KöMaL 2010 A512

We have $n=\frac{3^k-3}2$ coins, each one indistinguishable from the other in terms of size and appearance, but only $n$-1 of them have the same weight, whilst one of them is a counterfeit, being either lighter or heavier than the rest. Using a balance we have to find the counterfeit coin. Show that by $k$ measurements it is possible to find the counterfeit coin and to determine whether it is heavier or lighter than the rest; moreover, it is possible to arrange the measurements in advance, without knowing the results of the measurements.