits out of 9.


Next we looked at a specific case of choosing 3 digits: 3, 6, and 7. What numbers have these digits? How many of those numbers have increasing digits?

At the end of this video the boys are trying to figure out whether or not order matters when we select the three digits.


In the last movie we sort out the counting issues – the 84 ways to select 3 numbers out of 9 actually gives the solution to the original problem! I really loved hearing they boys ideas as this approach to the problem suddenly made sense to them.


So, a really nice and instructive counting problem. Problems like this show that why counting can be such a fun subject – there’s sometimes much more going on that you might think!

Introduction to counting with combinations

This morning we started looking at combinations and “choosing” numbers. I think this is one of those topics that seems much easier after you’ve learned it than it seemed when you were learning it. After a quick introduction to the idea of choosing number, the boys worked on an example problem that involved counting the number of games in a round robin tournament:

The were able to work through that problem, but I thought that thinking through it again would still be helpful, so we reviewed the problem from start to finish.

First – how did we think about this problem before we talked about choosing numbers?


After that quick review we began to talk about choosing numbers. I let the boys explain their ideas about these numbers. One fun thing that happened in this part of our project is the boys discovered that there is a little bit of symmetry in these numbers:


Now we looked at patterns that arise in the choosing numbers. There’s a little trick that I mistakenly thought we’d already talked about – that 0! = 1. After telling them that was simply a definition, we moved on to finding a fun pattern that comes up in the choosing numbers – Pascal’s triangle!


Finally, as a way of confirming the connection to Pascal’s triangle, we looked to see if the addition relationship between two rows of the triangle also shows up in the choosing numbers. This is one of the first examples of a combinatorial proof that the boys have seen!


So, a really fun project showing that counting has some surprising connections to other topics in math. It was fun to hear their ideas (and their surprise) when they found the connection to Pascal’s triangle. Showing them a basic combinatorial proof at the end was fun, too – those proofs can be absolutely amazing!

A great problem for kids posted by Tina Cardone

[sorry for the hasty write up – I had some problems downloading the video from the camera, and the rest of my day is a little busy. Just wanted to get this one out the door because it is so fun.]

Saw this wonderful problem posted on Twitter by Tina Cardone yesterday:

For clarity, here’s the picture rotated so it is easier to read:

Math Problem

I thought it would be fun to try out the problem for our Family Math project today, but I didn’t really think through the problem before getting started. That laziness on my part led to a huge surprise – this is an amazing problem to talk through with kids! There are so many opportunities to talk about arithmetic, patterns, and why the sequences evolve in the way that they do.

We started by talking through the problem and making sure that the kids understood the way the pay evolved over time:


Next up – after choosing Lego figures to represent some of the participants in this problem – we began to look more carefully at the patterns. One of the ideas that begins to emerge in this video is that this problem provides a great opportunity to talk a little bit about arithmetic and rounding with kids.


We continued looking at the initial patterns in this video, but then my younger son noticed something interesting about the numbers in the first column – they were related to powers of 5. We then talked a little bit about how those same numbers were also decimal representations of powers of 2. This conversation went a little longer than usual, but there were some fun observations about the numbers from the boys.


Now we moved on to the middle column – this is where the payment amounts are rounded up. Turns out the pattern here is fairly simple (eventually), but it gives us a great opportunity to talk a little bit about how that pattern relates to the original payment formula.


Finally we look at the last column – where the pay is rounded down. This part of the project provides a great opportunity to talk about arithmetic and rounding with negative numbers. A great way to finish this project.


So, a super project for kids – one of my favorites ever. Thanks so much to Tina for sharing this problem!

Two fun introductory probability questions for kids

Saw this tweet from Nassim Taleb yesterday:


It reminded me of an old and unfortunate mistake from the 1998 Minnesota state high school math contest. Here was that question:

You go to visit a friend who has two children. However, you cannot remember the gender of either child. When you arrive at the house, one of those children answers the door. That child is a boy. What is the probability that the other child is a boy. Warning, the answer is not 1/2.

And, yes, that warning was part of the original question.

We were a little tight for time this morning, so I decided to use these two problems for a quick Family Math project. Here’s my kids taking a look at Taleb’s question:


Here’s my version of the second question (without the warning) and the thoughts my kids had thinking through it:


Although it isn’t all that difficult to understand the statement of either of these two questions, understanding why the two situations are different is sometimes no so easy. Taleb compares the question in his tweet to the famous Monty Hall problem. Another potentially good comparison – though much harder to understand mathematically – come from a recent Andrew Gelman blog post:

Hey Guess What? There Really is a Hot Hand

So, even though this was a quick little project, it still is both fun and instructive!

Playing around with the odds of winning Powerball

We started chapter 4 in our Introduction to Counting and Probability book today. The topic is combinations. After a few examples and a few problems I introduced them to a “real world” application of this math – lotteries. The specific lottery we looked at was Powerball.

First up, how the game works and what you have to do in order to win the big prize:

After writing down an expression for the odds of winning, we turn to our calculator to see if our expression matches the odds on Powerball’s website. We also spend of time trying to figure out how long you’d have to play before you’d expect to win once:

Finally we explore one of the prizes where the odds look wrong – the odds of matching only the Powerball. Seems like this should be a 1 out of 35 chance, but the website says roughly 1 in 55. Why?

So, a nice – and fairly straightforward – application of permutations and combinations. Definitely a fun way to start off this new chapter.

Counting arrangements of 10 people around a table

Today was our last day looking at Chapter 3 of Art of Problem Solving’s Introduction to Counting and Probability and I picked one of the challenge problems for our wrap up project. The problem has two parts and asks about counting arrangements of 10 people sitting around a table with a few constraints.

Here’s the boys approach to the first part of the problem where the constraint is that two people insist in sitting directly opposite from each other:

At the end of the last movie my older son had a question about reversing the position of the two people who insist on sitting opposite from each other. We spend a little extra time on that question because it is important to see that that switch is just a rotation of an arrangement that we’ve already counted.

The second piece of the challenge problem is that you now have two pairs of people who want to sit opposite each other. This problem was exactly at the right level for the kids – they couldn’t see the answer right away, but after talking about a few different approaches they were able to find their way to an answer in 5 minutes.

So, a nice little problem to wrap up the third chapter of our summer counting project. It is fun to hear kids talk through these counting problems. The problems are easy enough to understand, but also can be really challenging. I also like the way that all of the arithmetic inside of these counting problems help build up a little number sense, too.

A challenging counting example from Art of Problem Solving

We finished up the last section of chapter 3 in Art of Problem Solving’s Introduction to Counting and Probability today. The section is an introduction to counting with symmetry. The topic was one of the most eye-opening topics for me as an undergraduate, and some of the advanced ideas – Polya’s theory of counting, for example – are among the most beautiful ideas I ever saw in math.

For our project today we reviewed one of the exercises from the end of the chapter. The problem deals with arranging two groups of 5 people around a table. In the book the two groups are democrats and republicans, but we went with lego figures instead. Our groups are “nindroids” and “flood infection spores” . . . .

Part 1: How many distinct ways are there to arrange the 10 people if there’s no restrictions in the seating arrangement, but we’ll say that two arrangements are the same if they differ only by rotation?

[note: sorry for having 6 flood infection spores in the video – didn’t notice this problem until we started in on the next video.]


Part 2: In this next case we’ll add the seating restriction that each of the two groups of 5 people must sit together. My attempt to provide a “better” explanation of how to think about this problem was a disaster – sorry about that!


Part 3: The last case we looked at was the number of ways for the two groups to sit in an alternating pattern. This version of the problem caused a lot of difficulty for the boys when we were going through it the first time. Having them think through and explain this problem a second time was the main motivation for this project:


So, maybe not our most error free project, but still a fun one. I think that these “counting with symmetry” problems are great ways for kids to see some math that is both fun and challenging. These problems also show that counting arrangements can be a little more difficult that it might initially seem.