On his blog A Blank Slate, Vishal Patel posts a cute little brain teaser (with a hat tip to the Cosmic Variance blog):
Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?
(a) Yes
(b) No
(c) Can not be determined
This reminded me of one of my favorite little “zinger” math proofs. (If you think about the brain teaser long enough, you’ll see the connection.)
The proof demonstrates that an irrational number to an irrational power is at least sometimes rational.
Proof: Write x for √ 2 √ 2 . Then you can check that x√ 2 =2.
Is x rational or irrational? That’s a hard question, but fortunately we don’t need to know the answer. We only need to know that the question has an answer. Because:
a) If x is rational, then the equation √ 2 √ 2 =x demonstrates that an irrational to an irrational can be rational.
b) If x is irrational, then the equation x√ 2 =2 demonstrates that an irrational to an irrational can be rational.
Either way, we win. And the cool thing is that we can know we’ve won without having the slightest idea how we’ve won.
