The joy of x^x^x^. . . .

Back in 2013, Steven Strogatz posed this problem on Twitter:

Consider the sequence a, a^a, a^(a^a), etc., where x[1]=a and x[n+1]=a^(x[n]), n>1. For which a>0 does this sequence converge?

— Steven Strogatz (@stevenstrogatz) June 30, 2013

A month or so ago, Shecky Riemann suggested that I take a look at this book to find a few fun projects with my kids:

Today I finally got around to the book. We opened it up to a random page and found this problem on page 77:

Find the value of x that satisfies the equation

So, we stumbled on a variation of the problem that Strogatz was tweeting about 2 years ago. The fact that he was tweeting about it suggests that there might be more going on with this problem than initially meets the eye.

As an aside, I originally intended to talk about the next problem in the book, too, which was the Monty Hall problem. However, our discussion of the infinite tower of x’s was so interesting that it didn’t really make much sense to move on to the second problem. We’ll talk Monty Hall some other time.

We started the project by talking about each kid’s initial reaction to the problem My older son noticed that x would have to be between 1 and 2 because he thought that an infinitely tall tower of powers of 2’s would have a value equal infinity. My younger son was worried that, actually, any infinitely tall tower of powers of numbers greater than 1 would have a value equal to infinity.

Interesting observations – we decided to check and see what happens with 2’s. Estimating the value of was nice little number sense exercise.

 

Now that we’ve talked a little bit about how a tower of powers works, we returned to the original problem. It was fun to hear that my older son remembered that we played around with some similar towers of powers when we played around with Graham’s Number.

My younger son thought there might be a connection to the series 1 + 1/2 + 1/4 + 1/8 + . . . , which was also nice to hear.

Despite some interesting ideas like these ones, we didn’t seem to be getting closer to the answer in this part of the discussion.

 

At the beginning of this part of the project my younger son identifies the main difficulty in this problem: the tower of powers never ends, so how can you evaluate its value? After a little bit of thinking, my older son notices that one interesting thing about the tower is that it doesn’t matter where you start in the tower – you always have to go infinitely high up to find the value.

That’s the critical observation, but it still takes a little bit of discussion to figure out how to use that idea to solve the problem. But . . . eventually they notice that the equation seems to be equivalent to the much easier equation !

 

Having found that seems to solve the equation, we decided to explore that solution on a calculator. I probably should have gone to Wolfram Alpha or Mathematica for this part but I didn’t plan ahead so well. At least the calculator helped the boys see that the tower of powers of did seem to converge to 2.

 

For the last part of the project I wanted to show the boys what happened when you try out a few numbers other than 2 – the idea here is related to the Strogatz tweet I mentioned above. The boys were caught off guard by the seeming paradoxes we ran into. After we turned off the camera my older son said:

“Wait, that means that 5 is less than 2 and that 2 equals 4. I’ll have to re-write the number system.”

I love it when math ideas really bother the boys – it shows that the ideas in the project are really making them think!

So, thanks to Shecky Riemann for the book recommendation. Our first project from the book was definitely fun!