Although I'm not a professional mathematician by training, I felt I should have easily been able to answer straight away the following puzzle:
Three men go to a shop to buy a TV and the only one they can afford is £30 so they all chip in £10. Just as they are leaving, the manager comes back and tells the assisitant that the TV was only £25. The assistant thinks quickly and decides to make a quick profit, realising that he can give them all £1 back and keep £2.
So the question is this: If he gives them all £1 back which means that they all paid £9 each and he kept £2, wheres the missing £1?
3 x £9 = £27 + £2 = £29...??
Well, it took me over an hour of thinking before I finally knew what the correct answer to this puzzle was and, I'm embarrassed.
It reminds me of the embarrassement some professional mathematicians must have felt in not being able to give the correct answer to the famous Monty Hall problem answered by Marilyn Vos Savant:
Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?
Yes; you should switch.
It's also mentioned in the book: The Man Who Only loved Numbers, that Paul Erdos was not convinced the first time either when presented by his friend with the solution to the Monty Hall problem.
So what other simple puzzles are there which the general public can understand yet can fool professional mathematicians?