One way of solving this problem is to work out the distance between the 145th and 201st milestones (i.e. 56 miles) as a fraction of the total distance between the towns.

We do not know at which milestone the buses meet for the first time, so take the moment that the Veston bus (note that ‘Veston bus’ always refers to the bus starting in Veston at the beginning of the problem, and vica versa for ‘Shipton bus’) arrives in Shipton. Since the Veston bus has travelled 4/4 of the way to Shipton, we can infer that the Shipton bus has travelled 5/4 of the way to Veston; that is to say, it has already travelled 1/4 of the way back to Shipton, leaving 3/4 (or 9/12) of the total distance between the towns between the two buses. Again, applying the ratio tells us that before they meet, the Shipton bus must travel another 5/12 of the way to Shipton, while the Veston bus travels 4/12 of the way back to Veston. Since the Shipton bus has travelled in total 1/4 + 5/12 = 8/12 of the total distance between the towns since leaving Veston, we know that the 145th milestone is 2/3 of the way from Veston to Shipton.

Next we must find the location of the 201st milestone. This time, take the moment at which the Veston bus has travelled from Veston to Shipton and back to Veston again, meeting the Shipton bus twice. At this moment, the Veston bus has travelled 4/2 of the way between the towns; therefore the Shipton bus has travelled 5/2 of this distance, meaning that it is already halfway back to Veston when the Veston bus arrives there. Expressing the distance between them as 9/18 of the distance between Shipton and Veston, we can easily use the ratio of the buses speeds to see that when the third meeting takes place, the Shipton bus will have travelled another 5/18 of the distance between the towns, while the Veston bus will have travelled 4/18 of the total journey. This means that the 201st milestone must be 4/18 of the way from Veston to Shipton.

Expressing the locations of the milestones with the lowest common denominator, we get that the 145th milestone is 6/9 of the way from Veston to Shipton, while the 201st is 2/9 of the way from Veston to Shipton. The distance between the milestones is therefore 4/9 of the distance between Veston and Shipton. Since we know that the distance between the milestones is 201 – 145 = 56 miles, we can now find the distance in miles between any combination of two milestones or towns. Firstly, dividing 56 by four tells us the distance in miles equal to 1/9 of the distance between Shipton and Veston (14 miles). Since the distance between Veston and the 201st milestone is 2/9 of the total distance between the towns, or 28 miles, the milestone at Veston is thus the 201+28 = 229th milestone. The milestone at Shipton is of course 9/9 of the distance between the towns away, and this distance is equal to 9 x 14 = 126 miles. Therefore, the Shipton milestone is the 103rd.