An Evil and Odious exercise from Tanya Khovanova

Wandering around the internet this week, I somehow landed on this old blog post from Tanya Khovanova:

Thue-Morse Odiousness

I’d not heard of evil and odious numbers before but Khovanova’s blog post gave me some ideas about how to talk about them with kids. It seemed like a great topic, actually – kids would get to review a little bit about binary and also have an opportunity to make lots of observations / conjectures about patterns in the sequences that came up.

We’ve used a few ideas from Khovanova’s (incredible!) blog before. For example:

Using divisibility rules to build number sense

PISA and V. I. Arnold Questions from Twitter

Also, since evil and odious numbers were invented by John Conway, here are a few of our preview John Conway-inspired projects:

Sharing the Surreal numbers with kids

The various projects in the last link helped me come up with a fun Family Math night for the 4th and 5th graders at my younger son’s school. Watching the kids play around with the game checker stacks and learn about the surreal numbers was one of the most enjoyable moments in math that I’ve ever had.

The Collatz Conjecture and John Conway’s “Amusical” Variation

Playing John Conway’s Game of Life with Kids

So, with that as background, here’s today’s project. First up was introducing the idea of “evil” and “odious” numbers and finding the first couple of numbers in each set:

Next (off camera) we looked at the first 20 positive integers and then on camera we talked about some of the patterns we saw:

Next we studied the process that Khovanova described in her blog. Oversimplifying a little, that process is:

(1) Start with any sequence of positive intergers,
(2) Turn that into a sequence of 0’s and 1’s based on whether or not each integer is odious or evil,
(3) Convert the sequence of 0’s and 1’s back into a sequence of integers which is essentially the increasing sequence of integers with the smallest numbers which share the same odious / evil pattern

There were lots of fun ideas to discover here:

Finally, we went to the computer to investigate a few other sequences. Some short (and a little clumsy . . . sorry) Mathematica code allowed us to look at these sequences pretty quickly:

So, even though this project was a little longer than usual, this was really fun. There are some great opportunities for kids to form and check some conjectures in a setting where the patterns are almost surely different from anything that they’ve ever seen before.

I’m really happy to have stumbled on Khovanova’s great blog post this week!

Introduction to geometric series

Earlier in the week we looked at a problem that involved the series 1 + 2 + 2^2 + 2^3 + \ldots + 2^n. Each kid had a different and interesting approach to summing that series. For today’s project I wanted to review each of their approaches and then explore a few other simple geometric series to see if we could use the same ideas to add them up, too.

We started with a review of my older son’s approach to the original series. His approach was essentially mathematical induction, which was pretty cool since we’ve never talked about mathematical induction. Oddly, though, telling him that his approach had a name seems to have confused him a little bit, and it takes a little while for him to remember what he did. That little bit of confusion made me happy that we decided to revisit this topic today:


Next I had my younger son explain his approach to summing the series. His approach came from noticing a connection to binary numbers when he first saw the series. That connection to binary is a really clever way to think about this sum:


With that introduction and review of their prior ideas, we decided to see if we could apply those ideas to a new series: 1 + 3 + 3^2 + \ldots + 3^n. My older son tries his approach first – the connection isn’t obvious, but then we compare a few of the sums to the powers of 3 and make some progress.


Now my younger son takes a shot a the new series with his “trinary” ideas.


Finally, we wrap up the project by looking at the series 1 + 9 + 9^2 + \ldots + 9^n At this point they’ve seen each of the two prior sums in two different ways and they are able to see how the prior ideas apply to the new series. They even speculate about what a general formula would be!


So, a fun little exploration. I’m happy that the boys had a chance to review their prior solutions, especially since some of the ideas weren’t quite in the front of their mind anymore. Hopefully that review was helpful even if it wasn’t intended to be the main point of the project. By the end of the discussion today they seemed much more comfortable applying their ideas to a few new (though obviously similar) series. It is fun to show them how two ideas that seem pretty different help you work through these problems.

Counting, Pascal’s triangle, and binary numbers

In yesterday’s counting project the boys noticed a connection between a counting problem and binary numbers. Here’s that project:

Revisiting an AMC 10 Counting Problem

For today’s project I wanted to explore that connection an a little more depth. To start off we looked at the connection between counting arrangements and Pascal’s triangle:

In the first part of this project we saw a connection between counting pairings of tourists and guides has a interesting connection with Pascal’s triangle. Here we look more carefully at that connection by trying to understand how the rule that tells you how to construct the rows of Pascal’s triangle also shows up when you count these pairings.

We explored the connection here in two parts. In this first part we show the 6 ways that you can pair 4 tourists with 2 guides when each guide has 2 tourists. We also show the 3 ways to pair 3 people with 2 guides where the first guide gets 1 person and the 3 ways to pair 3 people with 2 guides where the first guide gets 2 people.

Now we are ready to find the connection between the two lists me made in the prior video. That connection is important because it shows that the same addition rule that gives the rows of Pascal’s triangle also applies to counting arrangements of certain sets, and therefore helps you understand why Pascal’s triangle helps you count those arrangements.

In the last part of the project we explore the connection between binary numbers and Pascal’s triangle. We do this using an example of 5 digit binary numbers (from 00000 to 11111). This connection allows you to see that the rows of Pascal’s triangle always add up to be a power of 2.

So, a nice little project showing some fun connections between Pascal’s triangle, counting, and binary numbers. Some of these connections are pretty deep and I certainly don’t expect that the boys will have understood every detail from this project. They did seem to have fun with it, though, and their understanding seems to have come a long way from when we worked through the AMC 10 problem earlier this week.

*Writing 1/3 in binary

We are nearly to the end of our Introduction to Number Theory book. In addition to providing us with many math project that are fun all on their own, I’ve been really happy to find so many great and unusual ways to secretly work number sense. The topic today was writing “decimals” in other bases which gave us lots of time to play with fractions, decimals, and place value.

When I got home from work tonight we reviewed a few of the homework problems and then I gave my son a shot at writing out 1/3 in binary. It was neat to watch him work through this problem, and so cool to see him smile at the analogy to 0.99999…. = 1 in binary at the end of the video. I think the fun of topics like this can draw kids into math while also providing lots of opportunities to build on their basic understanding of arithmetic and numbers.

Adding in binary with Duplo blocks

Several years ago I was talking about adding in binary with my older son.  On a whim we started using Duplo blocks to see how a “binary adding machine” would work.  It was a really fun exercise and I returned to it today with my younger son when we started the section in our book about adding in other bases.

I like talking about adding in binary with Duplo blocks for a couple of reasons.  First, it helps reinforce the idea of place value.  Second, it shows that you can add numbers in bases other than 10 without first converting them back to base 10.  Finally, both of these nice features happen in a setting that is pretty fun and surprising for the kids.

Our project from this morning went pretty well:

So well, in fact, that my son asked if we could do more tonight, so we did this second project.  This time we added 4 binary numbers, or was it 100 numbers for you binary fans 🙂

We’ll cover subtraction the same way, too, which is even more fun – you just need a new color to represent -1.  Can’t wait for that!

A great counting problem for kids involving binary

This summer I’m slowly working through Art of Problem Solving’s Introduction to Counting and Probability with the boys. I though it would be fun to see them working together and since I haven’t covered this subject with either of them previously I was hoping the age difference wouldn’t be that big of a deal. So far so good.

Flipping through the challenge problem section in Chapter 2 earlier this week I ran across this wonderful problem from an old ARML:

If you write the integers from 1 to 256 in binary, how many zeroes do you write?

Definitely too difficult for a homework problem for either of them right now, but it struck me as a great problem to use for one of our weekend Family Math projects. Some parts were more difficult than I expected, but the kids remained engaged all the way through. It was really fun to see them talking through some of the more challenging details as well as listing out binary numbers with the snap cubes. Although this one is a little more challenging and a little longer than a normal Family Math project, I’m really happy with how this project turned out. I think it would be incredibly fun to work through this problem with a large group of kids.

We began by simply discussing the problem which, not surprisingly, meant a short review of writing numbers in binary. My younger son understood that 256 was a power of two and would be written as 1 followed by a bunch of zeroes. How many zeroes exactly was the starting point to today’s discussion:


Next was a neat idea from my older son – to solve the problem we want to break it down into easier cases. His first guess at a way to break the problem down was to organize the numbers by the number of 1’s they have. An interesting idea for sure, and one that works well for numbers that only have one 1. Unfortunately breaking the problem into these cases gets complicated fast. After we do a bit more work with permutations and combinations it might be pretty fun to return to this idea, but for today we walked down this road a little bit and discovered it is a pretty tough way to go.

Since this video is a little long and has nothing to do with how we eventually solve the problem, it is ok to skip it. However, I think it is really important to try out ideas like my son had here. Learning how to identify when an idea is working out and when it isn’t is a really important lesson:

Next we moved to our kitchen to look a little more closely at what binary numbers look like. We used snap cubes to “write” a few numbers in binary. Looking at the list we decided to group the numbers by number of digits and see if that helped us count the zeroes.

At the end of the last video we formed a conjecture that the number of zeroes in our 4 digit binary numbers would be 9. In this next video we write out the numbers using our blocks and discover that we have 12 zeroes in the 4 digit binary numbers rather than 9. We then talked through how we could see the 12 zeroes as 3 groups of 4. The boys struggled a little to see the three groups of 4 that made these 12 zeroes, but eventually saw it and understood it. That led to another conjecture:

After the last video they were really engaged with this problem. We were guessing that we’d find 32 zeroes when we wrote out the 5 digit binary numbers. The boys spent about 10 minutes off camera building these numbers out of snap cubes, and we did indeed find 32 zeroes. We then talked for a bit about why the pattern we found makes sense. After the video was over I mentioned one extra reason that we could see why half of the digits (not counting the left most digit) were zero – the last 4 digits are the same forwards and backwards if you reverse the colors. That helps us see that for every yellow block we have a black block that can be paired with it.

Finally we went back to our big whiteboard to add up the results. Nothing super special going on here – we write down all of the cases and add up the numbers. The kids thought we’d need a calculator, but we somehow managed to add up the numbers without one!

As I said at the beginning, I thought this problem would make a great project to work through with the kids. Although it was a little long, I’m really happy that we worked through this one. Certainly this was one of the most challenging problems that we’ve gone through together, but since the kid remained engaged all the way through, I’m super happy that we gave this one a shot. Again, I’d love to go through this with a large group of kids.

What a fun start to the day today!

How would you teach this problem?

I’m doing a slow walk through Art of Problem Solving’s Introduction to Counting & Probability with the boys this summer. Mainly just for fun, but also because I thought that it would be neat to see them working together.

Peeking a little bit ahead in the book last night I found this problem in the Challenge Problem section at the end of chapter 2 (problem 2.33 on page 47). It is from an old ARML:

How many zeroes do we write when we write all the integers from 1 to 256 in binary?

I like this problem a lot and am planning on using it for one of our Family Math activities this weekend. Before we dive into the Family Math project, though, I thought it would be fun to ask around – how would you approach going through this problem with kids?

Any and all ideas are welcome!

The Joy of teaching my kids

A few weeks ago for our weekend Family Math project we talked about fractions and decimals in binary.    That blog post is here:

These family math project are just for fun.  These projects tend to cover either fun math we find around the house – see the paper folding example from all the way back in Family Math 1:

or, if not stuff from around the house, they are intended to be a fun overview of some advanced math.  The overview of fractions and decimals in binary was supposed to be in the second category, but it led to a really great surprise this morning.

Today with my younger son we moved on to a new chapter in our book – repeating decimals.   A few days ago we had started off talking about decimals and fractions by reviewing why .9999…. = 1, so I was hoping to play off of that to show why 1/3 = 0.33333….  However, when I sat down and asked my son what he thought the decimal expansion for 1/3 would be I got a little surprise:

“I don’t know, but I know what it is in binary.”

So fun that he remembered this talk from the Family Math project from a few weeks ago:

With that, we started down a totally new path – how does knowing what 1/3 is in binary help you understand the decimal expansion?

Such a fun morning!!

Fractions and Decimals in Binary

I returned from a work trip to London this week and started a new math chapter with my younger son – decimals.

One of the most interesting parts of home schooling over the last four years has been learning how to teach some basic math concepts.  Topics such as fractions and decimals are things that I hadn’t really given any thought to in 25 years, and obviously I had never taught either of these subjects previously.  There were definitely some false starts and mistakes the first time around with my older son, and I’m hoping to do a better job this time around.

Last year we did a project that my younger son really liked – creating a binary adding machine out of Duplo blocks.  On the plane back from London I decided that talking about decimals in binary might be a fun way to get him thinking about decimals in general.   So, for our Family Math project today we spent about an hour reviewing binary numbers and introducing fractions and decimals in binary.

The first step was talking about the usual base 10 representation of numbers and reminding the boys about how to represent numbers in binary.  We hadn’t talked about binary numbers in over a year, but it seemed as though they had some memory of the basic ideas:

The next step was reminding them about the binary adding machine.   Although the motivation for the project today was to write some fractions in binary, a quick reminder of some of the basic arithmetic rules in binary seemed like it would be a good idea.


With that review out of the way, we moved on to talking about decimals.  The starting point was base 10 decimals and specifically talking about what the digits after the decimal point represented.   With the base 10 representation written down on the board, it was easy to shift over to talking about decimals in binary.  Talking through the ideas here also turns out to be great review of powers and fractions (who knew . . . ha ha):

Although I promised in the last video that the next step would be looking at 1/3 and 1/5 in binary, that wasn’t actually the next step.   First we needed to do a quick review of how to convert base 10 numbers to binary (and also how to multiply by 2 in binary).  We did this step with our snap cubes:

With all of this review out of the way, we moved on to writing down 1/3 in binary.  Both of the kids thought this was a really fun exercise.  I think they were really surprised to see that the same procedure that worked for integers also worked for fractions:

The last step was converting 1/5 to binary.  The only difference between 1/5 and 1/3 in binary is that the fractions you encounter in the conversion from base 10 are tiny bit more complicated.

After we finished up here, the boys spent a little time figuring out the representation for 1/7.  There’s a neat trick involving multiplying by 8 that helps you see that the representation is right, and they managed to eventually find that trick, too, which was fun.

Definitely a fun morning and hopefully a neat way to introduce decimals.



Fun with James Tanton’s base 1.5

When I was working through basic arithmetic with my kids a few years ago I thought it would be fun to teach them about different bases.  One of the first different bases that we studied was binary, and the way we looked at it was to make a “binary adding machine” out of duplo blocks.

These are two example videos with my younger son:

The kids were amazed that you could use duplo blocks to do simple arithmetic.

While I was in London last week, I saw a really neat post on twitter from James Tanton.   The title alone made me want to see it:  “Exploding Dots:  On base One-and-a-Half”:

In this video he introduces the concept of base 1.5 and talks about a few fun open problems.  It is always such a thrill to see a new idea that is relatively easy to understand.  I couldn’t wait to get back to the US and go through it with the boys.


As an extra bonus, the “exploding dots” method of arithmetic that Tanton talks through is almost exactly how my kids taught themselves to do arithmetic.   Those videos will be fun to go through with the boys, too.