There's no "hard way" of solving a problem --- an anecdote about John von Neumann as told by Eugine Wigner.

The following problem can be solved either the easy way or the hard way.

Two cyclists 40 miles apart are riding toward each other on a straight track; each one is going at a speed of 20 miles per hour. A swallow starting above one of one of them flies back and forth between them at a rate of 50 miles per hour. It does this until the cyclists meet. What is the total distance the swallow has flown?

The swallow actually flies back and forth an infinite number of times before the cyclists meet, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the cyclists are 40 miles apart and each cyclist is going 20 miles an hour, it takes one hour for the cyclists to meet. Therefore the swallow was flying for one hour. Flying at a rate of 50 miles per hour, it must have flown 50 miles. That's all there is to it.

When this problem was posed to John von Neumann by Max Born, Neumann immediately replied, "50 miles!"
"It is very strange," said Born, "but nearly everyone tries to sum the infinite series."
"What do you mean, strange?" asked Von Neumann. "That's how I did it!"

Source: John von Neumann Documentary starting at approximately 18 minutes into the film.
Based on the version of the anecdote from "Math Jokes".

Mad Hatter Paradox.

DEFINITION: Mad as a Hatter  – someone is said to be mad as a hatter iff there exists a mental illness (madness) from which they suffer, and they’re ignorant of its existence.

Now, consider someone asserting “I know that I’m mad as a hatter”. Call it the Mad Hatter sentence, and denote it with MH.

Is MH true or false?

Suppose MH is true. So it’s true that the person uttering it knows that they’re mad as a hatter. But by knowing that they are mad as a hatter they’re aware of the illness they purport to be suffering from, i.e. being mad as a hatter and so by definition fail to satisfy the conditions for being mad as a hatter. Hence MH is false, it seems.

But if they are not mad as a hatter, and assert a knowledge of being such, the person asserting MH is oblivious of that ignorance (them in fact not being mad as a hatter), a delusion of sorts, and hence qualifies them for being mad as a hatter, thus rendering MH true - but we know where that leads.

Who is lying?

Here's a rather simple and fun logic puzzle.

We have three people; Alice, Bob and Cecil. One of them is a liar. You have to determine who is the liar, and give reasons for your choice, based on the following information. Alice claims that Bob is a liar. Bob claims that Cecil is a liar. Cecil claims that both Alice and Bob are liers. Who's lying and why?

Answer in comments.

Logic Puzzle: falsification optimization

Below we have four cards, where each has a number on one side and a letter on the other.

Now suppose we make a conjecture about those four cards: "Every card that has P on one side has 3 on the other". Now the question is what's the minimum number of cards we have to turn over in order to check the truth of the conjecture?


Some interesting statistics, concerning the number of people getting this right:
10% - General public
29% - Undergraduates
43% - Mathematicians

So if you got this correct, it means that you did better than an average mathematician :)

SOLUTION IN "COMMENTS"

There's a family of interesting theories concerning the reasons why people get this wrong, and why a more practical set up of this "Watson selection task" reduces the percentage of people who get it wrong.

"Masters of Logic" puzzle

Three Masters of Logic wanted to find out who was the wisest amongst them. So they turned to their Grand Master, asking to resolve their dispute. “Easy,” the old sage said. "I will blindfold you and paint either red, or blue dot on each man’s forehead. When I take your blindfolds off, if you see at least one red dot, raise your hand. The one, who guesses the color of the dot on his forehead first, wins." And so it was said, and so it was done. The Grand Master blindfolded the three contestants and painted red dots on every one. When he took their blindfolds off, all three men raised their hands as the rules required, and sat in silence pondering. Finally, one of them said: "I have a red dot on my forehead."
How did he guess?

SOLUTION IN "COMMENTS"