# Struggling thorough a challenging counting problem

I tried a pretty tough counting problem with the boys last night. For the first part of our discussion they were accidentally heading down the wrong path without realizing it. It is a good lesson on how easy it is to accidentally count the wrong thing.

Here’s the problem:

How many 4 digit integers are missing at least one of the digits 0 and 1?

Here’s the introduction to the problem and the walk down the wrong path:

At the end of the last video we came to the conclusion that we’d gone wrong. Our guess was around 700, but the answer definitely had to be bigger than 4,000. What happened?

I think the process we went through here is a good lesson – it is really easy to count the wrong thing by accident. You don’t always get as lucky as we did here and count *exactly* the wrong thing!, though. But it still is nice to see how going back and reviewing your solution can help you get back on the right track.

# Sharing Numberphile’s Collatz Conjecture video with kids

Numberphile published a beautiful video on the Collatz Conjecture today. I thought it would make for a fantastic project with the kids tonight:

We have looked at the Collatz Conjecture before, so we aren’t starting from scratch here. Two of our prior projects are here:

Revisiting the Collatz Conjecture

the Collatz Conjecture and John Conway’s Amusical Variation

I started the project tonight by asking the kids what they thought was interesting about the video:

Next we tried to recreate the “tree” that was in the video. This exercise was a nice way to check that the kids understood what was going on in Numberphile’s video:

To wrap up I wanted to walk through one example of how the Collatz conjecture plays out. Somewhat unluckily, though, my son chose 31 as the starting point. 31 takes more than 100 steps to converge!

BUT, this video shows why I think the Collatz conjecture is such a fun math idea to share with kids – you can sneak in a lot of arithmetic practice ðŸ™‚

So, we gave up after maybe 30 steps in the last video and went to check how long it would take to converge using Mathematica. Someday I’ll learn that when I zoom in too far on Mathematica the video gets super fuzzy . . . but today was not that day ðŸ˜¦

I’m really grateful to Numberphile for their video – I think videos like it will really help show off the beauty of math to a large audience.

# Returning to inclusion / exclusion

We’d taking a break from our inclusion / exclusion project but I wanted to return to it tonight. I picked a fairly challenging problem from one of my old math books:

How man 5 card poker hands have at least one card from each suit?

I didn’t have any idea how it would go . . .

We started by reviewing the ideas in inclusion / exclusion and the moved on to try to get our bearings in the problem:

Having formed a pretty good plan in the first video, we moved on to tackling the rest of the problem.

I’m really happy with how this went. It is fun to see the boys learning to break complicated problems down into problems that are slightly easier to deal with.

# Revisiting a counting project for kids that I learned from Jim Propp

For our math project today we returned a tiling idea that is a really fun idea for kids to explore. Here are a few of our prior projects on the subject:

A fun counting exercise for kids suggested by Jim Propp

Counting 2xN domino tilings

Today the plan was to look at 2xN tilings first and then move on to tilings of 3xN rectangles with 3×1 dominoes.

We stared by exploring some simple 2xN cases and looked for patterns:

In the first video we counted the number ways that we could tile 2×1, 2×2, 2×3, and 2×4 rectangles with dominoes. Now the boys noticed the connection with the Fibonacci numbers and we tried to find and explanation for why the Fibonacci numbers seemed to be showing up here. The nice thing is that the boys pretty much got the complete explanation all on their own.

Now we moved on to counting the tilings of a 3xN rectangle using 3×1 “dominoes” – what would be different? What would be the same?

One really interesting thing here is that my older son and younger son came up with different ideas for how to count the general arrangement.

So, in the last video my older son had a counting hypothesis that I couldn’t quite understand. In the beginning of this video I have him explain his process more carefully. The surprise was that for the 3×6 case we were looking at next both of their counting procedures predicted the same number of domino tilings.

In this part of the project we tried to follow both procedures to see how they worked.

Having sorted out the counting procedure in the last video, we now looked carefully at the 2xN and 3xN tiling procedures and saw that we could compute the number of tilings for the 2×100 and 3×100 cases if we wanted to.

I’d love to come up with more counting projects for kids. These projects are accessible to young kids and I think shows of some really fun ideas from advanced math that kids probably don’t usually see in school.

# Learning 3d geometry with Paula Beardell Krieg’s pyramids

Earlier in the week we got a nice surprise when we received a fun little pyramid puzzle from Paula Beardell Krieg:

Our initial project using the shapes is here:

Playing with an amazing present from Paula Beardell Krieg

I thought a follow up project would be fun, so I decided to try out a basic exploration in 3d geometry. The goal was to make these shapes ourselves using Mathematica, then to 3d print them, and finally to play with the new shapes to see that they were indeed the same.

We started by talking about the shapes in general and see if we could identify some very specific properties of the shape using coordinate geometry:

Next we talked about how to describe the planes that formed the boundary of the shape. It was fun hearing my 5th grader try to figure out how to describe the planes (and regions) we were studying here. One other challenge here is that we were also trying to describe the 3d regions above and below these planes.

Now came the special challenge of finding a mathematical way to describe the hard to describe plane in the shape. I had to guide the discussion a bit more than I usually do here, but the topic of finding the equation for a plane is pretty advanced and something that kids have not seen before.

Having written down the equations, we went up to look at the Mathematica code I’d used to make the shapes. The boys were able to see that the first shape had exactly the same equations we’d written down, and they were able to see that the equations for the 2nd shape were not any more difficult.

The shapes printed overnight and we had an opportunity to play with them this morning. It is pretty neat to hear them compare the shapes and see that, indeed, the shapes we made are really the same as the shapes Paula sent us.

So, there’s quite a lot we can study with Paula’s shapes. You’ve got the potential to study folding patters, basic 3d geometry, the volume formula for a pyramid, and even 3d printing! Fun how such a seeming simple idea can lead you in so many different directions.

# Playing with an amazing present from Paula Beardell Krieg

We occasionally get interesting packages in the mail from Paula Beardell Krieg. For example:

Paula Beardell Krieg’s Intersecting Squares

and

Paula Beardell Krieg’s Puff Boxes

Paula Beardell Krieg’s Puff Boxes Day 2

Today we got a new one!

I opened the top of the box before we got started, but only the top. I figured opening the actual box wouldn’t be that interesting on camera, but what was inside . . . .

There was a 2nd smaller box that we’d originally moved off to the side – what was in that??

Very fun! You definitely have to follow Paula on twitter to see where all of these amazing shapes come from!

# Playing with shapes of constant width

Tonight’s project was just for fun – they boys both had long days of school / music.

So, last night I downloaded and printed some shapes of constant width from Thingiverse:

Anenome’s Object of Constant Width on Thingiverse

With 4 of them on the table I asked each of the boys what they thought the shapes were and then let them play around with them. After they played for a bit I put a book on the shapes and asked them how they thought the book would move as the shapes rolled.

Here’s what my younger son thought:

Here’s what my older son thought:

I always find it fun to hear what kids think about complicated shapes. Lots of neat ideas and then a good “wow” when you learn the secret property!

# A neat counting problem I saw on Twitter today

Saw a fun problem in this sequence of tweets today:

Seemed like a great problem to cover with the boys tonight. They had some pretty good ideas right from the start:

Once they had the idea for the first 100 numbers down we moved on to trying to find the 1000th number with no 5’s or 7’s. Their approach was a little different from how I would have proceeded, but it was nice to see their idea. They also noticed and avoided a little trap right at the end ðŸ™‚

Really happy I saw this problem today. Excited to get the book when it comes out.

# A challenging but super instructive inclusion / exclusion example

My son had a really interesting problem as part of the homework for an enrichment math program he’s in. I’m writing this post from the road so I don’t have the exact statement of the problem in front of me, but it went something like this:

You are going to make 7 digit numbers using the digits 1, 2, 3, . . . , 7 exactly once. How many of these numbers have no consecutive digits with common divisors?

So, for example 1,234,567 is a perfectly fine number, but 2,413,567 doesn’t work.

My son’s solution was nice, but complicated. He found the number of ways to separate the even numbers (there are 10) and then found the ways to fill in the odd numbers in each of those cases.

I couldn’t find an easier solution and wondered on Twitter if there was one. One response I got pointed me to a similar problem that was discussed on the Art of Problem Solving problem forum:

Looking through the thread I stumbled on a really clever inclusion / exclusion solution. Since we’ve been taking a closer look at inclusion / exclusion ideas I thought it would be fun to step through this solution with the boys. I think this a really instructive inclusion / exclusion example. One thing that was a little tough for the boys to understand was that the elements we were “excluding” were pairs of integers.

Also, just to be clear, I’m not expecting the boys to have a complete understanding of this solution. Rather, I just wanted to show them an inclusion / exclusion example that had some interesting twists.

So, we started by introducing the problem because my younger son hadn’t seen it before:

Next we dove in to the inclusion / exclusion solution. The “no restrictions” case is easy! Seeing the way to express the restrictions is pretty challenging. Once we understood that case we looked at subtracting away the cases with 1 restriction.

Next we looked at the 2 restriction case. Now things get really tricky – the fact that we have now have pairs of pairs of numbers is one bit of confusion. Another bit of confusion comes because one pair of pairs is not like the others.

Finally we looked at the case with 3 restrictions. This part, I think anyway, is really cool. The surprise is that several of the cases are impossible!

Despite being a very challenging problem, I love this problem as an inclusion / exclusion example for kids. No individual piece is beyond their reach and if you walk through the problem slowly everything is accessible to them.

# Extending our arithmetic / geometry connection project to calculus

Yesterday we did a fun project connecting arithmetic and geometry:

Connecting Arithmetic and Geometry

While we were talking about the shapes my older son commented that one of the shapes looked like a pyramid. I thought it would be fun to make the shapes look even more like a pyramid and see what the kids thought.

We started by just talking about the shapes – the most interesting thing to me here was how challenging it was for them to compare the volumes of the shapes:

Because they were having a little bit of difficulty with the volumes I spent a little extra time on the idea. Things seemed to clear up a little bit, luckily:

Finally, I thought it would be interesting for the boys to see some of the math I used to create these shapes. Although this section goes on a little longer than I would have liked, I think this is a fun little introduction to functions and scaling even if we don’t define those ideas explicitly:

A fun little project. I think that some of the broad ideas from calculus are within the grasp of kids even if the underlying calculations probably aren’t. It was fun for me that a question from my older son led from us jumping from arithmetic to geometry to calculus ðŸ™‚