Five children had been playing outside all morning (presumably there must have been a power cut, and their Dreamcasts wouldn't work). Their mum, who insists on cleanliness but doesn't like naming names, noticed them coming in and pointed out, "At least one of you - AHEM - has got a dirty forehead". They all knew what this meant - they wouldn't hear the end of it until the offending person(s) washed the mud off again. But being very well brought up, none of them wanted to point the finger at any of the others.

Vanessa, the brightest of the five, looked at her four playmates and noticed that precisely one of them, Bill, had mud on his forehead. Why doesn't he realise it's him, she thought to herself. After all, he can see we've all got clean foreheads, so he must know it's him. Then she thought a bit more, and realised that the reason Bill hadn't worked this out must be because he couldn't see four clean foreheads. Since she could see for herself that the other three foreheads were clean, she realised that the dirty forehead Bill could see must be her own. "Oh sorry!" she announced. "I'll go and wash the mud off straight away."

Bill, who's almost as smart as Vanessa, was thinking as well. Why did she take so long to realise her forehead was dirty? She's usually very smart, and she should have realised at once that her forehead was dirty when she saw that ours were all clean.... Putting two and two together, he suddenly realised that she must have been confused because she thought someone else had the dirty forehead, and since the other three were all clean, that meant... "Oh sorry," said Bill, "I'll go and wash my forehead as well."

"Very good, children," exclaimed their mother, who was very proud of her children's precocious logical capabilities. She explained to the remaining three children how first Claire, and then Bill, had been able to deduce that their foreheads were dirty. But then she puzzled them completely by asking a rather awkward question. This is what she pointed out.

"Claire knew that Bill's forehead was dirty because she could see it was, so I told her nothing she didn't already know when I said 'At least one of you has got a dirty forehead.' She could see that at least one forehead was dirty, because she could see Bill's dirty forehead. But if I hadn't said anything, she wouldn't have been able to work out that her own forehead was dirty - so how on earth did my saying 'at least one of you has a dirty forehead' give her the ability to deduce something she couldn't deduce before, when I didn't actually tell her anything she didn't already know?"