Here are two puzzles with surprising solutions, and a paradox that has fooled many professional mathematicians:Puzzle no. 1
Connect the dots with four straight lines, without lifting the pen.
Puzzle no. 2
Form four equilateral triangles with six match sticks (without breaking them).
Monty Hall's Paradox
This paradox is named for the host of a famous gameshow in the U.S. in the 1960s. It has confounded a lot of people, including a number of professional mathematicians, because of its counterintuitive nature.
See this description, or any of a number of sites dedicated to the paradox.
Tit for tat
This is a strategy to handle the Prisoner's Dilemma, a well-known problem in Game Theory. The problem can be stated as follows:
Two prisoners A and B are being questioned separately. If one confesses, he goes free, while the other gets a 10-year sentence. If both confess, they each get a 5-year sentence. If both refuse to confess, they each get a 6-month sentence.
In real life, of course your choice would be heavily influenced by your assessment of the character of your fellow prisoner. But game theorists are more interested in the optimal strategy when both prisoners are completely rational (self-serving), and the game is repeated many times. Many strategies have been discussed and tried in computer simulations. It has turned out that the best strategy in the long run is to co-operate with your fellow prisoner. If he defects on you, you should immediately follow suit, hence the nickname "tit for tat".
During a management conference in the 1990s, I was subjected to a variation of this game, along with a number of my colleagues at the Swedish Space Corporation. As I recall, we played 8 rounds of the game. Obviously, the consultant who ran the seminar had an agenda: he wanted to demonstrate the value of trust and co-operation.
It did not turn out that way. The co-winners were a colleague of mine and myself. We had both read an article in Scientific American on the Prisoner's Dilemma problem and the Tit-for-Tat strategy. We both chose to defect in round 7, reasoning that several of our colleagues might defect in round 8, as there could be no penalty for doing so. The way points were calculated, no one could catch us in round 8, even when they tried to punish us by defecting themselves.
From the point of view of the seminar leader, this was of course a fiasco. He lamely reminded us that "next time" we would suffer the consequences, but of course there was no next time. The wrath of our colleagues only added to our pleasure!
