GK 2016 Q7
7. Four towns \(P\), \(Q\), \(R\) and \(S\) are connected by roads, as shown.
A race uses each road exactly once. The race starts at \(S\) and finishes at \(Q\). How many possible routes are there for the race?
- A 10
- B 8
- C 6
- D 4
- E 2
Grey Kangaroo 2016, Question 7. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 6
Any route starts by going from \(S\) to \(P\) or \(S\) to \(Q\) or \(S\) to \(R\). Any route starting \(S\) to \(P\) must then go to \(Q\) and then has the choice of going clockwise or anticlockwise round triangle \(QSR\), giving two possible routes. By a similar argument, there are two routes that start \(S\) to \(R\). For those routes that start by going from \(S\) to \(Q\), there is then the choice of going clockwise or anticlockwise round quadrilateral \(QPSR\), giving two more routes. Therefore there are six possible routes in total.
Solution © UK Mathematics Trust, from the GK 2016 solutions.