The sanctuaries can be solved on the upper floor, pressing the buttons plays the cry of the Pokémon you must have in your party.
Since the puzzle was too difficult, the Lunar and Solar Mail tell you which Pokémon those sounds belong to.
The numbers represent the Pokédex number of the Pokémon you must have in your team.
For example, for the Lunar Sanctuary, you must have Umbreon (197), Lunatone (337), Absol (359), Clefable (36), Murkrow (198), and Noctowl (164)