Solution: Qualifying for the Snail Olympics Puzzle

The Puzzle:

The Solution:

Unfortunately, Gertie has missed out this year. Maybe she'll make it for the next Olympiad.

Let's do some algebra to see why.

Let \(d\) cm be the length of one lap of the course, let \(t\) be the time (in hours) taken to complete a lap at \(12 \text{ cm/h}\), and let \(t'\) be the time (in hours) required to complete one lap of the course at this new, faster speed.

Then Gertie needs to travel a total distance of \(2d \text{ cm}\) to complete two laps of the course, and she will do this in \(t + t'\) hours. Since \(\text{speed} = \dfrac{\text{distance travelled}}{\text{time taken}}\), we require \(24 = \dfrac{2d}{t + t'}\), so \(24 t + 24t' = 2d\).

But, she completes one lap of the course at \(12 \text{ cm/h}\) in \(t\) hours. So, \(d = 12t\) and \(2d = 24 t\). Plugging this into the equation gives

So, Gertie would have to complete the second lap in no time at all! In other words, her speed would have to be infinite.

For example, let's suppose that the course is \(12 \text{ cm}\) long. Gertie completes the first lap of the course in one hour at \(12 \text{ cm/h}\). The total length of two laps is \(24 \text{ cm}\), so she would have to complete the two lap trial in \(1 \text{ hour}\) to have an average speed of \(24 \text{ cm/h}\). This time has already elapsed, and Gertie is only halfway through. So, there's no way that Gertie can possibly reach the qualifying speed for the Olympics. Better luck next time, Gertie!