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.
Hey I did that problem for HW one time! You could also have titled this post “predictably rational.”
Always fun to appropriate mathematical terms:
http://twocentsapoint.blogspot.com/2009/11/unions.html
Wow! Not only could I follow the math proof, and it’s been a long time…, in one of those enjoyable flashes of insight, I changed my mind on the answer to your original question!!
Wolfram Alpha seems to think that the result is transcendental…
(sqrt(2))^sqrt(2)
or did I misunderstand?
toss255: Yes, the result is transcendental (so it’s irrational). But that’s hard to prove. The point is that we don’t need to know this in order to get the result we want.
It’s been way too long – what does the line above the 2 mean? doesn’t it mean it repeats indefinately?
Sorry – been alot of drinks since high school Maths
Dave: The line above the 2 is part of a square root sign. I hope it shows correctly in all browsers.
Gotcha – looks weird on my browser. All makes sense.
Thanks Steve
PS will you be doing a book tour at all?
As a complete moron when it comes to math, what on God’s (the spaghetti monster of course) green earth does all this mean?
What is the answer? Is a married person looking at another married person OR NOT? My initial answer was “Oh well it’s a trick because they don’t tell you if Anne is married OR if George is looking at Jack.” Expecting to see something like that, there is this mind boggling (to me) math proof. What the heck does it mean in “common sense” terms?
and (sqrt(2n))^sqrt(2a)^sqrt(2b) is rational when ab is a perfect square?
Chris: you don’t need to know whether Anne is married, and George may as well have her eyes closed, it doesn’t affect the answer. Either Anne is married or she isn’t – in each case consider whether or not a married person is looking at an unmarried person.
Steven: If Anne is British, and in a civil partnership, is she (a) married; (b) unmarried; (c) either a or b, but the proof still works; or (d) cannot advise since there is no suitable precedent in law? ;-)
Actually, the correct answer is (c), since the problem does not require that Anne be a person. She could be a cocker spaniel, parakeet, etc. When she is one of those, a married person is not looking at an unmarried person.