# Using Richard Green’s post about complexity of a number with kids

Saw a another great Google+ post from Richard Green today:

It is so great to see ideas from research mathematicians that you can use with kids. We’ve actually used a couple of Green’s posts before:

Another great piece of math to share with kids from Richard Green

Using a Richard Green Google+ post to talk about geometry with my son

Tonight I was looking for a quick little project since I’ll be on the road for work tomorrow, and talking about the complexity of a few numbers seemed like it would be a lot of fun.

(oh, and sorry for the poor lighting, we were working on the floor of the study while my wife was watching football . . . )

I started with my younger son – he caught on to the idea of complexity quickly and even formed a nice little conjecture about the complexity of prime numbers. The conjecture turned out to be false, but it helped him understand a new way to compute complexity. He even wondered about how you’d compute the complexity of big numbers.

My older son also found computing the complexity of a number to be interesting. We ended up talking for about 8 minutes, so I split the video into two pieces.

He also caught on to the idea fairly quickly and chose to compute the complexity for 10 first. After that he tried 13 and, similar to my younger son, he thought you’d need 13 1’s because 13 is prime. However, with a little more thinking he found a way with 9 1’s and then eventually 8.

In the second half of the conversation I asked him to come up with a few other things to explore. He thought it would be tough to come up with a general formula. I decided to show him one thing that might be a little surprising – if a > b, the complexity of a isn’t necessarily greater than the complexity of b.

After that he wondered what the smallest possible value of the complexity for a two digit number. He had some ideas that led him to believe the smallest complexity would be 7.

To wrap up, we guessed at the value of the complexity for 100.

So, a great post from Richard Green that isn’t too hard to use with kids. It is so fun to be able to share ideas from math research with kids. This idea from Green was especially fun because it got the boys thinking about numbers in ways that are a little different than usual. Amazing what you can do (and what is unsolved!) with just addition and multiplication with 1’s ðŸ™‚

# Revisiting Factorials

Last week I did a fun project with my older son about this problem from the 2015 AMC 10b:

Problem 23 from the 2015 AMC 10b

Here’s the problem:

I wrote about the project here:

A Challenging Factorial Problem

Mostly I don’t miss academic math, but sometimes I do

Today I wanted to revisit the project with the boys so that we could answer one question that my older son had – are there infinitely many positive integers with the property mentioned in the AMC 10 problem. So, are there infinitely many positive integers n with the property that (2n)! ends in three times as many zeros as n! does?

Let’s see . . . we began by discussing the problem and discussing some basic ideas about factorials. Mostly it is my younger son speaking in this and the next video since my older son already worked through the problem:

At the end of the last video my younger son hit on the main idea you need to find the number of zeros at the end of n! – you just have to count the number of 5’s in the multiplication. He expands on this idea here:

Now we started to tackle the AMC 10 problem. Since this part wasn’t the main point of today’s project, we went through this section fairly quickly:

Next we dug in a little deeper to see if we could find other numbers where the number of zeros tripled as you moved from n! to (2n)!.

The boys had some great ideas here and began to think that we’d never see 3x again. In fact, their ideas was that we’d see roughly 2x the number of zeros most of the time.

To investigate the number of zeros in n! and (2n)! more carefully we moved to Mathematica. This part ran a little long (~7 min) but we were able to refine the ideas about the number of zeros roughly doubling.

Finally, just for a fun wrap up, I showed them some other fun factorial facts – what is (1/2)! for example:

So, a fun slightly deeper than usual dive into factorials. I’m happy that my son was wondering how many integers satisfied the conditions of the original problem. It was neat showing him that you could actually answer that question ðŸ™‚