This seems to refer to the drinking song "Es geht ein Rundgesang". volksliederarchiv.de traces it back to several late 19th century song books and aludes to an oder version "Es geht ein Saufcomment an unserem Tisch herum". Saufcomment or Bier-Comment is a toungue-in-cheek name for a beer-drinking evening ("Kneipe") at a student fraternity, but with a plethora of rules that have to be followed. Singing a silly song might well be among the rituals of such an evening.
An older man refering to some lines from this song can be a sentimental reference to the exuberance of youth, but also can be a hint the writer and his addressee might be members of the same Burschenschaft. (German academic fraternities are meant as lifelong brotherhoods, so they are still members in their older age.)
The text given on the website and the associated game (here translated to english) go like this:
Es geht ein Rundgesang
An unserm Tisch herum, herum
Es geht ein Rundgesang
An unserm Tisch herum.
Dreimal drei ist neune,
jeder singt das Seine
Es geht ein Rundgesang
An unserm Tisch herum.
After the common „Rundgesang“ one sings alone and, as he has mastered his performance, everyone sings either:
Hat's gut gemacht, hat's gut gemacht
drum wird er jetzt nicht ausgelacht
Hat's gut gemacht, hat's gut gemacht
drum lacht man ihn nicht aus
or
Hat's schlecht gemacht, hat's schlecht gemacht
drum wird er jetzt brav ausgelacht
Hat's schlecht gemacht, hat's schlecht gemacht
drum lacht man ihn brav aus
So it goes around, until everyone has sung.
See https://www.youtube.com/watch?v=Yz_Q2CHouZY for a rendering from an (unnamed) german musical film. The scene has it two times, at the beginning and at the end. Note that the first time, the rhyme goes "Dreimal drei ist neune, jeder singt das Seine, but at the end, it has "Drei mal drei ist neune, ihr wißt ja, wie ichs meine". Both variants seem to be well-known, for example here the second one is used.
There is a children's singing game Es geht eine Zipfelmütz that is directly derived from this song, but instead of drinking, it is changed into a circle song that reminds a bit of the better known "Bi-Ba-Butzemann". I find it relatively improbable that this variant is meant.