# A neat game shared by Tracy Johnston Zager

Saw this tweet from Tracy Johnston Zager earlier today:

The game looked fun, so I thought I’d turn it into a project with my younger son.

Also, as a note, I haven’t really studied much about games, but this game looks a lot like the game of Nim. It may even be exactly on online version of the game of Nim, I’m not sure. I didn’t look into Nim, though, rather I just watch my son play with this game.

I had him read through the rules while I set up the camera. Here are his initial attempts to play the game and his thoughts as he played. He begins to notice some losing positions and explores a few strategies to try to win.

In the second bit of the project I asked him to think about how to analyze this game. His first thought is to work backwards because he know the scenario with two piles of just one drop each is a losing situation if it is your turn.

He also thinks that watching how the computer plays would be good way to learn. The computer does seem to win a lot!

Then we got lucky – I was searching for a game with just two columns of drips and in the set up we found, he saw an easy way to win! It turns out the computer doesn’t win every time! Winning this game creates a few other ideas for how to win the two column game.

For the last part of the project we tried a few three column games. He tries out a few ideas at the beginning that turn out not to work. However, after a bit of experimenting he’s able to get a win!

So, a really fun game to play and a great game to get kids talking about strategy. I liked simplifying the game to two and three columns to narrow the potential strategies, but I’m sure that playing whatever board showed up on the screen would be fun, too.

# Divisor counting problems

Last night my older son asked me for a little help understanding divisor counting problems. He was struggling a little bit with that type of problem at his math club.

Luckily Art of Problem Solivng’s Introduction to Number Theory book has an entire section on divisor counting problems, so it wasn’t difficult at all to find some good problems. We worked through four of them.

Divisor counting is one of my favorite topics – I remember learning about it in high school and being amazed that you could know the number of factors of an integer without listing all of them. That, all by itself, is a neat basic counting idea. Another reason that I like the topic is that it helps me sneak in a little arithmetic practice with the kids, and you’ll see why I like that practice in the first two videos.

Here are the four problems:

(1) Find the number of positive divisors of 999,999.

(2) What is the sum of the three positive numbers less than 1,000 that have exactly 5 positive divisors?

(3) If $n$ has two prime divisors and 9 total divisors, how many divisors does $n^2$ have?

(4) How many divisors of 3,240 are (i) multiples of 3, and [separate question] (ii) how many are perfect squares?