Larry Guth’s “No Rectangles” problem

[note: sorry for the rushed write up – I wanted to get this written up before 9:00 this morning because of a work project that’s going to have me tied up for most of the rest of the day]

Yesterday I was lucky to attend a lecture that Larry Guth gave at MIT:

Larry Guth’s “No Rectangles” problem

It was sort of doubly lucky because I’d already set up a meeting in Cambridge and a little bit of juggling with some times allowed me to have the meeting and attend the lecture. The problem that Guth discussed has the wonderful property of being simultaneously accessible to kids and interesting to research mathematicians. His talk began with a discussion about a simple 3×3 grid and ended with a discussion of intersecting lines in the field Z mod p! Today I used some of the basic ideas in Guth’s talk as the basis for a fun little project with the boys.

I really loved talking through Guth’s problem with them. As often happens in our projects, we ended up going in a slightly different direction than I’d anticipated – but the discussion we had as a result was also much better than I’d anticipated!

We started with a quick introduction to Guth’s “no rectangles” problem and the boys tried to work through the 3×3 case:


The first part of the project ended with the boys finding a way to place 6 snap cubes on a 3×3 square without forming a rectangle. Now we try to see if 6 is the maximum number or is there a way to place 7?

They approached this question in a way that surprised me a little. The first thing they wanted to do was count the total number of rectangles in the grid (which has one subtle point). This approach turned out to be a pretty interesting way to look at the original problem. It is also a pretty interesting counting problem for kids all by itself:


So, having counted 9 rectangles in the 3×3 grid, the boys then tried to see if they could eliminate all 9 of these rectangles by taking away just two of the snap cubes. The conversation here was super fun – lots of great opportunities for the kids to talk about patterns and explore different ideas. For example, my younger son noticed that you are forced to take away one of the cubes on a corner of the 3×3 grid.

In the 2nd half of this video I show them a different way to see that you can’t have 7 cubes on the grid without forming a rectangle. This approach – which is the way Guth explained it in his talk – involves the pigeonhole principle.


For the next part of the project I thought it would be fun for the kids to try to figure out the total number of different ways to place snap cubes onto an NxN grid. Thinking through this problem helps you understand that trying to work through the “no rectangles” problem by brute force with a computer is going to take a long, long time.

Just like the rectangle counting problem the boys came up with in the earlier part of the project, this problem is a nice counting problem for kids all by itself.


Now that we’ve found that the total number of ways to put the snap cubes onto an NxN square is 2^{n^2}, we try to understand the size of some of these large powers of 2. We start by finding an approximation for 2^{25} – the number of ways to put snap cubes on a 5×5 grid – and then try to see if a computer could handle a number this large.

What about a 10×10 grid, though? This problem ends up bringing to the surface a little problem understanding the exponential notation, but that’s one of the great things about this project – you have a nice opportunity to talk about / review exponents while talking about a really interesting problem:


For the last part of the project we try to understand how long 10^{21} seconds is. It is actually so long that there’s probably no way for kids to have an intuitive understand of that many seconds. However, we can try to convert it to other units of time, and years seemed like a pretty natural choice. We found that if you can check 1 billion squares per second, searching through all of the possible ways to put snap cubes on a 10×10 grid would take about 300 trillion years!!

Since that’s more time than we’d probably want to spend on that problem, you need to look for a different approach. One really cool thing about Guth’s problem is that there’s a surprising (to me!) connection to geometry and number theory hiding beneath the surface. Guth ended his talk by explaining how to solve the problem using those ideas. That approach, by the way, is truly marvelous but there wasn’t enough room in the margin of my white board to explain it . . . . 🙂


I love being able to share projects like this one with the boys. There aren’t too many problems that are both interesting to research mathematicians and accessible to kids. The kids were able to understand the problem right way and had lots of interesting ideas about the 3×3 case. Exploring the 4×4 case would probably be lots of fun for kids, too. It always amazes me how much fun you can have and how far math conversations with kids can go when the problems keep them engaged. It really was a great bit of luck to have had the opportunity to attend Guth’s lecture yesterday.

A challenging number pattern problem

My younger son struggled with a number pattern problem from an old MOEMS test today. I enjoyed talking through it with him tonight because it was interesting to see how he approached the pattern in the numbers once he saw it – his approach was quite a bit different that what I was expecting.

Here’s an introduction to the problem and our initial talk that gets us on the path that surprised me:


So, my surprise in the last video is that he wanted to go to the end of a row and subtract a certain amount to get back to the beginning. I thought it would be interesting to see if he could see that you could also add 1 to the square at the end of the last row. This idea was hard for him to see, but eventually we got there.


At the end of the last video we talked about how the odd numbers relate to the perfect squares. The sequence of rows in the original problem hints at the relationship, though for me, at least, the connection doesn’t jump off the page. To get a better sense of that relationship we went to our kitchen table and looked at the relationship using snap cubes:


So, a fun little project starting from an old math contest problem. Ultimately the lesson I’m hoping to convey with my son here is about looking for patterns. The connection between arithmetic and geometry in the last part is also something that I hope he finds interesting. I always find it fun when geometry helps us understand arithmetic a little better.

One of my all time favorites – The McNugget problem!!

This morning my younger son had some trouble with a problem on an old MOEMs test that is similar to the famous Chicken McNugget problem. Similar enough, actually, that I decided to put off our second day of divisibility rules until tomorrow in order to spend our time today talking about this problem.

Given that we just finished up a section on modular arithmetic, there is a bit of a connection to what we’ve been studying, but I mostly just wanted to talk McNuggets!

First up was talking through his approach to the MOEMs problem this morning. In this video we do get to the answer by checking every number until we get a bunch in a row. I wanted to let him talk through this approach just to show him that he really could solve this problem on his own.


Next up was an approach to the problem that was a little more systematic and also brought in some ideas about pattern recognition. When we write out our grid of numbers my younger son notices that above 13 you can just keep adding 3’s to one of the rows and get all of the remaining numbers.


The last part of talking about the MOEMs problem was bringing in modular arithmetic. The ideas here help us see why we could just keep adding 3’s in the last step:


Finally, the punchline – the Chicken McNugget problem. This one is a tiny bit more difficult because there are 3 sizes rather than just the 2 from the MOEMs problem, but we do manage to get to the end using basically the same ideas we talked about earlier in the project.


So, an accidentally fun morning with one of my all time favorite little math puzzles 🙂

Angry Birds and Snap Cubes: Using Bryna Kra’s MoMath public lecture to talk math with kids

Last night I watched Bryna Kra’s public lecture at the Museum of Math:


I’m not quite sure how to talk through some of the simple dynamical system ideas in the lecture, but the earlier material about patterns and the pigeonhole principle are definitely fun topics to talk about with kids. We used our collection of snap cubes and Angry Bird stuffed animals as props 🙂

In the first part of the talk we introduce the pigeonhole principle and talk through a simple pattern with only single blocks based on one of the elementary patterns Kra uses in her talk. This simple pattern allows us to get a little bit of practice identifying the “pigeons” and the “pigeonholes” in a problem:


In the second talk we look at a slightly more complicated pattern – patterns you get with two blocks rather than one. For this pattern we consider the order of the birds to be important – so a (red, blue) group is different than a (blue, red) group. The example we look at in the last part of today’s talk will consider those two groups to be the same.

The boys were able to see the four different patters that we could make with the two birds / blocks. My older son even noticed a connection with Pascal’s triangle which was fun to see. We then talked about how to count the different types of pairs by looking at the number of choices we had for the first bird and for the second bird. That led my younger son to wonder if there would be a total of 9 groups of two birds if we allowed three different birds in the pattern. Pretty fun discussion:


At the end of the last talk my younger son wondered what would happen if we used three different colors of blocks rather than two. I hadn’t planned on discussing that problem, but what the heck! It was interesting to see the kids figure out how to group the blocks to make the 9 pairs. They were also now able to see how the patterns would continue if we varied the colors and/or number of blocks in the pattern. Fun little exercise. Watching this again I wish I would have spent a little time responding to my older son’s comment that there was no connection to Pascal’s triangle pattern anymore – oh well, next time!


Our last project was a slightly different twist on the Pigeonhole principle. We looked at a tournament involving 4 birds in which each game involves 2 birds. The question I had the boys look at was this: If there are 7 total games played in this tournament, show that at least two of the games must involve the same two players.

I liked their approach to solving this problem. Their instinct was to solve the problem by listing out all of the types of games that could happen. If we were at our whiteboard I would have drawn a square with its sides and diagonals, but their list of all of the types of games was good enough for this project. They had a little difficulty identifying the pigeons and pigeonholes here, but that’s ok, it isn’t always so obvious how to make that identification.


So, a fun project based on another MoMath talk. See here for our last project based on a MoMath lecture:

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids

I think the public lectures at the Museum of Math are a great way for kids to see some amazing math. There will surely be some lectures that are too advanced for young kids, but many of these lectures have ideas in them that are not hard at all for kids to understand. With Bryna Kra’s lecture, the ideas about patterns and the pigeonhole principle are topics that kids can play with and really enjoy. I’m super glad that MoMath is making these lectures available to the public. It is really fun to show kids some ideas that professional mathematicians use in their research, and hopefully also a great way to inspire a new generation of mathematicians!

Jo Boaler’s exercise with snap cubes

[note:  sorry this is a little rushed – trying to get this published before heading up to Boston for the weekend]

Earlier this week I saw a new video posted from Stanford Math Education professor Jo Boaler:

The video (which hopefully you can click to in the above link) includes a Fawn Nguyen-like counting exercise that I thought would be fun to try with the boys.  For many similar exercises, check out Fawn’s amazing site:

We started off just talking through the problem.  In retrospect I should have set up the shapes first, but these are the things you don’t think about at 6:30 in the morning!  It was interesting to hear both of the kids try to explain the pattern in words.

I’d intended to proceed as Boaler does in her video, but my older son happened to notice hat the number of blocks in each step was a perfect square.  My younger son also picked up on that fact quickly, so we jumped into talking about the squares right away:

Talking about the squares led to a slight diversion to see if there was a different geometric way to see that 1 + 3 + 5 + 7 + . . .  (odd integers) always adds up to be a perfect square.  This is something that we’d talked about previously, but talking through that point one more time felt pretty natural here:

Finally, we returned to the original problem and looked for new ways to think about the patterns we saw there.  I thought that the last videos sort of got the kids anchored to thinking about the pattern in rows, so I switched to having the shapes all have the same color blocks.  Probably should have been doing that from the beginning.  Sorry for the slight goof up at the end of this video, but at least we caught the mistake!

All in all, a fun problem.  We’ve done many similar little project before, so this type of problem isn’t completely new for the kids – see here for just one example:

Definitely a fun morning before heading out on a quick trip.


Fawn Nguyen’s influence

A few weeks ago I asked the boys what math they wanted to do for the summer.  They decided on a summer project that they could work on together rather than the separate algebra / prealgebra math we did during the school year.    Not the easiest task in the world given the 2.5 year age difference, but I ended up settling on a slow walk through Art of Problem Solving’s “Introduction to Counting and Probability.”    Some parts of the book might be pretty challenging (and skipped for now!), but it looks like we’ll have a fun summer of basic combinatorics.

Book Pic

I got a nice surprise right off the bat and it showed quite directly the value of all of the work we’ve been doing with Fawn Nguyen’s material.  Particularly the visual patterns:

The first section in the book discusses counting lists of numbers.  One of the slightly complicated problems was counting the numbers in this list:

3 2/3, 4 1/3, 5, 5 2/3, . . . , 26 1/3, 27

My older son’s solution had a really nice grouping strategy that surprised me and clearly showed Fawn Nguyen’s influence:

Definitely going to be a fun summer!

Pascal’s Triangle and Powers of 11

The most difficult thing about teaching my kids, by far, has been that I have no experience at all teaching elementary math.  When a concept is difficult for either of them to understand, quite often I struggle to work out exactly what it is that they struggling to understanding.  But we muddle along.

One obvious consequence (and sometimes it is a “feature” and other times a “bug” !!) is that I have no idea at all about the accepted ways to teach some of the most basic subjects.  Fairly often I just let them figure it out the elementary stuff on their own.  Basic arithmetic is a good example of a subject where my older son came up with his own methods (which have nothing to do with borrowing and carrying).  Since his ideas were perfectly fine mathematically, I just ran with ran with them.   Here are three examples that illustrate his method:

(1)  356 + 672 = 900 + 120 + 8 = 1028,

(2)  532 – 384 = 200 – 50 – 2 = 148

(3) 25 * 13 = 200 + 60 + 50 + 15 = 325

I think the main disadvantage of this way of doing arithmetic is that it is a little slow, but there are at least two nice advantages.  First, the approach highlights place value and therefore made it very easy to talk about arithmetic in other bases.  Second, this method looks pretty similar to the way we normally do arithmetic in algebra, so learning to multiply polynomials, for example, was not particularly difficult.

Yesterday I got a nice surprise when we stumbled on a new problem where this arithmetic method added a lot of value. We were doing some review work and one of the questions was simply to find the cube root of 1,331.  My son told me that he new the cube root of 1,331 was 11 because 1,331 was a row of Pascal’s triangle.

I love opportunities to talk math that come out of the blue, and this was as good an opportunity as any.  I’m not actually sure where the connection between powers of 11 and Pascal’s triangle came from since we haven’t talked about Pascal’s triangle in a long time.  However, as luck would have it, we’d spent the last couple of months talking about polynomials, so we were primed for a fun discussion.  It turned out to be even better than I’d hoped!

The first question I asked him was what he thought 11^4 was.  He drew out Pascal’s triangle and said he thought that 11^4 would be 14,641.  Fine, but the next row of Pascal’s triangle is 1 5 10 10 5 1, so what would 11^5 be?    Since the method of finding powers of 11 using Pascal’s triangle now appears to break down, he proceeded to calculate:

14,641 * 11 = 146410 + 14,641 = 100,000 + 50,000 + 10,000 + 1,000 + 50 + 1 = 161,051.

My plan, of course, why the connection to Pascal’s triangle has disappeared, but his unusual method of addition meant that the coefficients of Pascal’s triangle were right there on the board!  Ha ha, the joke was on me.  We revisited the calculation this morning:

Following that discussion this morning, we spent a few minutes connecting polynomials to Pascal’s triangle and showing why the powers of 11 are hiding inside the triangle.  Definitely a fun and surprising weekend of math!!

For me moments like these have always been the best part of teaching.  As I’ve said many times, I’m glad that I have the time and flexibility to teach my kids.

Another great problem from Fawn Nguyen (2 of infinity)

Earlier this week Fawn Nguyen posted a nice conjecture from one of her students on twitter:

I love Fawn’s approach to teaching kids math and try to blatantly steal incorporate her work into what I’m doing with my kids.  This problem had so many possibilities and I put it in my back pocket for the weekend.
The reason that I didn’t use it right away is the fraction / decimal component seemed like it would be too distracting for my 7 year old, so I wanted to take a day or two to think through how to present it.  There was, however, a similar problem that avoided the fractions that we played with right way -> multiplying 3 consecutive integers.  Here, instead of being nearly a square, you get a product that is nearly a cube:




We had a lot of fun building these shapes and talking about the pattern.  In retrospect, I wish I would have asked about 0 x 1 x 2 as well.

As an aside, building these shapes got me thinking a little more about connecting geometry and arithmetic and the next day we had a really fun talk related to the geometry of imaginary and non-commutative “numbers” :

Anyway, we circled back to Fawn’s problem this morning.  I’d toyed around with the idea of modifying the problem to talk about the Arithmetic Mean / Geometric Mean inequality, but thought that would be a little too ambitious.  Finally I decided that a better lesson than AM / GM, as well as the easy way to avoid the complexity of decimals, was to talk about multiplying integers that differed by 2 rather than 1.  We talked through the original problem, moved on to the slightly modified problem, and then talked through some of the algebra.  Made for a really fun morning.  Thanks for another great lesson, Fawn!