GK 2016 Q24

Let the numbers of passengers in the five carriages be \(p\), \(q\), \(r\), \(s\) and \(t\) respectively with \(p\), \(q\), \(r\), \(s\) and \(t\) all at least 1. Consider the neighbours of the passengers in the first and second carriages. Since each passenger has 5 or 10 neighbours, we have \(p - 1 + q = 5\) or 10 and \(p + q - 1 + r = 5\) or 10. Therefore \(p + q = 6\) or 11 and \(p + q + r = 6\) or 11. However, we know that \(r \geqslant 1\) and hence \(p + q \leqslant 10\) and therefore \(p + q = 6\) and \(r = 5\). Similarly, considering the neighbours of the passengers in the fourth and fifth carriages, we obtain \(s + t = 6\) and (again) \(r = 5\). Therefore, the total number of passengers in the train is 6 + 5 + 6 = 17. (Note that while the total number of passengers in the train is uniquely determined, the arrangement of these passengers in all but the centre carriage is not unique.)