Here is what you've all been waiting for - the inaugural Greenwich Maths Challenge. The intention is that a new question will be posted monthly during the University session. There will be a prize of a £10 token for the best answer received from a Greenwich student or a group of Greenwich students. Collaboration is encouraged. Your answers should be sent to A.Mann@gre.ac.uk. The judges' decision is final.

The first challenge is this. What is wrong with Scheherazade's argument in the following story by Raymond Smullyan? The clearest explanation submitted before midnight on Sunday November 29 by a Greenwich student or students will win the prize.

*"And now" said Scheherazade, "I have a paradox for you. There are three boxes labeled A, B and C. One and only one of the three boxes contains a gold coin; the other two are empty. I will prove to you that regardless of which of the three boxes you pick, the probability that it contains the gold coin is one in two."*

"That's ridiculous!" said the king. "Since there are three boxes the probability is clearly one in three."

"Of course it's ridiculous," said Scheherazade, "and that's what makes it a paradox. 1 will give you proof that the probability is one in two, and your problem is to find the error in the proof, since the proof must obviously contain an error."

"All right," said the king.

"Let's suppose you pick Box A. Now, the coin is with equal probability in any of the three boxes, so if Box B should be empty, then the chances are fifty-fifty that the coin is in Box A."

"Right," said the king.

"Also, if Box C is empty, then again the chances are fifty-fifty that the coin is in Box A."

"That's right," said the king.

"But at least one of the boxes, B or C, must be empty, and whichever one is empty, the chances are fifty-fifty that the coin is in Box A. Therefore the chances are fifty-fifty, period!"

"Oh, my!" said the king.

"That's ridiculous!" said the king. "Since there are three boxes the probability is clearly one in three."

"Of course it's ridiculous," said Scheherazade, "and that's what makes it a paradox. 1 will give you proof that the probability is one in two, and your problem is to find the error in the proof, since the proof must obviously contain an error."

"All right," said the king.

"Let's suppose you pick Box A. Now, the coin is with equal probability in any of the three boxes, so if Box B should be empty, then the chances are fifty-fifty that the coin is in Box A."

"Right," said the king.

"Also, if Box C is empty, then again the chances are fifty-fifty that the coin is in Box A."

"That's right," said the king.

"But at least one of the boxes, B or C, must be empty, and whichever one is empty, the chances are fifty-fifty that the coin is in Box A. Therefore the chances are fifty-fifty, period!"

"Oh, my!" said the king.