What is Augustine's original of Tolstoy's "Three Hermits" story? Leo Tolstoy's charming short story Three Hermits (Три старца) of 1886
has frequently been cited in academic and popular works about prayer.
I don't want to spoil the ending - the (short!) text can be readily found
online in English
and Russian.
There's even a film dramatization in Притчи 2 (2011)
(19m25 video). Tolstoy, a collector of folk tales, apparently
did not originate this story, but adapted or expanded it from one
he had heard.

In the 1961 translation by Gleb Struve
(Russian Tales, Bantam Books, 1961) he remarks:


  At the core of the legend lies a well-known subject, a variant of which can
  be found in St. Augustine. St. Augustine's version of the legend became known 
  in Russia in the sixteenth century. It is unlikely, however, that Tolstoy used 
  as his source any of the bookish versions; the legend probably
  came to him in an oral retelling. p165


Similarly, Donald Barthelme's 
short story At the Tolstoy Museum (New Yorker, 24 May 1969, p32ff),
later collected in City Life (Farrar, Straus and Giroux, 1970),
includes the lines:


  [Three Hermits] is written in a very simple style. It is said to originate in a folktale.
    There is a version of it in St. Augustine.


It is not hard to find other people mentioning an Augustinian connection, but I haven't found anything more specific.
"In St. Augustine" is not a very helpful citation, given the
size of his corpus.

Does anyone know where Augustine's version of the story can be found (or indeed any non-Tolstoy version)? 
How does Tolstoy's retelling differ from his Russian sources or from Augustine's original,
and do any of those differences relate to Tolstoy's specific Christian beliefs (for example, Christian anarchism and its rejection of church authority)?
Separating history from legend is often tricky. There is a legend stating that while Augustine of Hippo was writing his treatise De Trinitate (about the Trinity), he was walking along the beach and thinking about the Godhead. There he encountered a boy running back and forth between the ocean and a small hole in the sand, filling it with water. When Augustine asked the boy what he was doing, he replied, "I'm putting the sea into this hole." After Augustine commented on the absurdity of this notion, the boy retorted, "I will sooner empty the sea into this hole than you will manage to get the mystery of the Most Holy Trinity into your head!"

There is nothing in Augustine's writings containing this account, only legend. The story is told in the 15th century Legenda Aurea written by Jacobus de Voragine, and it is depicted in numerous works of art from that time until now.

It is possible that this legend influenced Tolstoy's telling of The Three Hermits, but that's somewhat of a stretch since these stories have different points they are trying to make. I am not aware of any other stories related to this associated with Augustine.