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.

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