Saw a fun problem in this sequence of tweets today:

One of our mathsjamers @puzzlecritic has brought their puzzle book along (not yet available in shops) it's hard, well test it ðŸ˜‰ pic.twitter.com/PzmzJIP1lB

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.

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.

We started the project today by examining the shape and comparing it to a few other shapes we printed. The comparison wasn’t planned – the other shapes just happened to still be on the table from prior projects . . . only at our house ðŸ™‚

Next we talked about permutations and the basic idea we were going to use to make the permutohedrons. We drew the 1 dimensional version on the whiteboard and talked about what we thought the 2 dimensional version would look like.

We used our zometool set to make a grid to make the 2 dimensional permutohedron. Lots of different mathematical ideas for kids in this part of the project -> coordinate geometry, permutations, and regular old 2d geometry!

Next we went back to talk about how PFF000’s shape was made. Here’s the description on Thingiverse in case I messed up the description in the video:

“The boundary and internal edges of a 3D permutahedron.

The 4! vertices are given by the permutations of [1, 3, 4.2, 7], with an edge connecting two vertices if they agree in two of the four coordinates. The 4D vertices live in a 3D hyperplane, namely the sum of the coordinates is 15.2.
Designed with OpenSCAD.”

This part of the project was a little longer, but worth the time as both the simple counting ideas on the shape and the combinatorial ideas in the connection rules are important ideas:

Finally we wrapped up by taking a 2nd look at the shape and also comparing it to Bathsheba Grossman’s “Hypercube B” which was also still laying around on our project table!

This was a really fun project that brought in many ideas from different areas of math. I’m grateful to Allen Knutson for the tip on this one!

For today’s Family Math project we talked through a classic probability puzzle:

An airplane has 100 seats.Â 100 passengers are going to board and each one has an assigned seat.Â The first person to board ignores the assigned seat requirement, though, and chooses a seat at random (including the 1/100 possibility of actually choosing the correct seat).Â After that, everyone else boards taking their assigned seat if it is open, or choosing a seat at random if their seat is taken.Â What is the probability that person #100 sits in their correct seat?

Here’s how I introduced the problem to the boys:

At the end of the last video my younger son suggested studying a smaller problem first. He picked though the case with 4 people would be easier – and it would! I suggested starting even lower than that, though, so we started with just one person:

After seeing a pattern in the smaller cases we went back to try to tackle the larger case. We had a little bit of confusion, though – and that confusion may have been only on me misunderstanding what my son was saying! – so we cut this movie a bit short to return to the 4 person case:

Returning to the smaller case with 4 people, my son clarified his argument. That argument was, essentially, an induction argument which was really cool! The boys were able to explain how you extend the same argument to the case with 100 people. Nice solution!

At the end we talked about another fun feature of this problem – what are the possible seats that the last person might sit in?

It is always fun to go over a famous problem. This time was an especially nice discussion surprise since the induction argument was an out of the blue surprise! I think this is a fun problem to talk through with kids.

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy ðŸ™‚ If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

[sorry for a hasty write up – had to be out the door by 8:15 this morning . . . ]

Yesterday I saw the latest video from Grant Sanderson, and it is incredible!

I couldn’t wait to share the “fair division” idea with the boys. I introduced the concept with a set of 8 yellow and 8 orange snap cubes. To start, we looked at simple arrangements and just talked about ways to divide them evenly:

Next we looked at the specific fair division problem. We made a random arrangement of the blocks and tried to find a way to divide the cubes evenly with 2 cuts:

To finish up we looked at a few more random arrangements. Some were a little trick, but we always found a way to divide the cubes with two cuts! We also found an arrangement where the “greedy” algorithm from the 2nd video didn’t work.

After we finished the project I had the boys watch Sanderson’s video and they loved it. So many people are making so many great math videos these days – how are you supposed to keep up ðŸ™‚

This Scrooge feels the need to point out that this only proves that the sum of the 3rd & 4th triangular numbers is a square #. #bahnumbughttps://t.co/KR4dALQIaL

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

Kelsey Houston-Edwards released a new math video last week:

So far we’ve been able to use all of her videos for great weekend projects. This video had a fun little surprise because we’d seen the problem she talks about in a (seemingly) totally different context – an old magic set! Once we dug out that out magic set from under my younger son’s bed we started the project ðŸ™‚

Here’s what the boys had to say about the video:

Before jumping to the challenge problems, we looked at the old magic trick:

Finally, we tried to answer the two challenge problems from Houston-Edwards’s video. I’m sorry this got a little rushed at the end – I’d not noticed that we were out of batteries! We finished with about 10 seconds to spare!

The two challenges are great problems for kids to think through – the boys found a few interesting patterns even though the relationship with powers of 2 was a little hard for them to see.

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

In the middle of the book I found a really nice Simpson’s Paradox example and tried it out today with the kids. For more on Simpson’s Paradox, the Wikipedia page is actually great:

So, here’s the first part of today’s project – we have 4 boxes that have fixed amounts of red and blue cubes inside of them. First we divide them into two groups of 2 and ask which one in each group gives you a better chance of selecting a red block. It turns out that this is also a good introductory fraction exercise for kids, too!

Next we see the “paradox.” We combine the two winning boxes into one box and combine the two losing boxes into one box. Now which of the two remaining boxes gives you a better chance of selecting a red block?

So a fun and strange example for kids to see. Again, the Wikipedia page linked above gives a few more fun (and famous) examples. Really happy to have found this example in Moscovich’s book last night!