There are many problems with no one answer.
For example, I remember a professor in college throwing this out: "how many ping pong balls would it take to fill this lecture hall?"
Some people shouted out things like "a million", "a billion", but he was trying to get us to think abou the problem, not simply guess.
"How big is a ping pong ball? About an inch in diameter. ABOUT how many in a cubic foot, then? 12 is close to 10 so 1,000 to a fair approximation. Don't worry too much about how they're packed."
"I'm about 6 feet tall, the ceiling is about three times my height, so how high is the ceiling? About 18 feet, call it 20. The floor tiles are about a foot square, so how wide and long is the room?" About 60 by 100 feet.
So 20 x 60 is about 1,000 (we over-estimated the height and under-estimated the width). Multiply that by 100 and we get 100,000 cubic feet. Each cubic foot holds about 1,000 ping pong balls, so we estimate about 100 million balls.
That's certainly not the exact answer but thinking about a problem like that helps one figure out when something just doesn't make any sense. I've see many "cool inventions" where a simple estimate blows holes in the claims... "Let's put spring loaded generators in roadways in front of turnpike toll booths. The cars driving over them will provide all the power for them..."
Or "solar freakin' roadways...", which seem to have recently been popularized on the web. Being able to estimate, not getting any one particular "correct" answer, but within an order of magnitude or so is a very useful skill.
In the article, being able to rank "unlikely" "50% chance" and "will I go to the beach?" is a good skill to have so one can visualize and discuss a problem.
Beyond all that, Kurt Gödel's Incompleteness Theorems mathematically demonstrated that some problems can provably have an answer that can't be determined, some statements are provably both true and false, and other mind blowing concepts. Douglas Hofstadter's excellent book Gödel's, Escher, Bach; an Eternal Golden Braid provides a wonderful explanation and contemplation of number theory and it's limitation.