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.

1. 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 $\alpha$
2. 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 $n$.
3. The point of the game is to approximate $\alpha$ using a sequence that looks like this:

$\displaystyle\frac{\pi^{n}}{b_{1}} + \frac{\pi^{n}}{b_{2}^{2}} + \frac{\pi^{n}}{b_{3}^{3}} + \dots$

where all the $b_{i}$’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:

$(\pi^4)/57 + (\pi^4)/67^2 + (\pi^4)/37^3 \approx \sqrt{3}$.

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 $\frac{\sqrt{2}}{2}$.”  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 $n$) what numbers can we make from these sequences?  What if we also set one of the $b_{i}$’s in the assumption (for example, making $b_{3} = 5$)?]

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).

3 Responses to “Two Turtles Playing with a Hammer: A Game.”

1. Anonymous said

You can get any positive real number, yo!

• James said

Whoops! You’re right. The original game started with $\pi^{n} / 1$ (to start with a zeroth power) and I just kind of left that out. Thanks for pointing that out!

2. Anonymous said

Intresting blog, cheers.