In their valuable book Board games around the world. Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia. One such investigation gives rise to the following problem:
Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression).
The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.