## Two Turtles Playing with a Hammer: A Game.

### January 16, 2012

While I’m finishing up the Baire Category post, let’s have a little fun. Here’s a game I like to play when I’m in my office hours and have literally nothing else to do. I call it, “**Two Turtles Playing With A Hammer.” **

The game is simple.

- Pick a number (or, have a friend pick one for you!). It’s more fun if you pick a number that’s a well-known irrational or transcendental number. Let’s call this number .
- Pick a natural number (for example, 4 or 12). Usually, 2, 3, 4, or 6 give the nicest looking results. Let’s call your number .
- The point of the game is to approximate using a sequence that looks like this:

where all the ’s are natural numbers. If this is not possible to do, you lose. Else, you win! [Of course, you can feel free to make your own game where this is an alternating sequence.]

**An example. **Let’s just do one that I played around with a bit:

.

Of course, the game is rather pointless, but you can say to your friends, “Look at this! This crazy sum of stuff is pretty dang close to .” They’ll be impressed.

[**Note:** It may be a good (undergraduate-level) exercise to think about the following: given that we only add terms, and given that the denominators must be elements of the naturals, then (given some ) what numbers can we make from these sequences? What if we also set one of the ’s in the assumption (for example, making )?]

**Edited: **I fixed the note above. Another good undergraduate-level question is: what if we allow only one term to be subtracted; can we make any real number? If they’re particularly good at programming, you could ask them to write a program that approximates any number to some number of places with such a sequence (either from the game or using a single subtracted term).

You can get any positive real number, yo!

Whoops! You’re right. The original game started with (to start with a zeroth power) and I just kind of left that out. Thanks for pointing that out!

Intresting blog, cheers.