NL-NWO 1987 Problem 3

There are two kinds of creatures living in the flatland of Pentagonia: the Spires ( $S$ ) and the Bones ( $B$ ). They all have the shape of an isosceles triangle: the Spiers have an apical angle of $36^o$ and the bones an apical angle of $108^o$ . Every year on Great Day of Division (September 11 - the day this Olympiad was held) they divide into pieces: each $S$ into two smaller $S$ 's and a $B$ ; each $B$ in an $S$ and a $B$ . Over the course of the year they then grow back to adult proportions. In the distant past, the population originated from one $B$ -being. Deaths do not occur. Investigate whether the ratio between the number of Spires and the number of Bones will eventually approach a limit value and if so, calculate that limit value.