# Today Puzzle #581

## Puzzle No. 581â€“ Thursday 3 October

The Queen Elizabeth Ocean Liner sets sail from New York to Liverpool and travels the 3300 miles directly at a constant speed of 20 miles per hour. At the same time one of the Liver birds sets out from Liverpool to meet it, travelling directly at the phenomenal speed of 100 miles per hour. When it gets to the ship it immediately turns round and returns to Liverpool. On returning to Liverpool it sets out once again to meet the ship. This to-and-fro continues until the ship reaches Liverpool 165 hours after it left New York. How far does the Liver bird travel in total?

Today’s #PuzzleForToday has been set by Damian Haigh, Headteacher, University of Liverpool Mathematics School.

The Liver Bird problem

This problem is an adaptation of a famous old puzzle know as the Bird-Train problem or the Two Trains Puzzle. http://mathworld.wolfram.com/TwoTrainsPuzzle.html

Like a good cryptic crossword puzzle it has a distracting surface but everything you need to solve it is there in the problem and the maths required does not actually go beyond Key Stage 3, i.e. any 14 year old should be able to solve it, if they were listening when they were taught about speed, distance and time.

Although it’s tempting to think about the many back and forth journeys the Liver Bird must make, it’s only really necessary to consider two facts: the bird was travelling at a constant 100 mph and it was travelling for 165 hours.

100 miles per hour means only “what it says on the tin” - every hour is another 100 miles done - so in 165 hours at that speed it must travel a total of 165 x 100 = 16500 miles.

A curious mathematician might like to consider how many miles were travelled in each direction and think a little about the series of journeys. It’s not too difficult to work out when and where the bird and the boat first meet. It gets a little more complicated after that, but it’s accessible to an A-Level maths student with some perseverance.