Going by the popularity of the little physics problem I posted last week, I thought some of you might be interested in this little conundrum which dates back to the 17th Century (to Galileo Galilei, in fact). It’s not a problem to which there’s a snappy answer, so no poll this time, but I think it’s quite a good one to think about – and please try to resist the temptation to google it!

The above figure shows a large circular wheel, which rolls from left to right, without slipping, on a flat surface, along a straight line from P to Q, making exactly one revolution as it does so. The distance PQ is thus equal to the circumference of the wheel.

Now, consider the small circle, firmly fixed to the larger circle with the two centres coincident. The small circle also makes one complete revolution as the wheel rolls, and it travels from R to S. Similarly, therefore, the distance RS must be the circumference of the small circle.

Since RS is clearly equal to PQ it follows that the circumferences of the large wheel and the small circle must be equal!

Since the radii of the large and small circles are different, this conclusion is clearly false so what’s wrong with the argument?