Right, you can stop furrowing your brows now - here are the answers to my Christmas quiz

There were some very impressive answers to my Christmas quiz, which just demonstrated once more that my readers are more intelligent than I am. Here are the answers:

I began with the story of a person walking around a tree trying to catch sight of a squirrel, but the clever squirrel kept scampering around the trunk so that he always remained out of sight. The person goes round the tree, but does he go round the squirrel? It depends what you mean by "going around". If you mean "going north, west, south and east of the squirrel", then he has gone round it. If you mean "being in front of the squirrel, to the right of him, behind him, to the left of him and then in front of him again" you haven't. (The philosopher William James tells the story in his classic book Pragmatism.)

Two: A friend of mine has two dogs. At least one of them is male. What are the chances that both are male? There are four possibilities: male, female; female, male; male, male; female, female. Strike out the last, because we know at least one is male. Of the other three pairs, two contain females, so the chances of both dogs being male is one in three.

Three: Another friend of mine has two dogs. I met him going for a walk one day with one of them. I noticed it was male, as one does. What are the chances that both are male? It sounds like the last case, but isn't. In this case we have met one of the dogs. The other dog has an equal chance of being male or female so the odds of both being male are one in two.

Four: This is my favourite. You're a contestant on a game show faced with three sealed boxes, two empty, one containing a prize. You select a box, but don't open it yet. The host (who knows where the prize is) then opens an empty box (obviously at least one of the two remaining boxes must be empty). At this point, you can change to the other sealed box if you want. Should you? The answer is yes, you are doubling your chances of winning. Even mathematicians have disputed this, but it's really quite simple. Imagine playing the game thousands of times. You will choose the prize about a third of the time and the empty box about two-thirds of the time. If you switch every time, then every time you first chose the prize (one third of the time), you will get nothing; but every time you first chose the empty box (two-thirds of the time) you will end up with the prize. So you have doubled your chances. Got it?

Then I gave you a trick. Think of a three-digit number. Then write it down followed by itself, thus creating a new six-digit number. Divide this six-figure number by seven, then by 11, then by 13 and you end up with the original three-digit number. Why? Because writing a three-digit number twice is the equivalent of multiplying it by 1,001. And seven times 11 times 13 makes 1,001. So all you're doing is multiplying your number by 1,001 and then dividing it by 1,001.

Five: Bill and Ben toss a coin 100,000 times. If it lands tails Bill gets a point; if heads, Ben gets a point. Some onlookers are interested not in who will win but how often the lead will change. I offered the following list of possibilities - 0, 5, 10, 50, 100, 500, 1,000, 10,000, 25,000, 50,000 - and asked you to rank them in order of likelihood. This perhaps isn't a fair question, but it emerged from something I read earlier this year and I've been looking for a way to work it into this column ever since.

The extraordinary answer is that 0 is the most likely, then 5, then 10 and so on (except that 50,000 is impossible). The mathematical proof of this is way beyond me, but the mathematician William Feller, author of the book where I found it, wrote as follows: "The results concerning fluctuations in coin tossing show that widely held beliefs about the law of large numbers are fallacious. They are so amazing and so at variance with common intuition that even sophisticated colleagues doubted that coins actually misbehave as theory predicts."

Six, and finally, I invited readers to consider the number described as follows: the smallest number not describable in fewer than 11 words. What's the problem?

The problem is that the description itself consists of ten words so that the smallest number not describable in fewer than 11 words is describable in ten words, which is a contradiction.

Warm greetings to the brave souls who tackled this rather unfair quiz (some prizes will follow), and I'd like to wish you all a happy final year of the second millennium.