NavList:

I wrote:   My clever cousin Cedrick rather overdid it last night while out celebrating Hogmanay.  What’s more, he borrowed my bike to travel with and arrived home in the early hours without it.  He’s still rather confused, and all I can get from him is that he remembers cycling along a straight road through four villages.  The villages in order were 6 miles, 8 miles, and five miles apart.  He remembers starting at one of the villages and going halfway to another of the villages. Then he went one third of the way to another village, and finally travelling one quarter to the way of another of the villages. He’s still got no idea in which order he visited the villages or what they were called.  How far from the first village in the road I should start looking for my bike? 

Answer:
Let the distances of the four villages along the road from some fixed point be a, b, c, and d in the order they are named in Cousin Cedrick’s instructions.  The first instruction would take me to a point a distance (a+b)/2 from the fixed point.  The second instruction would take me to (a+b)/2 +[C-(a+b)/2]/3=(a+b+c)/3.  Similarly, the final instruction would take me to (a+b+c+d)/4 from the fixed point.  This is Symmetric in the four distances, and I can calculate where my bike is without knowing which village is which, because it is synonymous with working out the centre of gravity of a weightless beam with equal masses placed in the positions of the villages along the road.  I should start looking for my bike [6+(8+6)+(8+6+5)]/4=39/4=9 ¾ miles from the first village along the road.  (Basic idea modified from question in Imperial/49) 

Can anyone think of an inter-planetary or inter-stellar equivalent?

I meant to ask if Navlisters would submit possible Christmas quiz questions to a central Editor to produce a suitable relevant quiz, but I left it far too late.  As a result, I ended up producing a rather weak solo effort for 2020/21.  Should we try and produce a combined effort for Christmas 2021/22 if we can find a way of keeping email addresses confidential?  DaveP